BPS Explained III: The Leading Order Behavior of the BBGKY Hierarchy in a Plasma
BBPS Explained III: Dimensional LeveragingorThe Leading Order Behavior of the BBGKY Hierarchyin a Plasma
Robert L Singleton Jr
University of LeedsSchool of MathematicsUK, LS2 9JT (Dated: 7/20/2020)
Abstract
This is the third in a series of lectures on the technique of dimensional continuation, employed byBrown, Preston and Singleton (BPS), for calculating Coulomb energy exchange rates in a plasma.Two important examples of such processes are the charged particle stopping power and the temper-ature equilibration rate between different plasma species. The first lecture was devoted to under-standing the machinery of dimensional continuation, and the second concentrated on calculating theelectron-ion temperature equilibration rate in the extreme quantum limit where the Born approxi-mation is fully justified. In this lecture, I will examine one of the main theoretical underpinnings ofthe BPS theory, namely, the dimensional reduction of the BBGKY hierarchy. There are two broadclasses of kinetic equations, applicable in complementary regimes, represented by the Boltzmannequation (BE) and the Lenard-Balescu equation (LBE). The BE describes the short-distance effectsof 2-body scattering, while the LBE models 2-point long-distance correlations. It is well knownthat the BE suffers a long-distance logarithmic divergence (in three spatial dimensions), confirmingthat it is indeed missing long-distance physics (correlations are being ignored). Conversely, theLBE suffers from a short-distance logarithmic divergence (in three dimensions), another indicationthat relevant physics is being overlooked (the scattering physics). There are multiple industriesin plasma physics devoted to regulating these infinities, thereby giving mathematical and physicalmeaning to the various calculations. To my knowledge, BPS is the only formalism that applies aregularization scheme systematically in a perturbative expansion of the dimensionless plasma cou-pling parameter g , while simultaneously treating short- and long-distance scales consistently and inthe same manner. A novel aspect of the BPS formalism is that it employs dimensional continuationto regulate the divergent integrals in the kinetic equations, a procedure first used in quantum fieldtheory to regulate divergent integrals during the renormalization program. The idea of dimensionalcontinuation is that one should perform the integrals in an arbitrary number of spatial dimensions ν , where, remarkably, the integrals become finite (except for ν = 3, where we happen to live). Theonly remembrance of the three dimensional divergences are simple poles of the form 1 / ( ν − g behavior of the BBGKY hierarchy as a functionof the spatial dimension ν , both above and below the critical dimension ν = 3. In these notes, Iwill prove that to leading order in g , the BBGKY hierarchy reduces to the BE for ν > ν <
3. We must eventually return to three dimensions, and the BPS formalism shows thatthe simple poles associated with the BE and the LBE exactly cancel, rendering the ν → ν is analytically continued from ν < ν >
3. This provides the leading and next-to-leading orderterms in g exactly, which is equivalent to an exact calculation of the so-called Coulomb logarithm with no use of an integral cut-off. Therefore, in this way, BPS takes all Coulomb interactions intoaccount to leading and next-to-leading order in g . a r X i v : . [ phy s i c s . p l a s m - ph ] J u l ontents I. Introduction II. The Coulomb Plasma in Arbitrary Dimensions
III. Coulomb Energy Transfer Rates in Arbitrary Dimensions
IV. The BBGKY Hierarchy in Arbitrary Dimensions ν ν < ν > V. The Boltzmann Equation from BBGKY in ν > VI. The Lenard-Balescu Equation from BBGKY in ν < VII. Conclusions Acknowledgments A. The Cross Section and Hyperspherical Coordinates
B. Center-of-Momentum Coordinates C. The Multi-component Poisson-Vlasov Equation References . INTRODUCTION
This is the third lecture on a novel technique for calculating the charged particle stoppingpower and the temperature equilibration rate in a weakly coupled fully-ionized plasma [1, 2].The method is exact to leading and next-to-leading order in the plasma coupling g , andtherefore calculates the Coulomb logarithm exactly. In Lecture I [3] of this series, I discussedthe basic theoretical machinery of dimensional continuation, and in Lecture II [4], as anexample of the method, I calculated the energy exchange rate between electrons and ionsin a hot plasma in using the BPS formalism. This formalism can be viewed in the light ofconvergent kinetic equations, and to my knowledge, it is the only formalism in the literaturethat uses a systematic expansion in powers of g . It is quite gratifying, therefore, that the BPSstopping power has recently been verified experimentally [5], and this has provided impetusfor another lecture. The purpose of these notes is to prove one of the primary claims uponwhich BPS is based, namely, that to leading order in the plasma coupling g , the BBGKYhierarchy [6] reduces to (i) the Boltzmann equation in dimensions ν >
3, and to (ii) theLenard-Balescu equation [7, 8] in dimensions ν <
3. However, for ν = 3 (the dimensionof interest), things are not so clean: the Boltzmann equation (BE) suffers a long-distancedivergence, and the Lenard-Balescu equation (LBE) contains a short-distance divergence. Inboth cases, the divergences are logarithmic, and this is a crucial observation in regularizingthem. Denoting the ν -dimensional Coulomb potential by φ ν ( r ), we see that the divergencesin ν = 3 arise because the potential φ ( r ) ∼ /r is the only potential φ ν ( r ) whose integralcontains both a short- and a long-distance divergence. The dimensional reduction of BBGKYis illustrated schematically in Fig. 1.The kinetic equations for systems interacting via the Coulomb force diverge in three spatialdimensions, and there have been many attempts to rectify this problem. In these notes, Iwill concentrate on the method of Brown, Preston, and Singleton (BPS) of Ref. [2]. Themethod relies on dimensional continuation, which is a regularization technique adopted fromquantum field theory calculations in arbitrary spatial dimensions ν . I will prove rigorouslythat the BBGKY hierarchy collapses to the Lenard-Balescu equation for ν < g . This is quite an involved calculation, and Clemmow andDougherty [9] is my primary reference. Their calculation breaks down in ν = 3 dimensions,but goes through unscathed in dimensions less than three. For completeness, I will alsoprove that to leading order in g , the BBGKY hierarchy reduces to the Boltzmann equationfor ν >
3. I will base this calculation on that of Huang in Ref. [10], which breaks down inthree dimensions, but becomes rigorous in dimensions greater than three.As we are concerned with short- and long-distance divergences, we must be clear in ournomenclature. In keeping with the standard usage of quantum mechanics, I will call ashort-distance divergence an ultra-violet (UV) divergence, and a long-distance divergencean infra-red (IR) divergence. This nomenclature arises from the well known episode in the3 (cid:38) (cid:36)(cid:37)
BBGKY(arbitrary ν ) (cid:0)(cid:0)(cid:0)(cid:9) (cid:64)(cid:64)(cid:64)(cid:82) ν < ν > (cid:39)(cid:38) (cid:36)(cid:37) Lenard-BalescuEquation (LBE) (cid:39)(cid:38) (cid:36)(cid:37) (cid:63) ν = 3Break Down (cid:31)(cid:30) (cid:28)(cid:29) FIG. 1: For ν > g . A similar reduction from the BBGKY hierarchy holds for the Lenard-Balescu equation in ν <
3, and the “textbook derivation” is also rigorous in these dimensions. In ν = 3, the derivationsof the Boltzmann and Lenard-Balescu equations break down for the Coulomb potential. history of physics in which classical physics spectacularly failed to calculate the observedblack-body spectrum. The classical calculation captured the long-distance infra-red partof the spectrum correctly, but it predicted that the short-distance ultra-violet part of thespectrum would diverge, which is absurd (and contrary to observation). In text books thisepisode is now called the ultraviolet catastrophe [11], although more precisely it might becalled the Rayleigh-Jeans catastrophe. As we will see, the Lenard-Balescu equation in threedimensions suffers its own UV catastrophe, and for similar reasons. Conversely, it turns outthat the Boltzmann equation in three dimensions suffers from an IR divergence. Both arerelated to the 1 /r behavior of the Coulomb potential.It might be of interest to review the history of the UV catastrophe in more details, andto bring out its role in the development of quantum mechanics. The UV catastrophe wasindeed a catastrophe for classical physics, and in retrospect can be marked as the birth ofquantum mechanics, although in a round about fashion [11]. The classical calculation of thespectral output of a black body is quite simple, involving a single integral over all black-bodyfrequencies ω . It was supposed to be a triumph of classical physics, but embarrassingly, thespectral integral turned out to diverge at small wavelengths or high frequencies. In otherwords, the classical integral possessed a UV divergence. This was completely unexpected,and is, as we know, cured by the discrete nature of quantum particles of light. Max Planckwas examining the divergent classical integral in 1900, and noticed that it became finiteif the integral were replaced by a sum over discrete energy states E n = n (cid:126) ω , where theangular frequency ω is that of the light or the electromagnetic radiation emitted from theblack body. One of the most radical things in Planck’s scheme is that it required a new4hysical constant h , sometime written as (cid:126) = h/ π , with units of action (energy times time,or equivalently momentum times distance). The constant h is now called Planck’s constant,but at the time, as far as I know, Planck attached no fundamental significance to it. Wecan, in a certain sense. think of Planck’s method as just another attempt at regulating adivergence, in this case a UV divergence. Then, in 1905, in his work on the photo-electriceffect, Einstein proposed that Planck’s energy quanta be taken literally. Einstein reasonedthat light with frequency ν is composed of discrete particles, now called photons , with energy E = hν = (cid:126) ω . The history of physics is rich in attempts to regulate infinite integrals, andthe unexpected consequences from doing so. Dimensional continuation is just one of manyregularization schemes, and is interesting that quantum mechanics has its roots in one suchregularization attempt.These notes are organized as follows. In Section II we discuss the Coulomb plasma inarbitrary dimensions, showing that the Coulomb force is short-range in dimensions ν > ν <
3. The dimension ν = 3 is the critical dimension in which long-and short-range contributions are comparable. This section also discusses the distributionfunction, and as a warm-up exercise we derive the standard result for the dielectric functionin a multi-component plasma. In Section III we discuss how to find energy transfer ratesusing dimensional continuation, and in Section IV we derive the BBGKY hierarchy in anarbitrary number of dimensions. We show how to define perturbation theory in the plasmacoupling constant g , and we calculate the BBGKY hierarchy accurate to order g . We showthat a complementary collection of 2-point correlations are dominant in ν > ν <
3, and this leads to the qualitative differences between the Boltzmann equation and theLenard-Balescu equation. In Section V we derive the Boltzmann equation from BBGKY in ν >
3, and in Section VI we derive the Lenard-Balescu equation in ν <
3. We conclude withSection VII, and cover some supplementary material in the appendices.5
I. THE COULOMB PLASMA IN ARBITRARY DIMENSIONS
We start with a plasma composed of multiple species labeled by an index a , the variousspecies being delineated by of a common electric charge e a and a common mass m a . Eachspecies is assumed to be in thermal equilibrium with itself at temperature T a , with a spatiallyuniform number density n a . Since we are working in ν spatial dimensions, the engineeringunits of the number density are L − ν , and the units of charge e a are energy times L ν . Wewill measure temperature in units of energy, setting Boltzmann’s constant to unity k B = 1,and we will employ the notation β a = 1 /T a for the inverse temperature. A. The Coulomb Potential
Before considering a plasma in a general number of dimensions, it is instructive to lookat the Coulomb field of a single point charge e in ν dimensions. Since Gauss’s law holds inan arbitrary number of dimensions, then for a point particle at the origin with charge e wehave (in cgs rationalized units) ∇ · E = e δ ν ( x ) , (2.1)where E = ( E , · · · , E ν ) is the electric field vector, with ∇ = ( ∂/∂x , · · · , ∂/∂x ν ) being the ν -dimensional spatial gradient, and δ ν ( x ) being the ν -dimensional Dirac δ -function centeredat the origin. This can be expressed in an integral fashion by integrating any spatial regionΣ containing the charge, (cid:90) Σ d ν x ∇ · E = e . (2.2)To find the electric field we will use Gauss’s theorem, (cid:90) Σ d ν x ∇ · E = (cid:90) ∂ Σ d A · E , (2.3)and exploit the usual symmetry arguments. Let Σ = B r be the ν -dimensional ball of radius r centered on the point charge e , and therefore the ( ν − ∂ Σ = S r . By symmetry, the field E ( x ) points radially outward with a magnitude E ( r ),along the direction ˆ x normal to S r . The length E ( r ) depends only upon the radial distance r = | x | and not upon its angular location along S r , and therefore (2.2) gives e = (cid:90) B r d ν x ∇ · E = (cid:73) S r d A · E = Ω ν − r ν − · E ( r ) with Ω ν − = 2 π ν/ Γ( ν/ . (2.4)The relation for the solid angle comes from (A9), and the electric field of a point particle atthe origin becomes E ( x ) = e Γ( ν/ π ν/ ˆ x r ν − , (2.5)6here ˆ x is a unit vector pointing in the direction of x , and the radial variable is r = | x | .Note that x = r ˆ x , which allows us to write (2.5) in a frequently used alternative form, E ( x ) = e Γ( ν/ π ν/ x r ν . (2.6)It will often be more convenient to work with the electric potential φ defined by E = − ∇ φ , (2.7)and upon integrating (2.5), we find φ ( x ) = Γ( ν/ − π ν/ er ν − , (2.8)where we have chosen the constant of integration so that the potential vanishes at radialinfinity (for ν > b at the origin is E b ( x ) = e b Γ( ν/ π ν/ ˆ x r ν − , (2.9)and the corresponding potential is φ b ( x ) = e b Γ( ν/ − π ν/ r ν − . (2.10)Note that E b ( x a − x b ) is the electric field at x a produced by a point charge e b at x b , andconsequently, the force acting on charge a from charge b is F ( b ) a = e a E b ( x a − x b ) = e a e b Γ( ν/ π ν/ x a − x b | x a − x b | ν . (2.11)This form will appear in the BBGKY kinetic equations for many-body Coulomb systems.The prefactor of the electric field in these units, known as cgs rationalized units, dependsupon the spatial dimension ν . In three dimensions we find the usual factor of 4 π , E ( x ) = e π ˆ x r (2.12) φ ( x ) = e π r , (2.13)where the numerical subscript denotes ν = 3. This potential will turn out to be special, inthat its integral diverges logarithmically at both small and short distances. It is the onlypotential whose integral diverges in the IR and the UV. It will be useful for our intuition tolook at the electric field and its potential for dimensions on either side of three. For example,in ν = 4, we have E ( x ) = e π ˆ x r (2.14) φ ( x ) = e π r , (2.15)7nd we see that the potential converges more quickly than φ for large values of r . The case ν = 2 must be handled with a little care, as we cannot simply substitute ν = 2 into (2.8).The electric field in ν = 2 dimensions is proportional to 1 /r , which integrates to a logarithmfor the potential, so that E ( x ) = e π ˆ x r (2.16) φ ( x ) = − e π ln( r/r ) , (2.17)where r is an arbitrary integration constant at which the potential is chosen to vanish. In ν = 2 something different has happened. The potential no longer asymptotes to a constantvalue at large r , but diverges logarithmically. We must therefore choose a finite but arbitraryradius r along which the potential vanishes. Note that the 2-dimensional potential alsodiverges logarithmically at small r . Since a logarithmic divergence is an integrable divergence,it is not as severe as the 1 /r divergence in ν = 3, and this is why the Lenard-Balescu equationin ν < ν ≥
3. We can alsoarrive at (2.17) by performing an analytic continuation in ν near the region ν = 2. In otherwords, define the small (continuous) parameter (cid:15) = ν −
2, and note that (2.8) takes the form φ ( x ) = e π Γ( (cid:15)/ √ π r ) (cid:15) . (2.18)Using the expansions Γ( (cid:15)/
2) = 2 (cid:15) − γ + O ( (cid:15) ) (2.19) a − (cid:15) = e − (cid:15) ln a = 1 − (cid:15) ln a + O ( (cid:15) ) , (2.20)and dropping linear and higher order terms in (cid:15) , we find φ ( x ) = e π (cid:32) (cid:15) − γ (cid:33)(cid:32) − (cid:15) ln π / r (cid:33) = e π (cid:20) (cid:15) − γ − ln π − r (cid:21) . (2.21)In the limit ν →
2, this potential contains the infinite constant 2 /(cid:15) − γ − ln π . The physicalreason for this (harmless) divergence is that the zero of potential energy in (2.8) vanishes inthe asymptotic limit r → ∞ , while the logarithmic potential for ν = 2 does not vanish atlarge r , but instead diverges. This is not a problem, as we are free to subtract a constant(even an infinite constant) from any potential. Indeed, as we move from ν = 3 to ν = 2, thereis no discontinuity in the r -behavior of the electric field. Let us therefore define a shiftedpotential ¯ φ ( x ) = φ ( x ) − φ ( x ), that is to say, we subtract the constant value φ ( x ) atan arbitrary x , and we find ¯ φ ( x ) = − ( e/ π ) ln( r/r ), which is just (2.17). Thus, there areno physically measurable discontinuities as we dimensionally continue from ν = 3 to ν = 2.Since we are eventually interested in analytically continuing ν to complex values in a small8eighborhood around ν = 3, and then taking the limit ν →
3, we may continue to formallyuse (2.8). It is, however, important that the electric field and its potential are well definedfor all positive integer values of ν . Finally, let us examine ν = 1, for which we obtain E ( x ) = ( e/
2) sign( x ) (2.22) φ ( x ) = − ( e/ r , (2.23)where r = | x | . The unit vector pointing away from the origin, ˆ x , can point only left or right,and can therefore be thought of as the sign function sign( x ): plus one for positive x andminus one for negative x , located at x = 0. Since x = | x | sign( x ), we can still express thespatial point x in the form x = r ˆ x . B. The Coulomb Potential and Dimensional Regularization
Let us explore in more detail how dimensional continuation acts as a regulator for divergentintegrals, rendering them finite and therefore algebraically amenable. We shall start byconsidering the Boltzmann equation for a plasma in three dimensions. We do not requirethe actual equation at the moment, but we only need to recall that it suffers a long-distanceIR divergence. In contrast, it is interesting to note that the far more idealized hard-sphere scattering model is finite. This hard-sphere model is based on the idea that particles in adilute gas behave like billiard balls. This is of course incorrect, or at least a highly idealizedpicture, but the model still provides useful insight. The reason that hard-sphere scatteringis finite is that it is short-range : the force acts only during the collision, after which theparticles move freely in a constant potential (until the next collision). There are no long-distance effects in this model. A gas of neutral particles acts somewhat like billiard balls, sowe expect the Boltzmann equation to be finite for a gas. And indeed it is. This is becausethe force between neutral particles is short-range, and they do not see one another at largedistances. In fact, the Boltzmann kernel is finite for any short-range force in three dimensions.The irony, however, is that we are not interested in short-range forces. Instead, it is theCoulomb force that is of relevance to plasma physics, and in three dimensions this is a long-range force. Consequently, we find a long-distance logarithmic divergence in the Boltzmannscattering kernel. This IR divergence essentially arises from the integration of the potential φ ( r ) ∼ /r at large- r . Since the divergence is only logarithmic, any potential that falls offfaster than 1 /r at large- r will not produce a divergence in the Boltzmann scattering kernel.We will return to this point in the next paragraph. In summary, even though the Boltzmannequation gets the short-distance physics right, it gets the long-distance physics wrong, and As stated in the introduction, we are using the quantum mechanical nomenclature in which IR stands for infra-red long-distance physics, UV stand for short-distance ultra-violet physics.
9e pay the price through a logarithmic IR divergence. Conversely, when we capture long-distance collective effects in a plasma using the Lenard-Balescu equation, we find a short-distance UV divergence (in three dimensions). The Lenard-Balescu equation captures thelong-distance physics correctly, but models the short-distance physics incorrectly, and onceagain we pay a price, this time introducing a logarithmic UV divergence. In this case, anypotential that diverges less severely than φ ( r ) ∼ /r at small- r will not suffer a divergence.But as before, we are not interested in such forces. This reasoning, however, will be essentialto understanding why the Coulomb potential in ν dimensions regulates the various kineticequations.We can now show why working in an arbitrary number of dimensions ν acts as a regulator.It turns out that the Boltzmann equation (BE) becomes IR finite for ν >
3, and that theLenard-Balescu equation (LBE) becomes UV finite for ν <
3, with the only reminder of pastdivergences being simple poles of the form 1 / ( ν − φ ν ( r ) ∼ /r ν − for a positive charge at the origin for ν = 1 , , , ν = 2 is actually logarithmic). For aesthetic reasons, the arbitraryintegration constants of the potentials have been adjusted so that the graphs for ν = 2 , , ν ,a potential φ ν ( r ) can be selected with the appropriate short- and long-distance behavior.The potential φ ( r ) ∼ /r is special, in that it produces logarithmic divergences in theUV and IR, indicating that short- and long-distance physics are equally important in threedimensions. Thus, φ ( r ) is a borderline case, and this is the reason that the BE and the LBEsuffer IR and UV divergences, respectively. However, and this is the key point, for ν < φ ( r ) for small- r , and thisis what renders the LBE finite in the UV. Conversely, for ν > φ ( r ) for large r , and this renders theBE finite in the IR. This is the reason dimensional continuation works as a regulator.As we have emphasized, the divergences in question are only logarithmic (rather thanlinear or higher order), and can therefore be rendered finite by slightly adjusting the rateof convergence of the potential φ ν ( r ) in the offending region of r (either at large or smallvalues of r ). This takes us into the domain of convergent kinetic equations. The integral ofany potential that diverges less slowly that 1 /r as r → ∞ , even by an infinitesimal amount,will in fact converge at large- r . For example, the potential φ ( r ) ∼ /r δ with δ >
0, butotherwise δ can be as close to zero as we wish, gives a convergent integral in the IR, (cid:90) dr r δ ∼ r − δ → r → ∞ , (2.24)and the BE does not possess an IR divergence for such a potential. Conversely, the integralof any potential that diverges less slowly that 1 /r as r → r .10 IG. 2: The Coulomb potential φ for a positive charge at the origin as a function of radius r ,for dimensions ν = 1 , , ,
4. The zeroes of potential energy have been adjusted for visual clarity.Short-distance ultraviolet (UV) physics is emphasized in dimensions ν > ν < ν = 3, the UV and IRphysics are equally important, and the energy rate diverges logarithmically at both large and smalldistances in three dimensions. The left panel illustrates that for ν <
3, the Coulomb potentialdiverges less severely that 1 /r as r gets small, and consequently the LBE does not suffer a short-distance divergence in ν <
3. Similarly, the right panel illustrates that the Coulomb force convergesto zero more rapidly than 1 /r as r gets large, and this renders the Boltzmann kernel finite at largedistances when ν > For example, consider a potential of the form φ ( r ) ∼ /r − δ , where δ > δ as closeto zero as we wish). Then the integral of the potential is finite in the UV, (cid:90) dr r − δ ∼ r δ → r → , (2.25)and the LBE does not suffer a UV divergence for such a potential. This is the essence of thetechniques of convergent kinetic theory, of which the BPS formalism is an example. One mustbe exceedingly careful, however, as the same physical regularization scheme must be used atshort- and long-distances. The quantities δ of (2.24) and δ of (2.25) are not independent!If they are treated independently, then one can produce spurious unphysical constants inthe Coulomb logarithm. Another benefit of the BPS formalism is that it treats long andshort distances in the same manner. See Lecture I regarding the Lamb Shift, in which thesignificance of using the same regularization was first realized. To my knowledge, BPS is theonly convergent kinetic scheme that treats the long- and short-distance divergences in thesame way. 11 . The Fourier Transform of the Coulomb Potential Unlike the spatial representation of the potential, we will show that the Fourier represen-tation takes the same form in any dimension. This is quite useful for calculations. There anumber of conventions for the spatial Fourier transform, and I employ φ ( x ) = (cid:90) d ν k (2 π ) ν e i x · k ˜ φ ( k ) (2.26)˜ φ ( k ) = (cid:90) d ν x e − i x · k φ ( x ) . (2.27)As a general rule, the factors of 2 π will always be placed with the k -integral, as this isanalogous to placing factors of 2 π (cid:126) with the p -integral, a convention based in quantummechanics that we shall also follow. As we now show, the ν -dimensional Coulomb potential(2.8), which is repeated here for convenience, φ ( x ) = Γ( ν/ − π ν/ er ν − , (2.28)has the Fourier transform˜ φ ( k ) = ek where k ≡ k · k = ν (cid:88) (cid:96) =1 k (cid:96) . (2.29)As emphasized above, the form of ˜ φ ( k ) does not depend upon the dimension of space, exceptin a trivial way though the length of k .Expression (2.29) for the Fourier transform of the potential (2.28) can be established ina number of ways. Perhaps the easiest is just to use Laplace’s equation, ∇ φ ( x ) = − e δ ( ν ) ( x ) , (2.30)which is obtained by substituting (2.7) into (2.1) . Upon inserting (2.26) for φ ( x ) into (2.30),and using the integral representation of the δ -function, we can write Laplace’s equation inthe form − (cid:90) d ν k (2 π ) ν e i x · k k ˜ φ ( k ) = − e δ ν ( x ) = − e (cid:90) d ν k (2 π ) ν e i x · k , (2.31)or (cid:90) d ν k (2 π ) ν e i x · k (cid:104) k ˜ φ ( k ) − e (cid:105) = 0 . (2.32)The quantity in square brackets must vanish, and solving for ˜ φ ( k ) indeed gives (2.29).It is also informative to prove this result by taking the Fourier transform directly. Theformula to remember is 1 a = (cid:90) ∞ ds e − as , (2.33)12here Re a >
0, from which we can take p derivatives to obtain another useful expression,1 a p = 1Γ( p ) (cid:90) ∞ ds s p − e − as . (2.34)We can analytically continue to complex values of p , in particular to p = ( ν − /
2. Wecan also prove (2.34) by changing variables in the s -integral to u = as , which producesthe correct scaling 1 /a p , while the remaining u -integral exactly cancels the Gamma-functionΓ( p ). We now use relation (2.34) to rewrite the term 1 /r ν − in the potential. First note that r = ( x · x ) / , which we write as r = ( x ) / , and this allows us to express1 r ν − = 1( x ) ( ν − / = 1Γ( ν/ − (cid:90) ∞ ds s − ( ν − / e − s x . (2.35)The Fourier transform of the Coulomb potential is therefore˜ φ ( k ) = e Γ( ν/ − π ν/ (cid:90) d ν x e − i x · k r ν − (2.36)= e π ν/ (cid:90) d ν x (cid:90) ∞ ds s ( ν − / e − sx − i x · k . (2.37)We can interchange the s and x integrals because the integrand is uniformly convergent. Wethen perform the x -integrals by completing the square, and we find˜ φ ( k ) = e π ν/ (cid:90) ∞ ds s ( ν − / (cid:90) d ν x e − s ( x + i k / s ) e − k / s (2.38)= e π ν/ (cid:90) ∞ ds s ( ν − / (cid:16) πs (cid:17) ν/ e − k / s (2.39)= e (cid:90) ∞ ds s − e − k / s = e (cid:90) ∞ dt e − tk / = ek . (2.40) D. The Distribution Function
For each component a of the plasma, there is a distribution function f a define by f a ( x , p , t ) d ν x d ν p (2 π (cid:126) ) ν ≡ number of particles of type a in a hypervolume d ν x about x and d ν p about p at time t , (2.41)where where h is Planck’s constant, and (cid:126) = h/ π . The phase-space factor (2 π (cid:126) ) ν = h ν ensures that f a counts the number of semi-classical quantum states in a phase-space volume d ν x d ν p . The factor (cid:126) makes the volume element in (2.41) dimensionless, thereby rendering f a dimensionless. Using (cid:126) in place of h is merely a convention. More critically, the factor of (cid:126) makes the classical to quantum transition more transparent, and the normalization (2.41)implies (cid:90) d ν p (2 π (cid:126) ) ν f a ( x , p , t ) = n a ( x , t ) , (2.42)13here n a ( x , t ) is the number density of a -type particles at position x and time t . That is tosay, n a d ν x is the number of particles of species a in a hypervolume d ν x about position x attime t . When performing long calculations, it is often convenient to combine the space andmomentum variables into a single phase-space variable X = ( x , p ). Distribution functionsare then written f = f ( X, t ), and the corresponding integration measure becomes dX = d ν x d ν p (2 π (cid:126) ) ν . (2.43)Definition (2.41) now means that f a ( X, t ) dX is the number of particles of type a in a volumeelement dX about phase-space location X at time t . This notation will be particularly usefulwhen considering multi-particle distributions.Throughout these notes, we primarily consider plasmas in local thermodynamic equilib-rium, so that n a is a slowly varying function of space and time , with engineering units of L − ν .We will often consider special cases in which the plasma is completely uniform, and the num-ber density is constant in space. As a general rule, we will leave time dependence implicit.From (2.42), we see that a normalized Maxwell-Boltzmann distribution at temperature T a and number density of n a takes the form f a ( p ) = n a (cid:18) π (cid:126) β a m a (cid:19) ν/ exp (cid:26) − β a p m a (cid:27) = n a λ νa e − β a E a , (2.44)where β a = 1 /T a is the inverse temperature in energy units, and E a = p a / m a is the kineticenergy of an individual particle of species a . The thermal de Broglie wavelength for particle a is defined to be λ a = (cid:126) (cid:18) πβ a m a (cid:19) / = h √ π m a T a . (2.45)Expression (2.44) shows explicitly that f a is dimensionless in accordance with our normal-ization, as n a λ νa is dimensionless. However, one might rightly ask why would a quantumparameter, (cid:126) , appear in a classical distribution? In fact, physical averages do not dependon (cid:126) , as the factors of (cid:126) in the integration measure (2.43) are exactly canceled by those inthe normalization term λ νa . However, this normalization is more than a mere convention, for n a λ νa counts the number of quantum states available to system a . Therefore, this normal-ization is the only one that gets questions about entropy correct. In fact, upon writing thedistribution function in terms of the chemical potential µ a , f a = e − β a ( E a − µ a ) , (2.46)we see that e β a µ a = n a λ νa . (2.47)14his gives the correct chemical potential of a free gas, µ a = T a ln n a λ ν = T a ln (cid:40) n a (cid:34) (cid:126) (cid:18) πm a T a (cid:19) / (cid:35) ν (cid:41) , (2.48)a standard result from quantum statistical mechanics, with a trivial generalization to multipledimensions.Two-particle correlations are described by a two-component correlation function, definedby f ab ( x a , p a , x b , p b , t ) d ν x a d ν p a (2 π (cid:126) ) ν d ν x b d ν p b (2 π (cid:126) ) ν ≡ number of particles of type a in a hypervolume d ν x a about x a and d ν p a about p a , and d ν x b about x b and d ν p b about p b , at time t . (2.49)We often write this as f ( x , p , x , p , t ) in a single-species plasma. We can go on to definehigher order distribution functions, and in fact show that they satisfy a system of connectedequations known as the BBGKY hierarchy. We will discuss this in much more detail infuture sections. For now, let us stick to the basic properties of a multi-component plasma,and let us see what we can do with just the single-particle distribution. E. The Dielectric Function of a Plasma
As these notes are also a tutorial, it is useful to take a detour and to describe how kinetictheory allows us to calculate the induced charge density ρ ind of a plasma when a smallexternal electric field D is applied. This in turn allows us to calculate the dielectric functionof the plasma. Consider a multi-species plasma in which species b has charge e b and mass m b .Furthermore, suppose that each species b is in thermal equilibrium with itself at temperature T b and Maxwell-Boltzmann distribution f b . We now use basic kinetic theory to prove thatthe dielectric function in a general number of dimensions takes the form (cid:15) ( k , ω ) = 1 + (cid:88) b e b k (cid:90) d ν p b (2 π (cid:126) ) ν ω − k · v b + iη k · ∂f b ( p b ) ∂ p b , (2.50)where the limit η → + is understood, and p b = m b v b . Reference [12] derives this well knownresult in three spatial dimensions, and this section is a simple extension to a general numberof spatial dimensions ν . It serves mainly as a refresher to the reader who is not an expert,and it is very beautiful physics. It is quite astounding that one can get so much from solittle. 15 . The Induced Charge Density and the Dielectric Function Let us start with a neutral equilibrium plasma, and apply a small external electric field D . Since the charges in the plasma are free to move, even the smallest field creates aninduced charge density ρ ind , as the ions and electrons are separated by the field. The chargeseparation creates an induced field P that weakens the applied field. The observed field E is a combination of both of these [12], E = D − P where (2.51) ∇ · D = ρ ext (2.52) ∇ · P = − ρ ind . (2.53)We are free to set the density ρ ext to anything that creates the desired applied field D . Theobserved field E satisfies Gauss’s law, ∇ · E = ρ , (2.54)where ρ is the total charge density of the medium, which consists of the external chargedensity and the induced charge density ρ = ρ ext + ρ ind . (2.55)The field D ( x , t ) = E ( x , t ) + P ( x , t ) depends not just on the electric field E ( x , t ) at time t ,but also on the electric fields at earlier times as well, through the polarization P ( x , t ) = (cid:90) t −∞ dt (cid:48) (cid:90) d ν x (cid:48) χ ( x − x (cid:48) , t − t (cid:48) ) E ( x (cid:48) , t (cid:48) ) . (2.56)It is understood that the kernel satisfies causality, χ ( x , t ) = 0 when t <
0, and we cantherefore extend the t (cid:48) integral from t to ∞ , giving D ( x , t ) = E ( x , t ) + (cid:90) ∞−∞ dt (cid:48) (cid:90) d ν x (cid:48) χ ( x − x (cid:48) , t − t (cid:48) ) E ( x (cid:48) , t (cid:48) ) . (2.57)Using the convolution theorem, the Fourier transform takes a particularly simple form,˜ D ( k , ω ) = (cid:15) ( k , ω ) ˜ E ( k , ω ) (2.58) (cid:15) ( k , ω ) = 1 + χ ( k , ω ) , (2.59)where the spatial and temporal Fourier transform of the susceptibility is˜ χ ( k , ω ) = (cid:90) ∞−∞ dt (cid:90) d ν x χ ( x , t ) e − i k · x + iωt , (2.60)16nd the inverse transform is χ ( x , t ) = (cid:90) ∞−∞ dω π (cid:90) d ν k (2 π ) ν ˜ χ ( k , ω ) e i k · x − iωt . (2.61)This sign convention is consistent with the sign conventions of quantum mechanics: − iωt and i k · x in (2.61). Returning to (2.51), we can write the Fourier transform of the polarizationvector as ˜ P ( k , ω ) = ˜ D ( k , ω ) − ˜ E ( k , ω ) = χ ( k , ω ) ˜ E ( k , ω ) . (2.62)The spatial form of Gauss’s law ∇ · P = − ρ ind translates into i k · ˜ P = − ˜ ρ ind in Fourier space,and this allows us to write˜ ρ ind ( k , ω ) = − iχ ( k , ω ) k · ˜ E ( k , ω ) = − χ ( k , ω ) k ˜ φ ( k , ω ) = − χ ( k , ω ) ˜ ρ ( k , ω ) . (2.63)The last form has been expressed in terms of the potential φ , defined by E = − ∇ φ , which inFourier space becomes ˜ E = − i k ˜ φ ( k , ω ). The Fourier transform of the dielectric is therefore χ ( k , ω ) = − ˜ ρ ind ( k , ω ) k ˜ φ ( k , ω ) , (2.64)and the problem reduces to calculating ˜ ρ ind ( k , ω ). The tool for performing such calculationsis kinetic theory.
2. Calculation of the Dielectric Function of a Plasma
Let us concentrate on an individual plasma component a . In the absence of an appliedfield, we assume that species a is in thermal equilibrium with itself, specified by a Maxwell-Boltzmann distribution ¯ f a ( p ) with inverse temperature β a = 1 /T a and charge e a . When anexternal electric field is applied, this induces a charge density ρ ind . The distribution functionconsequently departs from equilibrium, and the system is then specified by a new distribution f a ( x , p , t ). Because collisions are unimportant to the induced charges, the distribution f a satisfies the collisionless Maxwell-Boltzmann equation ∂f a ∂t + v · ∂f a ∂ x + e a E · ∂f a ∂ p = 0 , (2.65)where v = p /m a , and E = D − P is the total electric field seen by a . The electric field E is the sum of the applied field D and an induced contribution − P . Note that the kinetic The quantum energy and momentum operators are ˆ E = i (cid:126) ∂/∂t and ˆ p = − i (cid:126) ∂/∂ x , so that ˆ E e − iωt = (cid:126) ω e − iωt and ˆ p e i k · x = (cid:126) k e i k · x ; therefore, the energy and momentum Eigenvalues are E = (cid:126) ω and p = (cid:126) k . ρ a ( x , t ) = (cid:90) d ν p (2 π (cid:126) ) ν e a f a ( x , p , t ) (2.66) J a ( x , t ) = (cid:90) d ν p (2 π (cid:126) ) ν e a v a f a ( x , p , t ) , (2.67)then these quantities satisfy the continuity equation ∂ρ a ∂t + ∇ · J a = 0 . (2.68)This is because the last term in the kinetic equation (2.65) is a total divergence in momen-tum space, and therefore integrates to zero by the divergence theorem. It is reassuring tosee charge conservation arising directly from the kinetic equation. Indeed, all of hydrody-namics can be recovered from kinetic theory, although this would take us well beyond thescope of these notes. For this calculation, we start by expressing f a in terms of a small perturbation h a , f a ( x , p , t ) = ¯ f a ( p ) + h a ( x , p , t ) . (2.69)Upon substituting (2.69) back into (2.65) and work to first order. Since the induced electricfield E is first order, we can neglect the small second-order term e E · ∂h/∂ p , and write thekinetic equation as ∂h a ∂t + v · ∂h a ∂ x + e a E · ∂ ¯ f∂ p = 0 . (2.70)It is often convenient to express the electric field in terms of a potential, E = − ∇ φ ≡ − ∂φ∂ x , (2.71)in which case the transport equation (2.70) takes the form ∂h a ∂t + v · ∂h a ∂ x = e a ∂φ∂ x · ∂ ¯ f∂ p . (2.72)To solve this equation we take the space and time Fourier, thereby giving − iω ˜ h a + i k · v ˜ h a = e a ˜ φ ( k , ω ) i k · ∂ ¯ f∂ p , (2.73)where ˜ h a is a function of k , p , and p . We can now solve (2.73) for the perturbation,˜ h a ( k , p , ω ) = − e a ˜ φ ( k , ω ) ω − k · v k · ∂ ¯ f ( p ) ∂ p . (2.74)18ote that ˜ φ ( k ) is the Fourier transform of the applied potential, and it not given by (2.29).In performing the inverse Fourier transform to recover the correlation function h ( x , p , t )in space and time, we must integrate over k and ω . By convention, we hold k fixed andintegrate over the variable ω first. The integration contour for ω lies in the complex ω -planeslightly above the real axis. This avoids the pole at ω = k · v when integrating over ω ,and establishes the proper causality for h ( x , p , t ). This choice of contour is equivalent tointegrating over real values of ω , but adding a small complex term iη to the numerator in(2.74). We can therefore write the Fourier transform of the correlation function as˜ h a ( k , p , ω ) = − e a ˜ φ ( k , ω ) ω − k · v + iη k · ∂ ¯ f ( p ) ∂ p , (2.75)where limit η → + is understood. We can always restore the correlation function to spaceand time variables by performing the inverse Fourier transform, h a ( x , p , t ) = − (cid:90) d ν k (2 π ) ν dω π e i k · x − iωt e a k ω − k · v a + iη k · ∂ ¯ f a ( p ) ∂ p × k ˜ φ ( k , ω ) , (2.76)where we have factored out the term k ˜ φ ( k ) for convenience. Note that the form of ˜ φ ( k )is unknown, but it will cancel from the dielectric function. The induced charge densitytherefore becomes ρ ind ( x , t ) = (cid:88) b (cid:90) d ν p b (2 π (cid:126) ) ν e b h b ( x , p b , t ) . (2.77)It is actually more convenient to continue working in Fourier space, and using (2.75) allowsus to express the induced charge density as˜ ρ ind ( k , ω ) = (cid:88) b (cid:90) d ν p b (2 π (cid:126) ) ν e b ˜ h b ( k , p b , ω ) (2.78)= − (cid:88) b (cid:90) d ν p b (2 π (cid:126) ) ν e b k ω − k · v b + iη k · ∂ ¯ f b ( p b ) ∂ p b × k ˜ φ ( k , ω ) . (2.79)The susceptibility is therefore˜ χ ( k , ω ) = − ˜ ρ ind ( k , ω ) k ˜ φ ( k , ω ) = (cid:88) b e b k (cid:90) d ν p b (2 π (cid:126) ) ν ω − k · v b + iη k · ∂ ¯ f b ( p b ) ∂ p b , (2.80)which gives (2.50) for the dielectric function (cid:15) ( k , ω ) = 1 + χ ( k , ω ).19 II. COULOMB ENERGY TRANSFER RATES IN ARBITRARY DIMENSIONS
This section is a review of the basic BPS formalism, presented here for completeness, withan emphasis on the role of analytic continuation of the spatial dimension ν . We turn now tocalculating Coulomb energy exchange rates in the multi-component plasma described in theprevious section. The charged particle stopping power and the temperature equilibrationrate between plasma species of different temperatures are the two canonical examples I havein mind. A. Coulomb Energy Exchange
The single-particle distribution function for plasma species a satisfies a general kineticequation of the form ∂f a ∂t + v a · ∂f a ∂ x + F a · ∂f a ∂ p = (cid:18) ∂f a ∂t (cid:19) c ≡ (cid:88) b K νab [ f ] , (3.1)where the velocity is given by v a = p /m a , and F a is the total force acting on a at x , e.g. F a = e a E ( x ) in the case of an external electric field. The scattering rate ( ∂f a /∂t ) c is a genericexpression that accounts for the effects of scattering or collisions. It is calculated in kinetictheory text books under various conditions, the most relevant being for the Boltzmann kernel B ab [ f ] and the Lenard-Balescu kernel L ab [ f ]. For now, we will keep the form of the kernelgeneric and simply write K νab [ f ]. For stopping power calculations and other Coulomb energyexchange processes, we will set the external force to zero, so the distribution function f a satisfies ∂f a ∂t + v a · ∂f a ∂ x = (cid:88) b K νab [ f ] . (3.2)The kinetic energy density of plasma species a is defined by E a = (cid:90) d ν p a (2 π (cid:126) ) ν p a m a f a ( p a , t ) , (3.3)where f a is the corresponding distribution function. The stopping power is related to therate of energy loss by d E a dx = 1 v a d E a dt = 1 v a (cid:90) d ν p a (2 π (cid:126) ) ν p a m a ∂f a ( p a , t ) ∂t . (3.4)Using the kinetic equation (3.2), the divergence over x integrates to zero, and we find d E a dt = (cid:90) d ν p a (2 π (cid:126) ) ν p a m a ∂f a ( p a , t ) ∂t = (cid:88) b (cid:90) d ν p a (2 π (cid:126) ) ν p a m a K νab [ f ] . (3.5)We can therefore identify the rate of change in the kinetic-energy density of species a resultingfrom its Coulomb interactions with species b by d E ab dt = (cid:90) d ν p a (2 π (cid:126) ) ν p a m a K νab [ f ] . (3.6)20 . Dimensional Reduction of BBGKY As we have seen, moving to an arbitrary dimension ν acts as a regulator, renderingthe kinetic equations, both the Boltzmann equation (BE) and the Lenard-Balescu equation(LBE), finite in their respective dimensional regimes. Dimensional regularization, however,does far more than this. A more subtle advantage of working in a general dimension is thatit acts as a “physics sieve”, in that it selects the proper scattering kernel to leading order(LO) in g in the dimension at hand:BBGKY in ν > ⇒ ∂f a ∂t + v a · ∂f a ∂ x = (cid:88) b B ab [ f ] to LO in g , (3.7)where B ab is the ν -dimensional Boltzmann scattering kernel, andBBGKY in ν < ⇒ ∂f a ∂t + v a · ∂f a ∂ x = (cid:88) b L ab [ f ] to LO in g , (3.8)where L ab [ f ] is the ν -dimensional scattering kernel for the Lenard-Balescu equation. Provingthis statement, which I call the dimensional reduction theorem , is the main purpose of thesenotes. Figure 1 serves as a useful pictorial representation of the theorem.
1. The Boltzmann Kernel
In this subsection I will review the Boltzmann scattering kernel in some detail. In formalwork I will write the Boltzmann equation in schematic form as ∂f a ∂t + v a · ∂f a ∂ x = (cid:88) b B ab [ f ] : ν > , (3.9)or in calculations I will use the form B ab [ f ] = (cid:90) d ν p b (2 π (cid:126) ) ν | v a − v b | dσ ab (cid:26) f a ( p (cid:48) a ) f b ( p (cid:48) b ) − f a ( p a ) f b ( p b ) (cid:27) . (3.10)BPS included the quantum effects of two-body Coulomb scattering by replacing the clas-sical cross section by the corresponding quantum transition amplitude T ( ab → a (cid:48) b (cid:48) ) ≡ T a (cid:48) b (cid:48) ; ab ( W, q ), where W is the center-of-mass energy and q is the square of the momen-tum exchange during the collision. The cross section dσ ab and the square of the scatteringamplitude | T a (cid:48) b (cid:48) ; ab ( W, q ) | are related by | v a − v b | dσ ab ≡ (cid:90) d ν p (cid:48) a (2 π (cid:126) ) ν d ν p (cid:48) b (2 π (cid:126) ) ν (cid:12)(cid:12) T a (cid:48) b (cid:48) ; ab ( W, q ) (cid:12)(cid:12) (2 π (cid:126) ) ν δ ν (cid:16) p (cid:48) a + p (cid:48) b − p a − p b (cid:17) × (2 π (cid:126) ) δ (cid:16) E (cid:48) a + E (cid:48) b − E a − E b (cid:17) , (3.11)21nd the Boltzmann equation can then be written B ab [ f ] = (cid:90) d ν p (cid:48) a (2 π (cid:126) ) ν d ν p (cid:48) b (2 π (cid:126) ) ν d ν p b (2 π (cid:126) ) ν (cid:12)(cid:12) T a (cid:48) b (cid:48) ; ab ( W, q ) (cid:12)(cid:12) (cid:26) f a ( p (cid:48) a ) f b ( p (cid:48) b ) − f a ( p a ) f b ( p b ) (cid:27) (2 π (cid:126) ) ν δ ν (cid:16) p (cid:48) a + p (cid:48) b − p a − p b (cid:17) (2 π (cid:126) ) δ (cid:16) E (cid:48) a + E (cid:48) b − E a − E b (cid:17) . (3.12)The latter expression is more useful for formal manipulations, even in the classical regime,where one can define a classical “transition amplitude” T a (cid:48) b (cid:48) ; ab from (3.11) by using theclassical Rutherford cross section for dσ ab . Surprisingly, the classical amplitude is identicalto quantum Born amplitude. When ν >
3, expression (3.9) allows us to write the rate ofchange of the energy density resulting from the now finite Boltzmann kernel as d E > ab dt = (cid:90) d ν p a (2 π (cid:126) ) ν p a m a B ab [ f ] : ν > . (3.13)I have used a “greater than” superscript to remind us that we should calculate (3.13) indimensions greater than three.As we have discussed, in dimensions greater than three the derivation of the Boltzmannequation for Coulomb scattering is rigorous and finite. This is because the short distancephysics of the Coulomb potential is dominant in dimensions ν >
3, and the Boltzmannequation is designed to capture short distance scattering physics. Furthermore, the longdistance physics, where the Boltzmann equation breaks down in three dimensions, falls offfaster than 1 /r at large distances, thereby rendering the scattering finite for ν >
3. It shouldnot be a surprise that a simple scaling argument shows why B ab [ f ] is finite for ν >
3. Write (cid:15) = ν − >
0, and note that the amplitude scales as | T | ∼ /q , for momentum transfer q .Finally, a δ -function in q contributes a power q − , so that ν > B ab ∼ (cid:90) dq q ν − · q · q ∼ (cid:90) dq q ν − ∼ (cid:90) dq q − (cid:15) (3.14) ∼ q (cid:15) → q → . (3.15)The momentum transfer is related to the corresponding wavenumber by q = (cid:126) k , and wesee that small values of q correspond to large distances. This means that the Boltzmannequation does not possess an IR divergence for ν > . The Lenard-Balescu Kernel I usually write the Lenard-Balescu equation in schematic form, ∂f a ∂t + v a · ∂f a ∂ x = (cid:88) b L ab [ f ] : ν < , (3.16)although for calculations, we will use the explicit form L ab [ f ] = − ∂∂ p a · J ( p a ) (3.17) J ( p a ) = (cid:90) d ν p b (2 π (cid:126) ) ν d ν k (2 π ) ν k (cid:12)(cid:12)(cid:12)(cid:12) e a e b k (cid:15) ( k , k · v a ) (cid:12)(cid:12)(cid:12)(cid:12) π δ ( k · v a − k · v b ) (cid:20) k · ∂∂ p a − k · ∂∂ p b (cid:21) f a ( p a ) f b ( p b ) , (3.18)where v a = p a /m a is really an integration variable. The dielectric function (cid:15) is given by(2.50), which we repeat here for convenience with a change in summation index, (cid:15) ( k , ω ) = 1 + (cid:88) c e c k (cid:90) d ν p c (2 π (cid:126) ) ν ω − k · v c + iη k · ∂f c ( p c ) ∂ p c , (3.19)and the prescription η → + is implicit, defining the correct retarded time response. There-fore, when ν <
3, the rate (3.8) allows us to express d E < ab dt = (cid:90) d ν p a (2 π (cid:126) ) ν p a m a L ab [ f ] : ν < . (3.20)I have used a “less than” superscript to remind us that we should calculate (3.20) in dimen-sions less than three, where it is finite and well defined.In dimensions less than three one finds a complementary situation to the Boltzmannequation, namely, the derivation of the Lenard-Balescu equation is rigorous and finite when ν <
3. This is because the long distance physics of the Coulomb potential is dominant indimensions ν <
3, and the Lenard-Balescu equation is designed to capture such long distancephysics. Furthermore, the Coulomb potential falls off faster than 1 /r at large distances,where the LBE breaks down in three dimensions, and this renders the kernel finite in ν < L ab [ f ] is finite for ν <
3. Since (cid:15) = ν − <
0, we will workwith the quantity | (cid:15) | >
0. From (6.3), the kernel contains an obvious linear term k , and afactor k − arising from the Fourier transform of the potential. Note that (cid:15) ( k , k · v ) → k , so the dielectric function does not change the k → ∞ scaling behavior. Note that Eq. (3.57) for L ab [ f ] in Ref. [4] contains a spurious integration over the momentum p a . Fortu-nately, this typo was innocuous and did not affect the results that followed. δ -function gives a factor k − , and the p -derivative terms provide a compensating factorof k , so that ν < L ab ∼ (cid:90) dk k ν − · k · k · k · k ∼ (cid:90) dk k ν − ∼ (cid:90) dk k − −| (cid:15) | (3.21) ∼ k −| (cid:15) | → k → ∞ . (3.22)Since large values of k corresponds to small distances, this means that the LBE is finite atsmall distances for ν < C. Completing the Picture: The Rate and Analytic Continuation
As a matter of completeness, let us finish the calculation of the rate d E ab /dt . Recall thatwe have calculated the rates the rates d E > ab /dt and d E < ab /dt to leading order (LO) in g , andthey contain simple poles 1 / ( ν − d E > ab dt = H ( ν ) g ν − O ( ν −
3) is LO in g when ν > d E < ab dt = G ( ν ) g ν − − ν + O ( ν −
3) is LO in g when ν < , (3.24)where H ( ν ) and G ( ν ) are coefficients that depend upon ν . The heavy lifting for a realprocess is in calculating the functions H ( ν ) and G ( ν ) using the exact expressions for B ab and L ab . Once these calculations have been completed, in order to compare the rates (3.23)and (3.24), we must then analytically continue to a common value of the dimension ν (andthen take the limit ν → d E > ab /dt and d E < ab /dt as functions of a complex parameter ν , even though they were only calculated for positive integer values of ν . This is analogous toanalytically continuing the factorial function on the positive integers to the Gamma functionon the complex plane. For definiteness, I will analytically continue (3.24) to ν >
3, in whichcase g ν − = g ν − becomes subleading relative to the g dependence of (3.23), so that d E < ab dt = − G ( ν ) g ν − ν − O ( ν −
3) is NLO in g when ν > . (3.25)This is illustrated in Fig. 3. To finish calculating the rates, we need to work consistently tolinear order in the small parameter (cid:15) = ν −
3; therefore, we should expand H ( ν ) and G ( ν )to first order in (cid:15) , H ( ν ) = − A + (cid:15) H + O ( (cid:15) ) (3.26) G ( ν ) = − A + (cid:15) G + O ( (cid:15) ) . (3.27)24 ν • d E < ab /dtg ν − = g − (3 − ν ) ⇓ g −| ν − | LO: large when g (cid:28) d E < ab /dtg ν − = g ν − ⇓ g | ν − | NLO: small when g (cid:28) (cid:39) (cid:36) (cid:63) analytically continuearound the ν = 3 poleFIG. 3: The analytic continuation of d E < ab /dt from ν < ν > ν -plane. Thesame expression can be used for d E < ab /dt throughout the complex plane since the pole at ν = 3 can easilybe avoided. The quantity d E < ab /dt ∼ g ν − is leading order in g for ν <
3. However, upon analyticallycontinuing to ν > d E < ab /dt ∼ g | ν − | is next-to-leading order in g relative to d E > ab /dt ∼ g . It is crucially important here that H ( ν ) and G ( ν ) give the same value at ν = 3, a term thatI have called − A in (3.26) and (3.27), otherwise the divergent poles will not cancel. Finally,upon writing g (cid:15) = exp { (cid:15) ln g } in (3.25), and expanding to first order in (cid:15) , we find g (cid:15) (cid:15) = 1 (cid:15) + ln g + O ( (cid:15) ) . (3.28)This is where the nonanalyticity in g arises, i.e. the ln g term, and we can now express therates as d E > ab dt = − Aν − g + H g + O ( ν − g ) ν > d E < ab dt = Aν − g − G g − A g ln g + O ( ν − g ) ν > . (3.30)These expressions hold in the common dimension ν >
3, and to obtain the leading andnext-to-leading order result in three dimensions, we add and take the limit: d E ab dt = lim ν → + (cid:20) d E > ab dt + d E < ab dt (cid:21) + O ( g ) = − Ag ln g + Bg + O ( g ) , (3.31)with B = H − G . This gives the energy exchange rate from Coulomb interactions betweenplasma species, accurate to leading order and next-to-leading order in g , in terms of theCoulomb logarithm, d E ab dt = − Ag ln C g + O ( g ) , (3.32)where ln C = − B/A . The quantity L = ln Cg is known as the Coulomb logarithm .25
V. THE BBGKY HIERARCHY IN ARBITRARY DIMENSIONS
Now that we have reviewed statistical mechanics and Coulomb physics in ν -dimensions,we can start addressing the main claim of these notes, namely, that to leading order in theplasma coupling g , the BBGKY hierarchy reduces to the Boltzmann equation in ν >
3, andto the Lenard-Balescu equation in ν <
3. To prove this, we first derive the BBGKY hierarchyin a general number of dimensions, primarily to establish notation, and because it will bethe starting point in our derivation. We then develop perturbation theory in powers of thecoupling g , and we calculate the BBGKY equations to order g . To solve these equations, wemust make some approximations, and we discuss how the two regimes ν < ν > A. Liouville’s Theorem and Ensemble Averages
For simplicity, we consider a plasma with a single species of particle. We can add a chargeneutralizing background if desired; for example, one might consider an electron plasma withfixed ions as the background. Multiple plasma species can (and soon will) be added. Thereare several ways of representing the state of many-particle systems such as plasmas. Thefirst is through the use of a phase-spaced called µ , which is the 2 ν -dimensional phase space( x , p ) of a single particle. The state of a plasma with N particles is given by specifyingthe ν -dimensional positions x i and momenta p i for every particle i = 1 , , · · · , N in theplasma. Each particle is represented by a point ( x i , p i ) in µ -space, and the system lookslike a swarm of N particles, each interacting with all of the particles of the system. The ν -dimensional spatial slice of µ -space is the part of phase space that we observe with oureyes in the laboratory. The single particle distribution function f ( x , p ) lives in µ -space,and specifies the number of particles within a phase space element d ν x d ν p . A second way ofrepresenting the state of a multi-particle system is through the 2 νN -dimensional phase spacedefined by ( x , p , · · · , x N , p N ). This larger phase space is called Γ-space, and by design, theentire system is represented by a single point in Γ, rather than the swarm of points in µ . Tosimplify notation, we denote the 2 ν -dimensional phase space for particle i by X i = ( x i , p i ),so that points in Γ-space are specified by coordinate values ( X , · · · , X N ). Once the initialcondition of the N -body system is specified in Γ-space, that is to say, once the locationsand velocities of all N particles are specified at some initial time t = 0, then the subsequentevolution at any future time t is uniquely determined. The system therefore traces out a26ath in Γ-space as it evolves in time. There are only two possible types of paths: either thepath is periodic or it never intersects itself. This is related to the ergodic properties of thesystem.Let us now consider a large ensemble of systems in Γ-space, each evolving in time, tracingout a unique path for every member of the ensemble. We can think of building the ensembleby initially populating Γ-space uniformly among all possible initial conditions. None of thesystems in Γ-space interact, and every system evolves along its own private non-intersectingtrajectory in this 2 νN -dimensional space. The world-lines in Γ look like a tangle of non-intersecting spaghetti, with each strand oriented forward in time, never looping back onitself. Since we have populated Γ-space with all possible initial configurations, then by theergodic principle, ensemble averages in Γ-space give time averaged quantities as measuredby experiment. For an ensemble of systems in Γ-space, the ensemble density is defined by ρ ( X , · · · , X N , t ) dX · · · dX N ≡ probability that a system selected from the ensemblelies within dX i of X i for i = 1 , · · · , N, at time t, (4.1)where the measures are defined by dX i = d ν x i d ν p i (2 π (cid:126) ) ν . (4.2)The density is of course normalized to unity, (cid:90) dX · · · dX N ρ = 1 , (4.3)and ρ is symmetric in each of its arguments X i (again, this relates to the ergodic mixingproperties of the system).Since the individual systems in Γ are non-interacting, the density satisfies the conservationequation ∂ρ∂t + N (cid:88) i =1 ∇ X i · (cid:16) ρ ˙ X i (cid:17) = 0 . (4.4)It is more convenient to break the variables X i = ( x i , p i ) into their space and velocitycomponents, thereby giving ∂ρ∂t + N (cid:88) i =1 ∂∂ x i · (cid:16) ρ ˙ x i (cid:17) + N (cid:88) i =1 ∂∂ p i · (cid:16) ρ ˙ p i (cid:17) = 0 , (4.5)which we write in the form ∂ρ∂t + N (cid:88) i =1 ˙ x i · ∂ρ∂ x i + N (cid:88) i =1 ˙ p i · ∂ρ∂ p i + N (cid:88) i =1 ρ (cid:18) ∂∂ x i · ˙ x i + ∂∂ p i · ˙ p i (cid:19) = 0 . (4.6)27sing Hamilton’s equations of motion, ˙ x i = ∂H∂ p i (4.7)˙ p i = − ∂H∂ x i , (4.8)where H is the Hamiltonian, we see that the last term in Eq. (4.6) vanishes, (cid:18) ∂∂ x i · ˙ x i + ∂∂ p i · ˙ p i (cid:19) = ν (cid:88) (cid:96) =1 (cid:18) ∂ H∂x (cid:96)i ∂p (cid:96)i − ∂ H∂p (cid:96)i ∂x (cid:96)i (cid:19) = 0 . (4.9)The density in Γ-space therefore satisfies Liouville’s equation ∂ρ∂t + N (cid:88) i =1 v i · ∂ρ∂ x i + N (cid:88) i =1 F i · ∂ρ∂ p i = 0 , (4.10)where we have substituted v i = ˙ x i and F i = ˙ p i . The force F i includes the effects from all theother particles, and therefore F i depends upon all of the coordinates X , · · · , X N . Note that(4.10) takes the same form as the collisionless Boltzmann equation. This is because for non-interacting particles, the function f ( x , p , t ) acts like the ensemble density ρ ( X , · · · , X N ). B. The Hierarchy of Distribution Functions
We now show that there is a hierarchy of distributions functions that measure successivelyhigher-order correlations in the plasma. Note that we can define the average value of ageneral quantity Q = Q ( X , · · · , X N ), where we have not allowed for a possible explicit timedependence in Q (although we could), by (cid:104) Q ( X , · · · , X N , t ) (cid:105) = (cid:90) dX · · · dX N ρ ( X , · · · , X N , t ) Q ( X , · · · , X N ) . (4.11)Note that the time dependence of the average is due to that in ρ ( t ). Let us now define theone-particle function by F ( X ) = N (cid:88) i =1 δ ( X − X i ) , (4.12)where the particles are located at X , · · · , X N . Writing X = ( x , p ), we can recover thesingle-particle distribution by performing the ensemble average f ( x , p , t ) = (cid:104) F ( X ) (cid:105) . (4.13)To see this, we explicitly perform the ensemble average, f ( x , p , t ) = (cid:90) dX dX · · · dX N ρ ( X , X , · · · , X N , t ) N (cid:88) i =1 δ ( X − X i ) (4.14)= N (cid:90) dX · · · dX N ρ ( X, X , · · · , X N , t ) . (4.15)28he factor of N occurs because ρ is symmetric in its arguments, and therefore each term inthe sum over δ -functions is identical. Using the normalization (4.3) we see that (cid:90) d ν x d ν p (2 π (cid:126) ) ν f ( x , p , t ) = N (cid:90) dXdX · · · dX N ρ ( X, X , · · · , X N , t ) = N , (4.16)as required. Therefore the single-particle distribution f is automatically normalized cor-rectly. This motivates the definition of the s -particle correlation function f s ( X , · · · , X s ) = N !( N − s )! (cid:90) dX s +1 · · · dX N ρ ( X , · · · , X s , X s +1 , · · · , X N ) , (4.17)which has the normalization (cid:90) dX · · · dX s f s ( X , · · · , X s ) = N !( N − s )! . (4.18)We have now defined a hierarchy of distribution functions f , f , · · · , f s , · · · , f N − , f N , where f N ≡ ρ . In the next section we will find a set of coupled kinetic equations for each f s byintegrating over successive sub-spaces of the Liouville’s equation. C. The BBGKY Hierarchy of Kinetic Equations
Let us express the forces F i in Liouville’s equation (4.10) in terms of external forces F (0) i and the Coulomb interaction force F ( j ) i , F i = F (0) i + N (cid:88) j =1 F ( j ) i = N (cid:88) j =0 F ( j ) i , (4.19)where the Coulomb force on i from j takes the form F ( j ) i = e Γ( ν/ π ν/ x i − x j | x i − x j | ν . (4.20)We must exclude the j = i term from the sum in (4.19); or equivalently we can includethe value j = i in the sum by formally setting F ( i ) i = 0. We now show that the distribu-tion functions f s satisfy the following coupled set of kinetic equations called the BBGKYhierarchy, ∂f s ∂t + s (cid:88) i =1 v i · ∂f s ∂ x i + s (cid:88) i =1 s (cid:88) j =0 F ( j ) i · ∂f s ∂ p i = − s (cid:88) i =1 (cid:90) dX s +1 F ( s +1) i · ∂f s +1 ∂ p i , (4.21)where s = 1 , · · · N −
1. The BBGKY hierarchy is the the “ F = ma ” of kinetic theory. Theseequations are completed by Liouville’s equation (4.10) for x = N , ∂f N ∂t + N (cid:88) i =1 v i · ∂f N ∂ x i + N (cid:88) i =1 N (cid:88) j =0 F ( j ) i · ∂f N ∂ p i = 0 . (4.22)29e thus have a complete set of N equations in N variables f , f , · · · , f s , · · · , f N , and aunique solution will exist. It should be emphasized that the BBGKY hierarchy is timereversal invariant. It is only upon closing the equations at some level, usually s = 1 or s = 2, that we introduce time non-invariance. In other words, it is closing the hierarchyof kinetic equations that introduces the arrow of time. Except under the most contrived ofconditions, we cannot hope to find an exact solution, or even a numerical solution, as N is macroscopically large. However, the BBGKY hierarchy is still an extremely useful pieceof theoretical machinery, particularly in more formal arguments, and provides for a deeperunderstanding of kinetic theory.To prove (4.21), let us integrate (4.10) over dX s +1 · · · dX N , and multiply by the normal-ization factor of f s , thereby giving the exact equation N !( N − s )! (cid:90) dX s +1 · · · dX N (cid:32) ∂ρ∂t + N (cid:88) i =1 v i · ∂ρ∂ x i + N (cid:88) i =1 N (cid:88) j =0 F ( j ) i · ∂ρ∂ p i (cid:33) = 0 . (4.23)The first two terms of (4.23) are rather trivial to evaluate, and they correspond to the firsttwo terms of (4.21),term1 = N !( N − s )! (cid:90) dX s +1 · · · dX N ∂ρ∂t = ∂∂t N !( N − s )! (cid:90) dX s +1 · · · dX N ρ = ∂f s ∂t (4.24)term2 = N !( N − s )! N (cid:88) i =1 (cid:90) dX s +1 · · · dX N v i · ∂ρ∂ x i = s (cid:88) i =1 v i · ∂f s ∂ x i . (4.25)In expression (4.25), note that the sum over i has been truncated from N to s . This isbecause the terms i = s + 1 , · · · , N vanish by the use of divergence theorem, and the factthat ρ vanishes on the distant surface at infinity. To see that such terms explicitly vanish,let i ≥ s + 1, and consider the integral (cid:90) dX s +1 · · · dX N v i · ∂ρ∂ x i = (cid:90) dX s +1 · · · dX N ∂∂ x i · (cid:16) ρ v i (cid:17) (4.26)= (cid:90) dX s +1 · · · dX i − dX i +1 · · · dX N (cid:90) d p i (2 π (cid:126) ) ν (cid:90) V d ν x i ∂∂ x i · (cid:16) ρ v i (cid:17) = (cid:90) dX s +1 · · · dX i − dX i +1 · · · dX N (cid:90) d p i (2 π (cid:126) ) ν (cid:73) ∂V d S i · ρ v i = 0 , which vanishes because ρ vanishes on the surface at infinity, the boundary ∂V . Note that wehave enclosed the ν -space space x i in a very large but finite volume V (the volume will betaken to infinity in the limit). The boundary of V , denoted ∂V , is often called the surfaceat infinity. This is all standard, but it might be useful for the novice to have seen such acalculation all the way through. 30t this point in the derivation, the equation for f s is ∂f s ∂t + s (cid:88) i =1 v i · ∂f s ∂ x i + N !( N − s )! N (cid:88) i =1 N (cid:88) j =0 (cid:90) dX s +1 · · · dX N F ( j ) i · ∂ρ∂ p i = 0 . (4.27)We must now consider the last term in (4.27), which we decompose about the i = s contri-bution, N (cid:88) j =0 F ( j ) i = s (cid:88) j =0 F ( j ) i + N (cid:88) j = s +1 F ( j ) i . (4.28)The first sum in (4.28) is handled as before, and we can write (4.27) in the form ∂f s ∂t + s (cid:88) i =1 v i · ∂f s ∂ x i + s (cid:88) i =1 s (cid:88) j =0 F ( j ) i · ∂f s ∂ p i + N !( N − s )! N (cid:88) i =1 N (cid:88) j = s +1 (cid:90) dX s +1 · · · dX N F ( j ) i · ∂ρ∂ p i = 0 . (4.29)The final step in the calculation is to address the last term in (4.29). Recall that thedistribution function ρ is symmetric in its arguments X , · · · , X N . This means that everyterm of the j -sum in the last term of (4.29) is identical. Therefore, let us represent the sumby arbitrarily choosing the first term j = s + 1, and multiplying by ( N − s ) to account forthe remaining terms in the sum. This allows us to express the last term in (4.29) as N ! ( N − s )( N − s )! s (cid:88) i =1 (cid:90) dX s +1 · · · dX N F ( s +1) i · ∂ρ∂ p i , (4.30)where we have, for the usual reasons, truncated the i -sum at i = s . We can express (4.30) as s (cid:88) i =1 (cid:90) dX s +1 F ( s +1) i · ∂∂ p i N !( N − s − (cid:90) dX s +2 · · · dX N ρ (4.31)= s (cid:88) i =1 (cid:90) dX s +1 F ( s +1) i · ∂f s +1 ∂ p i , (4.32)and substituting (4.32) back into (4.29). This establishes the BBGKY hierarchy (4.21),which is repeated again for convenience, ∂f s ∂t + s (cid:88) i =1 v i · ∂f s ∂ x i + s (cid:88) i =1 s (cid:88) j =0 F ( j ) i · ∂f s ∂ p i = − s (cid:88) i =1 (cid:90) dX s +1 F ( s +1) i · ∂f s +1 ∂ p i , (4.33)for s = 1 , · · · , N −
1. The system is closed with Liouville’s equation, ∂f N ∂t + N (cid:88) i =1 v i · ∂f N ∂ x i + N (cid:88) i =1 N (cid:88) j =0 F ( j ) i · ∂f N ∂ p i = 0 . (4.34)31 . Dimensionless Variables We have now developed the BBGKY hierarchy (4.33) and (4.34) in the quite generalsetting of a non-equilibrium but single-component plasma. The plasma could be generalizedto have multiple components, but at the expense of increasing the complexity of the countingarguments and the simplicity of the formalism. In these notes, it turns out to be quite easy togeneralize the results of a single-component calculation to that of a multi-component plasma.For the sake of simplicity, we continue with a single-component plasma, whose constituentshave charge e and mass m . In equilibrium, the plasma is characterized by temperature T and number density n . We measure T in energy units, while n is the number of particles perunit hypervolume. The Debye wavenumber κ , and the plasma frequency ω p , are given by κ = e nT (4.35) ω p = e nm . (4.36)By dimensional analysis, these expressions hold in any spatial dimension ν , and we cantherefore use κ and ω p as defined by (4.35) and (4.36) in any dimension under consideration,as the electric charge absorbs any dimensional factors involving ν . Let us generalize theequilibrium system by imposing a small non-equilibrium background on the equilibriumplasma. This new quasi-equilibrium system is still described by the BBGKY hierarchy, andit is quite informative to express the BBGKY kinetic equations in terms of dimensionlessvariables. This is possible because the background equilibrium plasma provides naturallength and time scales. We shall see that the coupling constant g emerges quite naturally,and that a consistent perturbation theory in powers of g can be developed.We first express the basic kinematic variables in dimensionless form. We do this withthe following scale transformation, where the over-bar denotes the dimensionless form of thecorresponding variable, x = ¯ x /κ t = ¯ t/ω p (4.37) v = ( ω p /κ ) ¯ v p = ( mω p /κ ) ¯ p = ( T κ/ω p ) ¯ p (4.38) dX = (cid:16) mω p (cid:126) κ (cid:17) ν d ¯ X F (0) = κT ¯ F (0) (4.39) f s ( X , · · · , X s , t ) = (cid:18) (cid:126) κ mω p (cid:19) νs ¯ f s ( ¯ X , · · · , ¯ X s , t ) . (4.40)Motivated by the scaling κT for the external force F (0) , we are immediately led to expressthe Coulomb force as F ( j ) i = e Γ( ν/ π ν/ x i − x j | x i − x j | ν (4.41)= g κT ¯ F ( j ) i , (4.42)32here the dimensionless Coulomb force is defined by¯ F ( j ) i = ¯ x i − ¯ x j | ¯ x i − ¯ x j | ν , (4.43)and the remaining factors combine to form the plasma coupling constant, g = Γ( ν/ π ν/ e κ ν − T . (4.44)We see that the expansion parameter g simply falls out of the algebra. Finally, the BBGKYhierarchy (4.33) and (4.34) can be expressed in the dimensionless form ∂ ¯ f s ∂ ¯ t + s (cid:88) i =1 ¯ v i · ∂ ¯ f s ∂ ¯ x i + s (cid:88) i =1 ¯ F (0) i · ∂ ¯ f s ∂ ¯ p i + g s (cid:88) i =1 s (cid:88) j =1 ¯ F ( j ) i · ∂ ¯ f s ∂ ¯ p i = − g s (cid:88) i =1 (cid:90) d ¯ X s +1 ¯ F ( s +1) i · ∂ ¯ f s +1 ∂ ¯ p i , (4.45)for i = 1 , · · · , N −
1, in the square bracket along with the i = N equation ∂ ¯ f N ∂ ¯ t + N (cid:88) i =1 ¯ v i · ∂ ¯ f N ∂ ¯ x i + N (cid:88) i =1 ¯ F (0) i · ∂ ¯ f N ∂ ¯ p i + g N (cid:88) i =1 N (cid:88) j =1 ¯ F ( j ) i · ∂ ¯ f N ∂ ¯ p i = 0 . (4.46)It should be emphasized again that the coupling constant g as defined by (4.44) emerges quitenaturally, and when ν = 3, the coupling takes the usual form g = e κ/ πT (in rationalizedcgs units).In the next section, we will develop a method that permits us to solve the BBGKYequations perturbatively as an expansion in powers of g . Before doing this, it is instructiveto work through the algebra establishing (4.45) and (4.46). We start by measuring space inunits of inverse κ and time in units of inverse ω p , x = ¯ x /κ (4.47) t = ¯ t/ω p , (4.48)where the bared quantities are dimensionless. Note that ω p /κ = (cid:112) T /m = v th (4.49)is the thermal velocity of the plasma, which we use to form a dimensionless velocity v = ( ω p /κ ) ¯ v . (4.50)Since ω p /κ has units of velocity, we see that mω p /κ has units of momentum. The relation forthe thermal velocity (4.49) implies that ω p /κ = κT /ω p , and we can thus scale the momentumin two separate but equivalent ways p = ( mω p /κ ) ¯ p = ( κT /ω p ) ¯ p . (4.51)33oth forms of the momentum scaling will be used interchangeably. Since the temperature T has energy units, the quantity κT has units of force, and we define the dimensionless externalforce as F (0) = κT ¯ F (0) . (4.52)This motivates expressing the Coulomb force by F ( j ) i = e Γ( ν/ π ν/ x i − x j | x i − x j | ν = e Γ( ν/ π ν/ κ ν − T · κT · ˆ x i − ˆ x j | ˆ x i − ˆ x j | ν (4.53)= g κT ¯ F ( j ) i . (4.54)The next quantity that we consider is the phase space measure, and it transforms as dX = d ν x d ν p (2 π (cid:126) ) ν = (cid:16) mω p (cid:126) κ (cid:17) ν d ν ¯ x d ν ¯ p (2 π ) ν = (cid:16) mω p (cid:126) κ (cid:17) ν d ¯ X . (4.55)The final quantity to consider is the distribution function itself, which transforms by aconstant factor f s ( X , · · · , X s , t ) = N s ¯ f s ( ¯ X , · · · , ¯ X s , ¯ t ) . (4.56)We can find N s by the requirement that (cid:90) d ¯ X · · · d ¯ X s ¯ f s ( ¯ X , · · · , ¯ X s ) = (cid:90) dX · · · dX s f s ( X , · · · , X s ) (4.57)= (cid:16) mω p (cid:126) κ (cid:17) νs N s (cid:90) d ¯ X · · · d ¯ X s ¯ f s ( X , · · · , X s ) , (4.58)which implies N s = (cid:18) (cid:126) κ mω p (cid:19) νs . (4.59)We will later require the ratio N s +1 N s = (cid:18) (cid:126) κ mω p (cid:19) ν = N , (4.60)but for now we will express our results in terms of N s . Upon changing to dimensionlessvariables, the first two terms of the dimensional BBGKY hierarchy (4.33) becometerm1 ≡ ∂f s ∂t = ω p N s ∂ ¯ f s ∂ ¯ t (4.61)term2 ≡ s (cid:88) i =1 v i · ∂f s ∂ x i = s (cid:88) i =1 (cid:16) ω p κ ¯ v i (cid:17) · (cid:18) κ N s ∂ ¯ f s ∂ ¯ x i (cid:19) = ω p N s s (cid:88) i =1 ¯ v i · ∂ ¯ f s ∂ ¯ x i . (4.62)34he third term of (4.33) can be decomposed into an external force and an internal Coulombcontribution,term3a ≡ s (cid:88) i =1 F (0) i · ∂f s ∂ p i = s (cid:88) i =1 (cid:16) κT F (0) i (cid:17) · (cid:18) ω p N s κT ∂ ¯ f s ∂ ¯p i (cid:19) = ω p N s s (cid:88) i =1 ¯ F (0) i · ∂ ¯ f s ∂ ¯p i (4.63)term3b ≡ s (cid:88) i =1 s (cid:88) j =1 F ( j ) i · ∂f s ∂ p i = ω p N s g s (cid:88) i =1 s (cid:88) j =1 ¯ F ( j ) i · ∂ ¯ f s ∂ ¯ p i . (4.64)The BBGKY equations now become ∂ ¯ f s ∂ ¯ t + s (cid:88) i =1 ¯ v i · ∂ ¯ f s ∂ ¯ x i + s (cid:88) i =1 ¯ F (0) i · ∂ ¯ f s ∂ ¯ p i + g s (cid:88) i =1 s (cid:88) j =1 ¯ F ( j ) i · ∂ ¯ f s ∂ ¯ p i +1 ω p N s s (cid:88) i =1 (cid:90) dX s +1 F ( s +1) i · ∂f s +1 ∂ p i = 0 . (4.65)Finally, the last term in (4.65) involving f s +1 can be writtenterm4 ≡ ω p N s s (cid:88) i =1 (cid:90) dX s +1 F ( s +1) i · ∂f s +1 ∂ p i (4.66)= 1 ω p N s +1 N s · (cid:16) mω p (cid:126) κ (cid:17) ν · g κT · ω p T κ s (cid:88) i =1 (cid:90) d ¯ X s +1 ¯ F ( s +1) i · ∂ ¯ f s +1 ∂ ¯ p i (4.67)= g s (cid:88) i =1 (cid:90) d ¯ X s +1 ¯ F ( s +1) i · ∂ ¯ f s +1 ∂ ¯ p i , (4.68)where we have used (4.60) for N s +1 / N s = N . We have now established (4.45) and (4.46),which we reproduce below for convenience, ∂ ¯ f s ∂ ¯ t + s (cid:88) i =1 ¯ v i · ∂ ¯ f s ∂ ¯ x i + s (cid:88) i =1 ¯ F (0) i · ∂ ¯ f s ∂ ¯ p i + g s (cid:88) i =1 s (cid:88) j =1 ¯ F ( j ) i · ∂ ¯ f s ∂ ¯ p i = − g s (cid:88) i =1 (cid:90) d ¯ X s +1 ¯ F ( s +1) i · ∂ ¯ f s +1 ∂ ¯ p i , (4.69)for s = 1 , · · · , N −
1, and ∂ ¯ f N ∂ ¯ t + N (cid:88) i =1 ¯ v i · ∂ ¯ f N ∂ ¯ x i + N (cid:88) i =1 ¯ F (0) i · ∂ ¯ f N ∂ ¯ p i + g N (cid:88) i =1 N (cid:88) j =1 ¯ F ( j ) i · ∂ ¯ f N ∂ ¯ p i = 0 . (4.70)35 . Perturbation Theory As expressed in the form (4.69) and (4.70), it is unclear how to solve the BBGKY hier-archy perturbatively in powers of g . This is because the relation between the distributionfunctions ¯ f s and the coupling constant g is not straightforward. The proper procedure isto expand in powers of the so called reduced distribution functions ¯ h s = ¯ h s ( ¯ X , · · · , ¯ X s ).We define the reduced distribution ¯ h s by subtracting all possible lower order correlationsfrom ¯ f s , a procedure that will be made more precise in just a moment. Consequently, thedistribution ¯ h s is also called the correlation function , as it encodes the full complement of s -body correlations. Perturbation theory is then constructed by expanding in powers of ¯ h s .We start this recursive procedure by first constructing the 2-point correlation function ¯ h .To do this, let us briefly return to dimensional variables, and define h ( X , X ) = f ( X , X ) − f ( X ) f ( X ) . (4.71)It is clear that h ( X , X ) captures the 2-body correlations, as the uncorrelated piece f ( X ) f ( X ) has been subtracted from the full 2-body distribution f ( X , X ): the remain-der can only be the correlations. We will assume that h is of order g , and more generallythat gh s ∝ g s . In dimensionless coordinates, we can therefore express the 2-point functionby the expansion ¯ f ( ¯ X , ¯ X ) = ¯ f ( ¯ X ) ¯ f ( ¯ X ) + g ¯ h ( ¯ X , ¯ X ) . (4.72)We will justify this perturbative assumption by proving that we can expand (4.69) and(4.70) to second order in g (in principle we could work to any desired order in g ). In asimilar manner, the reduced 3-point function ¯ h is defined by the expansion¯ f ( ¯ X , ¯ X , ¯ X ) = ¯ f ( ¯ X ) ¯ f ( ¯ X ) ¯ f ( ¯ X ) + g (cid:104) ¯ h ( ¯ X , ¯ X ) ¯ f ( ¯ X ) + ¯ h ( ¯ X , ¯ X ) ¯ f ( ¯ X ) +¯ h ( ¯ X , ¯ X ) ¯ f ( ¯ X ) (cid:105) + g ¯ h ( ¯ X , ¯ X , ¯ X ) . (4.73)We have removed the following lower order correlations from ¯ f : (i) a completely uncorrelatedpiece consisting of the product of three 1-point functions ¯ f × ¯ f × ¯ f , and (ii) three 2-pointcorrelations involving ¯ h × ¯ f , evaluated on the cyclic permutations of X , X , and X , and(iii) the 3-point correlation ¯ h . Note that ¯ h is of order g , or in dimensional form, gh ∝ g .To the order g in which we are working, the ¯ h term must therefore be dropped from(4.73) for consistency. Although we will not do so in these notes, one may press onward andcalculate the order g terms. To do this, we would keep the g ¯ h contribution to ¯ f . Wewould also need to construct the 4-point correlation function ¯ h by subtracting off the lowerorder correlations from ¯ f , which schematically takes the form g ¯ h ( ¯ X , ¯ X , ¯ X , ¯ X ) = ¯ f ( ¯ X , ¯ X , ¯ X , ¯ X ) − ¯ f ( ¯ X ) ¯ f ( ¯ X ) ¯ f ( ¯ X ) ¯ f ( ¯ X ) − (4.74) g (cid:104) ¯ h ( ¯ X , ¯ X , ¯ X ) ¯ f ( ¯ X ) + · · · (cid:105) − g (cid:104) ¯ h ( ¯ X , ¯ X )¯ h ( ¯ X , ¯ X ) + · · · (cid:105) − g (cid:104) ¯ h ( ¯ X , ¯ X ) ¯ f ( ¯ X ) ¯ f ( ¯ X ) + · · · (cid:105) .
36s usual, we subtract off the completely uncorrelated piece, this time consisting of theproduct of four 1-point functions. At this order, there are several more combinations oflower-order correlations that must be removed. For example, there are terms like ¯ h × ¯ f , inaddition to pair-wise 2-point contributions like ¯ h × ¯ h . Finally, there are contributions ofthe form ¯ h × ¯ f ¯ f . The next higher-order contribution would have even more combinationsof lower-order correlations, and we see that higher-order calculations become quite involvedvery rapidly.We now show that one can work consistently to order g , dropping terms of order g andhigher. We must prove that the s = 1 and s = 2 equations contain terms of order g orlower, and that the s ≥ g and higher. For consistency,we must therefore work only with the s = 1 equation, and a truncated version of the s = 2equation (as we have seen, we must also drop the h -contribution in the ¯ f term, as thiscontribution is of order g ). Writing the factors of g explicitly, we will show that the s = 1equation takes the form (cid:20) ∂∂t + V A + gV B [ ¯ f ] (cid:21) ¯ f = g K [ ¯ h ] , (4.75)where K [ h ] is a homogeneous integration kernel, while V A and V B [ ¯ f ] are differential andintegro-differential operators on ¯ X space, respectively. Note that the operator V B [ ¯ f ] con-tains a functional dependence on ¯ f . The truncated s = 2 equation can be expressed in theform (cid:20) ∂∂t + V C + gV D [ ¯ f ] (cid:21) g ¯ h = gS [ ¯ f ] + O ( g ) , (4.76)where S [ ¯ f ] is a source term depending upon ¯ f , while V C is a differential operator, and V D [ ¯ f ] is an integro-differential operator. Both operators act on ¯ X - ¯ X space, of which ¯ h is afunction. The precise form of the source term, the operators, and the kernel are not importantto this perturbative argument, although we shall calculate these quantities explicitly in thenext paragraph. The point here is that both (4.75) and (4.76) are of order g , and thathigher- s equations are of order g and higher. Since the kernel K [¯ h ] is homogeneous, notethat the g -dependence on the right-hand-side of (4.75) may be recast in the more suggestiveform gK [ g ¯ h ], so that (4.75) and (4.76) is a system of coupled integro-differential equationsfor ¯ f and g ¯ h . These equations are accurate to order g , with error of order g . With a lotof work, one can show that the s = 3 equation is of order g , and consistency demands thatwe neglect it as well (and all higher order equations). This justifies the assumption that ¯ h is of order g , and that we are indeed working consistently with an accuracy of order g , andan absolute error of order g . 37 . General Number of Spatial Dimensions ν Let us now verify equations (4.75) and (4.76). We shall drop the bar from the dimension-less quantities for ease of notation, and the s = 1 equation of (4.69) becomes (cid:18) ∂∂t + v · ∂∂ x + F (0)1 · ∂∂ p (cid:19) f ( X ) = − g (cid:90) dX F (2)1 · ∂∂ p f ( X , X ) . (4.77)When working with the Boltzmann equation in ν >
3, this form will be particularly useful.For the perturbative analysis, however, it is better to expand f (and f in the s = 2 equation)in terms of the 2-point correlation h . For convenience we repeat here the expansions (4.72)and(4.73), but in an annotated form, f = f ( X ) f ( X ) (cid:124) (cid:123)(cid:122) (cid:125) uncorrelated + g h ( X , X ) (cid:124) (cid:123)(cid:122) (cid:125) . (4.78) f = f ( X ) f ( X ) f ( X ) (cid:124) (cid:123)(cid:122) (cid:125) uncorrelated + (4.79) g (cid:104) h ( X , X ) f ( X ) (cid:124) (cid:123)(cid:122) (cid:125) + h ( X , X ) f ( X ) + h ( X , X ) f ( X ) (cid:124) (cid:123)(cid:122) (cid:125) (cid:105) + higher-order . Using (4.78) in (4.77) gives the coupled integro-differential equation ∂f ( X ) ∂t + v · ∂f ( X ) ∂ x + F (0)1 · ∂f ( X ) ∂ p + g (cid:90) dX f ( X ) F (3)1 · ∂f ( X ) ∂ p = − g (cid:90) dX F (3)1 · ∂h ( X , X ) ∂ p . (4.80)We have replaced the integration variable X in (4.77) by X to avoid conflicts with thevariable X when we turn to the s = 2 equation. We can recast the above equation in amore compact form by defining the self-consistent electric field at x by F [ f ] = (cid:90) dX f ( X ) F (3)1 = (cid:90) dX f ( X ) F ( x − x ) , (4.81)so that (4.80) becomes (cid:32) ∂∂t + v · ∂∂ x + F (0)1 · ∂∂ p (cid:124) (cid:123)(cid:122) (cid:125) V A + g F [ f ] · ∂∂ p (cid:124) (cid:123)(cid:122) (cid:125) V B [ f ] (cid:33) f ( X ) = − g (cid:90) dX F (3)1 · ∂h ( X , X ) ∂ p (cid:124) (cid:123)(cid:122) (cid:125) K [ h ] . (4.82)We have identified the quantities V A , V B [ f ], and K [ h ] in (4.75) by the under-braces. Notethat there is a factor of g for every Coulomb interaction F (3)1 , and a factor of g for the38orrelation ¯ h . In dimensionless units, there is no difference between the electric force and theelectric field, as the factors of electric charge have been collected in the coupling constant g .Let us now turn to the the s = 2 equation of the BBGKY hierarchy (4.69), which wewrite in the form (cid:34) ∂∂t + (cid:88) i =1 (cid:18) v i · ∂∂ x i + F (0) i · ∂∂ p i (cid:19)(cid:35) f + g F (2)1 · (cid:20) ∂∂ p − ∂∂ p (cid:21) f = − g (cid:88) i =1 (cid:90) dX F (3) i · ∂f ∂ p i . (4.83)Note that we have expanded the 1-2 scattering term as (cid:88) i =1 2 (cid:88) j =1 g F ( j ) i · ∂f ∂ p i = g F (2)1 · (cid:20) ∂∂ p − ∂∂ p (cid:21) f (4.84)by using Newton’s third law F (2)1 = − F (1)2 . Expression (4.83) can be recast into an equationfor h by expanding f and f in terms of the 2-point correlation h . We will do this instages, emphasizing the role played by the spatial dimension ν at each step, showing howthe physics changes depending upon whether ν < ν >
3. Using (4.78) for f in the 1-2scattering term allows us to express (4.83) as (cid:34) ∂∂t + (cid:88) i =1 (cid:18) v i · ∂∂ x i + F (0) i · ∂∂ p i (cid:19)(cid:35) f + g F (2)1 · (cid:20) ∂∂ p − ∂∂ p (cid:21) gh + g (cid:88) i =1 (cid:90) dX F (3) i · ∂f ∂ p i = gS [ f ] , (4.85)where the source term is defined by S [ f ] = − F (2)1 · (cid:20) ∂∂ p − ∂∂ p (cid:21) f ( X ) f ( X ) . (4.86)It is instructive to contrast equation (4.83) with (4.85). The latter form is more amenableto the perturbative analysis we are performing. It will also be used in deriving the Lenard-Balescu equation for ν <
3, where the Coulomb forces become long-range and 2-body scat-tering becomes soft. In this regime, we can drop the correlation term g F (2)1 · (cid:20) ∂∂ p − ∂∂ p (cid:21) gh (4.87)from (4.85). In contrast, this term must be kept when ν >
3. This is because the scatterbecomes short-range, and momentum exchange can be become quite large. In this case, it isbest not to make the substitution for f in the 1-2 scattering term, and to use (4.83) instead.39his is justified, however, only after perturbation theory has been established. We will havemore say about this in the next section. For now, we retain all terms for completeness.Let us return to the general perturbative argument. I will present the detailed algebraicmanipulations, since this calculation provides a template for proving that the s = 3 kineticequation is indeed higher order. Upon expanding the remaining f -term in (4.85), the s = 1equation can now be written as (cid:34) ∂∂t + (cid:88) i =1 (cid:18) v i · ∂∂ x i + F (0) i · ∂∂ p i (cid:19)(cid:35) gh + g F (2)1 · (cid:20) ∂∂ p − ∂∂ p (cid:21) gh + (4.88) (cid:20) ∂∂t + (cid:88) i =1 (cid:18) v i · ∂∂ x i + F (0) i · ∂∂ p i (cid:19) (cid:21) f ( X ) f ( X ) + g (cid:88) i =1 (cid:90) dX F (3) i · ∂f ∂ p i = gS [ f ] . The contribution from the uncorrelated piece of f is written in the second line of (4.88),which breaks up into two collections of terms, one proportional to f ( X ) and the otherproportional to f ( X ): (cid:20) ∂f ( X ) ∂t + v · ∂f ( X ) ∂ x + F (0)1 · ∂f ( X ) ∂ p (cid:21) f ( X ) + (4.89) (cid:20) ∂f ( X ) ∂t + v · ∂f ( X ) ∂ x + F (0)2 · ∂f ( X ) ∂ p (cid:21) f ( X ) . Our strategy will be to expand f in (4.88) using (4.79), and then to collect terms thatreproduce the s = 1 equation (4.82) within the square brackets. This equation will be evalu-ated at X and X in each square bracket, respectively, but they will otherwise vanish. Thisleaves only an equation involving gh on the left-hand-side (which is explicitly of order g ).To perform this calculation, we express the f scattering term by g (cid:88) i =1 (cid:90) dX F (3) i · ∂f ∂ p i = (4.90) g (cid:88) i =1 (cid:90) dX F (3) i · ∂∂ p i (cid:104) f ( X ) gh ( X , X ) + f ( X ) gh ( X , X ) (cid:105) + g (cid:88) i =1 F i [ f ] · ∂∂ p i f ( X ) f ( X ) + g (cid:88) i =1 F i [ f ] · ∂∂ p i gh ( X , X ) , The second line of (4.90) can be traced to the 2-3 and 1-3 correlations in (4.79), whilethe terms in the third line come from the uncorrelated piece of f and the 1-2 correlation,40espectively. We have generalized to definition of the self-consistent field to any position x i , F i [ f ] = (cid:90) dX f ( X ) F (3) i = (cid:90) dX f ( X ) F ( x i − x ) . (4.91)We now express equation (4.88) in the form (cid:34) ∂∂t + (cid:88) i =1 (cid:18) v i · ∂∂ x i + F (0) i · ∂∂ p i + g F i [ f ] · ∂∂ p i (cid:19)(cid:35) gh + g F (2)1 · (cid:20) ∂∂ p − ∂∂ p (cid:21) gh + (cid:88) i =1 (cid:90) dX g F (3) i · ∂∂ p i (cid:104) f ( X ) gh ( X , X ) + f ( X ) gh ( X , X ) (cid:105) + (cid:20) ∂f ( X ) ∂t + v · ∂f ( X ) ∂ x + F (0)1 · ∂f ( X ) ∂ p + g F [ f ] · ∂f ( X ) ∂ p (cid:21) f ( X ) + (4.92) (cid:20) ∂f ( X ) ∂t + v · ∂f ( X ) ∂ x + F (0)2 · ∂f ( X ) ∂ p + g F [ f ] · ∂f ( X ) ∂ p (cid:21) f ( X ) = gS [ f ] . Note that there are terms directly proportional to f ( X ), and others proportional to f ( X ),which have been grouped together in the square brackets. As mentioned above, each of thesquare brackets will turn out to vanish upon using the s = 1 equation at X and X ,respectively. Note that i = 1 term of the sum in the second line (from the 2-3 and 1-3correlations) takes the form g (cid:90) dX F (3)1 · ∂f ( X ) ∂ p h ( X , X ) + g (cid:90) dX F (3)1 · ∂h ( X , X ) ∂ p (cid:124) (cid:123)(cid:122) (cid:125) kernel for s = 1 equation at X × f ( X ) , (4.93)and the i = 2 term is g (cid:90) dX F (3)2 · ∂f ( X ) ∂ p h ( X , X ) + g (cid:90) dX F (3)2 · ∂h ( X , X ) ∂ p (cid:124) (cid:123)(cid:122) (cid:125) kernel for s = 1 equation at X × f ( X ) . (4.94)The second term in (4.93) marked by an under-brace is the kernel of the s = 1 equation(4.82), evaluated at the default position X . When combined with the terms in the firstsquare bracket, those proportional to f ( X ), we find the s = 1 equation evaluated at X ,and these terms vanish. Note that (4.82) is evaluated at the phase space position X ,and since X is just a free variable (in the formal mathematical sense), we can make thereplacement X → X in (4.82). Thus the s = 1 equation can also be evaluated at X . Wesee that the second term in (4.94) contains the kernel of the s = 1 equation at X , and thesecond square bracket also vanishes. The truncated s = 2 equation therefore becomes41 ∂∂t + (cid:88) i =1 (cid:18) v i · ∂∂ x i + F (0) i · ∂∂ p i + g F i [ f ] · ∂∂ p i (cid:19)(cid:35) gh + (4.95) g (cid:90) dX F (3)1 · ∂f ( X ) ∂ p gh ( X , X ) + g (cid:90) dX F (3)2 · ∂f ( X ) ∂ p gh ( X , X ) + g F (2)1 · (cid:20) ∂∂ p − ∂∂ p (cid:21) gh = gS [ f ] , which is in the form given by (4.76). Also note that the absolute error incurred by droppingthe h -contribution from gf is of order g .To fully complete the argument, we must show that the s = 3 equation, expressed herefor completeness, (cid:32) ∂∂t + (cid:88) i =1 (cid:34) v i · ∂∂ x i + F (0) i · ∂∂ p i + g (cid:88) j =1 F ( j ) i · ∂∂ p i (cid:35) (cid:33) f ( X , X , X ) (4.96)= − g (cid:90) dX (cid:88) i =1 F (4) i · ∂∂ p i f ( X , X , X , X ) , is of order g or higher. This is performed in complete analogy to the s = 2 case justpresented. We first express (4.96) in terms of gh , up to order order g . The definition of h and h ensure that the accuracy of (4.96) is of order order g . There will be terms analogousto the square brackets in (4.92), proportional to factors of f , but these terms will vanishby using the lower-order equations for f and gh . The final result will be of order g , andmust therefore be dropped for consistency.
2. Coulomb Physics in ν < and ν > We have expanded the BBGKY hierarchy to order g in a general number of spatialdimensions ν , with little regard to the behavior of the Coulomb physics as a function ν .The equations have been quite general, but we must make some approximations to proceed,and the validity of the approximations depends upon whether the scattering is hard or soft,that is to say, upon whether ν > ν <
3, respectively. We have already addressed the1-2 correlation and how it must be kept in ν >
3, and how can it can, in part, be droppedin ν <
3. It will turn out that complementary collections of 2-body correlations dominatein dimensions ν < ν >
3, and vice verse. In ν = 3, both collection of termscontribute equally to the transport equations, but they have the misfortune of diverging42ogarithmically in at long- and short-distances. Recall from Section II A, that the electricelectric field at position x from a point charge e at the origin is E ( x ) = e Γ( ν/ π ν/ ˆ x r ν − , (4.97)where ˆ x is the unit vector at the origin, and r = | x | is the distance to x . It is often moreconvenient to work with the Coulomb potential φ ( x ) = e Γ( ν/ − π ν/ r ν − . (4.98)These two expressions are just equations (2.5) and (2.8), and they produce a qualitativedifference in the Coulomb field for ν > ν <
3. The reader is encouraged to revisitFig. 2 for details. Short-distance ultraviolet (UV) physics is dominant in dimensions ν > ν <
3. The dimension ν = 3 is a critical case, in which the UV and IR physics are equally dominant. For ν < /r as r →
0. In this regime, the Lenard-Balescu scattering kernel does not suffer a UV divergence. In like manner, for ν > /r as r → ∞ . In this regime, and theBoltzmann scattering kernel does not suffer an IR divergence.The primary results from the previous section are the s = 1 equation (4.82), repeatedhere for convenience, (cid:32) ∂∂t + v · ∂∂ x + F (0)1 · ∂∂ p + g F [ f ] · ∂∂ p (cid:33) f ( X ) = − g (cid:90) dX F (3)1 · ∂gh ( X , X ) ∂ p , (4.99)and the truncated s = 2 equation (4.95), (cid:34) ∂∂t + (cid:88) i =1 (cid:18) v i · ∂∂ x i + F (0) i · ∂∂ p i + g F i [ f ] · ∂∂ p i (cid:19)(cid:35) gh + (4.100) g (cid:88) i =1 F i [ gh ] · ∂f ∂ p i + g F (2)1 · (cid:20) ∂∂ p − ∂∂ p (cid:21) gh = gS [ f ] . We have written (4.100) in a compact form, using the self consistent field F i [ f ] defined in(4.91), and the self-consistent field induced by h , F i [ h ] = (cid:90) dX h ( X , X j ) F (3) i . (4.101)Here, j = 2 when i = 1, and j = 1 when i = 2. Note that long-distance 2-body Coulombscattering is soft in dimensions ν <
3, and the momentum exchange is small. We cantherefore drop the 1-2 correlation term g F (2)1 · (cid:20) ∂∂ p − ∂∂ p (cid:21) gh ( X , X ) , (4.102)43hich we assume is formally of order g . This assumption does not mean that the momentumdifference in the collision is being neglected, i.e. we are not dropping the term − g F (2)1 · (cid:20) ∂∂ p − ∂∂ p (cid:21) f ( X ) f ( X ) = gS [ f ] . (4.103)In ν <
3, we can therefore neglect (4.102) from (4.100), giving (cid:34) ∂∂t + (cid:88) i =1 (cid:18) v i · ∂∂ x i + F (0) i · ∂∂ p i + g F i [ f ] · ∂∂ p i (cid:19)(cid:35) gh ( X , X ) + (4.104) (cid:90) dX gh ( X , X ) g F (3)1 · ∂f ( X ) ∂ p + (cid:90) dX gh ( X , X ) g F (3)2 · ∂f ( X ) ∂ p = gS [ f ] , where we have expanded the F i [ h ] term for clarity. The equation for h can be recast inthe form ∂h ∂t + V h + V h = S [ f ] , (4.105)with source S [ f ] = − F (2)1 · (cid:20) ∂∂ p − ∂∂ p (cid:21) f ( X ) f ( X ) . (4.106)Here, V is a linear integro-differential operator defined in X -space by V h ( X , X ) = v · ∂h ∂ x + F (0)1 · ∂h ∂ p + g F · ∂h ∂ p + g (cid:90) dX h ( X , X ) F (3)1 · ∂f ( X ) ∂ p , (4.107)and V is the corresponding operator in X -space, V h ( X , X ) = v · ∂h ∂ x + F (0)2 · ∂h ∂ p + g F · ∂h ∂ p + g (cid:90) dX h ( X , X ) F (3)2 · ∂f ( X ) ∂ p . (4.108)These expressions simplify marginally in the case of a uniform plasma, and this will be ourstarting point in Section VI on the Lenard-Balescu equation.In dimensions ν >
3, the behavior of the Coulomb field is quite different. The potentialbecomes short-range, and 2-body scattering is not always soft. This means that we cannotdrop the 2-body momentum exchange term in (4.102). This term arose from the expansionof f , so it is convenient not to make this expansion, and to express the s = 1 equation as (cid:18) ∂∂t + v · ∂∂ x + F (0)1 · ∂∂ p (cid:19) f = − g (cid:90) dX F (2)1 · ∂f ∂ p . (4.109)44e can, however, drop the short-range contributions to the scattering in f , and use thetruncated s = 2 equation (cid:34) ∂∂t + (cid:88) i =1 (cid:18) v i · ∂∂ x i + F (0) i · ∂∂ p i + F i [ f ] · ∂∂ p i (cid:19)(cid:35) f + g F (2)1 · (cid:20) ∂∂ p − ∂∂ p (cid:21) f = 0 . (4.110)This will be our starting point for Section V on the Boltzmann equation. Note that the term f implicitly contains a functional dependence on the 2-point correlation h .
3. The Uniform Plasma
In the calculations that follow, we make one more significant assumption, namely, wetake the plasma to be spatially uniform in that the 1-point function f = f ( p ) dependsonly upon the momentum. By Galilean invariance, the 2-point function can only be ofthe form h = h ( x − x , p , p ). This plasma conforms to the experimental situationsinvolved in inertial confinement fusion (ICF), the testing ground of charged particle stoppingpower. I did not wish to introduce this assumption sooner, as I wanted to prove the validityof perturbation theory for the BBGKY hierarchy in a more general setting. The kineticequations simplify somewhat, in that the self-consistent forces F i [ f ] vanish. Keeping theexternal force F (0) for generality, in ν <
3, the s = 1 equation becomes (cid:32) ∂∂t + F (0)1 · ∂∂ p (cid:33) f ( p , t ) = − g (cid:90) dX F (3)1 · ∂h ( X , X , t ) ∂ p . (4.111)We shall change spatial integration variables from x to x = x − x . Setting p = p and X = ( x , p ), we can express (4.111) as (cid:32) ∂∂t + F (0)1 · ∂∂ p (cid:33) f ( p , t ) = − g (cid:90) dX F ( x ) · ∂∂ p gh ( x , p , p , t ) . (4.112)In a similar manner, the truncated s = 2 equation in ν < ∂h ∂t + V h + V h = S [ f ] , (4.113)with source S [ f ] = − F (2)1 · (cid:20) ∂∂ p − ∂∂ p (cid:21) f ( p ) f ( p ) . (4.114)45ere, V is a linear integro-differential operator defined in X -space by V h ( X , X ) = v · ∂h ∂ x + F (0)1 · ∂h ∂ p + g (cid:90) dX h ( X , X ) F (3)1 · ∂f ( p ) ∂ p , (4.115)and V is the corresponding operator in X -space, V h ( X , X ) = v · ∂h ∂ x + F (0)2 · ∂h ∂ p + g (cid:90) dX h ( X , X ) F (3)2 · ∂f ( p ) ∂ p . (4.116)This will be our starting point when ν < ν > (cid:18) ∂∂t + F (0)1 · ∂∂ p (cid:19) f = − g (cid:90) dX F (2)1 · ∂f ∂ p (4.117)and (cid:34) ∂∂t + (cid:88) i =1 (cid:18) v i · ∂∂ x i + F (0) i · ∂∂ p i (cid:19)(cid:35) f + g F (2)1 · (cid:20) ∂∂ p − ∂∂ p (cid:21) f = 0 . (4.118)The latter equation is just the s = 2 equation in which f → . THE BOLTZMANN EQUATION FROM BBGKY IN ν > In this section will prove that in spatial dimensions ν >
3, the Boltzmann equation (BE)follows from the BBGKY hierarchy to leading order in the plasma coupling g . For simplicitywe work with a single-component plasma, for which the Boltzmann equation takes the form ∂f∂t + v · ∂f∂ x = B [ f ] , (5.1)with scattering kernel B [ f ] = (cid:90) d ν p (2 π (cid:126) ) ν | v − v | dσ (cid:26) f ( p (cid:48) ) f ( p (cid:48) ) − f ( p ) f ( p ) (cid:27) , (5.2)where p = m v . The ν -dimensional cross section dσ is defined in Appendix A 3. Theargument of this section is based on that of Huang from Section 3.5 of Ref. [10]. Huang’sargument in fact breaks down for the Coulomb force in ν = 3 spatial dimensions (the actualcase of physical interest) because of a long-distance infra-red (IR) divergence. However, theargument goes through unscathed when generalized to arbitrary spatial dimensions ν > /r ν − , which falls off faster than 1 /r at large r for ν >
3, thereby rendering finite any potential IR divergence. Since the short distancephysics of the BE is correct, the scattering kernel does not, as we expect, suffer a short-distance ultra-violet (UV) divergence.In Section IV E, we showed that for short-range interactions, in particular the Coulombforce in dimensions ν >
3, the BBGKY hierarchy to order g can be expressed as (cid:18) ∂∂t + v · ∂∂ x (cid:19) f ( X ) = − (cid:90) dX F (2)1 · ∂∂ p f ( X , X ) (5.3) (cid:32) ∂∂t + v · ∂∂ x + v · ∂∂ x + F (2)1 · (cid:20) ∂∂ p − ∂∂ p (cid:21) (cid:33) f ( X , X ) = 0 , (5.4)where f is the single-particle distribution function, and f is the 2-point distribution. Ex-pression (5.3) is the first BBGKY equation, while (5.4) is the second BBGKY equation,except that the 3-point function has been dropped from the right-hand-side of the full s = 2equation (4.83). To continue, let us express (5.4) in center-of-mass coordinates. Since ourfinal goal is to apply this formalism to a multi-species plasma, in this calculation let ustemporarily suppose particle-1 has mass m and particle-2 has mass m . We will denote thetotal mass by M = m + m , and the reduced mass by m = m m /M . We then define thetotal and relative momentum, and the center-of-mass and relative position by x = x − x p = m (cid:0) v − v (cid:1) (5.5) R = m x + m x M P = m v + m v . (5.6) We have restored a slowly varying spatial dependence to f . See Appendix B for the details of this coordinate transformation. P = 0 in the center-of-mass frame, in Appendix A 3 we show that v · ∂∂ x + v · ∂∂ x = (cid:0) v − v (cid:1) · ∂∂ x (5.7) ∂∂ p − ∂∂ p = ∂∂ p , (5.8)where p = m v and p = m v . To find the BE, we must consider the asymptotictime limit t → ∞ . This is because, as per Bogoliubov’s hypothesis, the 2-point correlation h comes into equilibrium much sooner than the single particle distribution f . We mayconsequently set the time derivative in (5.4) to zero, giving the static equation (cid:32)(cid:0) v − v (cid:1) · ∂∂ x + F (2)1 ( x ) · ∂∂ p (cid:33) f = 0 , (5.9)where we have used (5.7) and (5.8) to write (5.4) in terms of relative coordinates. Let usnow express the first BBGKY equation (5.3) in terms of a scattering kernel, (cid:18) ∂∂t + v · ∂∂ x (cid:19) f = B [ f ] , (5.10)where the kernel is defined by B [ f ] ≡ − (cid:90) dX F (2)1 · ∂f ∂ p (5.11)= − (cid:90) dX F ( x − x ) · (cid:20) ∂∂ p − ∂∂ p (cid:21) f = − (cid:90) dX F (2)1 ( x ) · ∂f ∂ p . (5.12)We have added zero in the form of the total derivative ∂/∂ p in (5.12), and we have used(5.8) to express the resulting difference in momentum derivatives in terms of the derivativeof the relative momentum. We can now use (5.9) to write the scattering kernel in the form B [ f ] = (cid:90) d ν p (2 π (cid:126) ) ν (cid:90) d ν x ( v − v ) · ∂f ∂ x . (5.13)It is understood that (5.13) is to be evaluated in the limit t → ∞ , or rather, at asymptotictimes compared to the time scale of the 2-point correlations h .We shall now express (5.13) in terms of the ν -dimensional cross section dσ . See Ap-pendix A 3 of these notes for a detailed treatment of the cross section in a general number ofdimension. As illustrated in Fig. 4, let the beam-line of the 1+2 collision define the x -axis,so that v − v = | v − v | ˆ x . In two-body scattering, the velocity vectors v and v aredirected toward one another along the beam-line, but they are offset (in a normal directionto x ) by a distance b called the impact parameter. Using expression (A17), the ν -dimensionalvolume element in cylindrical coordinates about particle-2 can be expressed as48 IG. 4: Two-body scattering for a short-range force. Particle-1 has velocity v and particle-2 hasvelocity v , although for simplicity particle-2 is pictured at rest. The particle velocities are directedtowards one another, with the beam-line defining the x -axis. Therefore, v − v = | v − v | ˆ x . Thecross section is given in terms of the impact parameter b by dσ = d Ω ν − b ν − db , and thereforethe volume element about particle-2 can be written d ν x = dσ dx . d ν x = d Ω ν − b ν − db dx . (5.14)Section A 3 of these notes proves that ν -dimensional differential scattering cross section takesthe form dσ = d Ω ν − b ν − db , (5.15)and therefore the spatial volume element can be written d ν x = dσ dx . (5.16)Since the Coulomb force in ν > r ∼ κ − ,we can choose points x and x on either side of x such that the force virtually vanishes for x < x and x > x , although the force cannot be neglected for x < x < x . In other words,we can choose the points x and x right after and right before the collision of interest. Thisis illustrated in Fig. 4. We can therefore write (5.13) in the form B [ f ] = (cid:90) d ν p (2 π (cid:126) ) ν (cid:90) dσ (cid:90) x x dx | v − v | ∂f ∂x (5.17)= (cid:90) d ν p (2 π (cid:126) ) ν | v − v | dσ (cid:104) f ( x ) − f ( x ) (cid:105) . (5.18)49ecall that the 2-point function f is the product of two factors of f and a correlationfunction h . Since the Coulomb force is short range in ν >
3, the function h vanishes at x and x , and we have f = f × f + h → f × f , (5.19)so that f ( x ) = f ( p ) f ( p ) (5.20) f ( x ) = f ( p (cid:48) ) f ( p (cid:48) ) . (5.21)Here, p = m v and p = m v are the momenta before the collision, and p (cid:48) and p (cid:48) are the momenta after the collision. In standard derivations of the Boltzmann equation,the assumption of molecular chaos is invoked at this juncture. This principle states thatthe momenta before and after a collision are uncorrelated, and we see that the short-rangenature of the Coulomb force in ν > B [ f ] = (cid:90) d ν p (2 π (cid:126) ) ν (cid:90) d Ω | v − v | dσ d Ω (cid:104) f ( p (cid:48) ) f ( p (cid:48) ) − f ( p ) f ( p ) (cid:105) . (5.22)In terms of a quantum transition amplitude T , we can express the Boltzmann scatteringkernel in the form B [ f ] = (cid:90) d ν p (cid:48) (2 π (cid:126) ) ν d ν p (cid:48) (2 π (cid:126) ) ν d ν p (2 π (cid:126) ) ν (cid:12)(cid:12) T (cid:48) (cid:48) ; 12 (cid:12)(cid:12) (cid:26) f ( p (cid:48) ) f ( p (cid:48) ) − f ( p ) f ( p ) (cid:27) (2 π (cid:126) ) ν δ ν (cid:16) p (cid:48) + p (cid:48) − p − p (cid:17) (2 π (cid:126) ) δ (cid:16) E (cid:48) + E (cid:48) − E − E (cid:17) . (5.23)The time non-invariance of the BE happens in two places in this argument: (i) using the s = 2equation at asymptotic times, and (ii) the molecular chaos assumption. The generalizationto multi-species is easy. Two-point functions are still uncorrelated at x and x , so that f ( x ) = f a ( p a ) f b ( p b ) (5.24) f ( x ) = f a ( p (cid:48) a ) f b ( p (cid:48) b ) , (5.25)and the Boltzmann scattering kernel becomes B ab [ f ] = (cid:90) d ν p (cid:48) a (2 π (cid:126) ) ν d ν p (cid:48) b (2 π (cid:126) ) ν d ν p b (2 π (cid:126) ) ν (cid:12)(cid:12) T a (cid:48) b (cid:48) ; ab (cid:12)(cid:12) (cid:26) f a ( p (cid:48) a ) f b ( p (cid:48) b ) − f a ( p a ) f b ( p b ) (cid:27) (2 π (cid:126) ) ν δ ν (cid:16) p (cid:48) a + p (cid:48) b − p a − p b (cid:17) (2 π (cid:126) ) δ (cid:16) E (cid:48) a + E (cid:48) b − E a − E b (cid:17) . (5.26)50 I. THE LENARD-BALESCU EQUATION FROM BBGKY IN ν < We now derive the Lenard-Balescu equation (LBE) from the BBGKY hierarchy in spatialdimensions ν <
3. We rely on Chapter 12 of Clemmow and Dougherty [9] as our primarysource in this section, since it is so clearly written and easily generalizes to multiple dimen-sions. The calculation is very long, but quite informative. The calculation of Ref. [9] actuallybreaks down in ν = 3 spatial dimensions because of a short-distance ultra-violet (UV) diver-gence. However, all quantities become finite when ν <
3, and the calculation can proceed aspresented. This is because the Coulomb force falls off like 1 /r ν − , and this renders the UVdivergence finite in ν <
3. Since the long distance physics of the LBE is correct, the kerneldoes not suffer a long-distance infra-red (IR) divergence. In a single-component plasma, theLBE takes the form ∂f∂t + v · ∂f∂ x = L [ f ] , (6.1)with scattering kernel L [ f ] = − ∂∂ p · J ( p ) (6.2) J ( p ) = (cid:90) d ν p (2 π (cid:126) ) ν d ν k (2 π ) ν k (cid:12)(cid:12)(cid:12)(cid:12) e k (cid:15) ( k , k · v ) (cid:12)(cid:12)(cid:12)(cid:12) π δ ( k · v − k · v ) (cid:20) k · ∂∂ p − k · ∂∂ p (cid:21) f ( p ) f ( p ) . As always, we take v i = p i /m for i = 1 ,
2. For a single component plasma, the dielectricfunction (cid:15) is given by (cid:15) ( k , ω ) = 1 + e k (cid:90) d ν p (2 π (cid:126) ) ν ω − k · v + iη k · ∂f ( p ) ∂ p , (6.3)and the prescription η → + is implicit, defining the correct retarded time response. A. Formal Solution to the Perturbative Equations
We now explicitly assume the plasma to be uniform, in the sense that the 1-point functionis constant in space, being a function only of momentum, f ( X, t ) = f ( p , t ) , (6.4)with X = ( x , p ). Galilean invariance then constrains the 2-point function to take the form h ( X , X , t ) = h ( x − x , p , p , t ) . (6.5)The time dependence t will often be left implicit, and we will employ a slight abuse ofnotation by writing h ( X , X ). The so called self-consistent fields F i [ f ] defined in (4.91)51anish under the condition of uniformity, and the kinetic equations take slightly simplerforms. In Section IV E, we showed that for long-range interactions, in particular for theCoulomb force in ν <
3, the the coupled system integro-differential equations is ∂∂t f ( p , t ) = − g (cid:90) dX F ( x ) · ∂∂ p gh ( x , p , p , t ) . (6.6)and ∂h∂t + V h + V h = S [ f ] , (6.7)where the source term is S [ f ] = − F (2)1 · (cid:20) ∂∂ p − ∂∂ p (cid:21) f ( p ) f ( p ) . (6.8)Note that F (2)1 = e E ( x − x ) is the Coulomb force at x from a point charge at x . Theseequations are accurate to order g in the plasma coupling. The quantity V is an integro-differential operator defined in X -space by V h ( X , X ) = v · ∂h ( X , X ) ∂ x + (cid:90) dX h ( X , X ) F (3)1 · ∂f ( p ) ∂ p , (6.9)and V is the corresponding operator in X -space, V h ( X , X ) = v · ∂h ( X , X ) ∂ x + (cid:90) dX h ( X , X ) F (3)2 · ∂f ( p ) ∂ p , (6.10)with F (3) i (for i = 1 ,
2) being the Coulomb force at x i from x . When the correlation function h is written without arguments, it is assumed to be h ( X , X ). The variable X in (6.9)“ just goes along for the ride”, and we may regard V as an operator in X -space acting on afunctions h ( X ). Similarly, the variable X is free in the operator V . The operators V and V therefore commute when acting on functions h ( X , X ) of two variables, and V and V may consequently be treated as numbers when solving the differential equation (6.7) for h ( t ).In deriving the Lenard-Balescu equation, we require the asymptotic time limit t → ∞ of h ( t ).This is because of Bogoliubov’s hypothesis, which states that h ( x , p , t ) quickly relaxes to itsasymptotic value h ( x , p , ∞ ), relative to f ( p , t ). We may therefore treat t as a parameter inthe source term S , and the operators V , and V .Let us now find a formal solution for h ( t ). We leave the phase-space variables x and p implicit. We will employ the method of Laplace transforms, where the Laplace transformand its inverse are related by ˜ h ( p ) = (cid:90) ∞ dt e − pt h ( t ) (6.11) h ( t ) = 12 πi (cid:90) C dp e pt ˜ h ( p ) , (6.12)52here the contour C runs parallel to the imaginary axis with all poles of ˜ h ( p ) lying to theleft of C . The analytic structure of ˜ h ( p ) in the complex p -plane determines the function h ( t )for all values of t greater than zero. Let us multiply (6.7) by e − pt and integrate over t , givingthe equation (cid:90) ∞ dt e − pt (cid:18) ∂h∂t + V h + V h (cid:19) = (cid:90) ∞ dt e − pt S = p − S . (6.13)Upon integrating by parts, (cid:90) ∞ dt e − pt ∂h∂t = e − pt h ( t ) (cid:12)(cid:12)(cid:12)(cid:12) ∞ + (cid:90) ∞ dt p e − pt h = − h (0) + p ˜ h ( p ) , (6.14)we can express this as ( p + V + V )˜ h ( p ) = p − S + h (0) . (6.15)Solving for the Laplace transform ˜ h gives the formal solution˜ h ( p ) = ( p + V + V ) − (cid:0) p − S + h (0) (cid:1) . (6.16)We can find the asymptotic value h ( ∞ ) from (6.16) in the following manner. From (6.16)we see that ˜ h ( p ) has a pole at p = 0, in addition to the other poles lying in the left half-planewith Re p <
0. As t → ∞ , the dominant contribution to the integral (6.11) comes from the p = 0 pole. This means we can replace the contour C by a circular contour C r of radius r about the origin, and we can evaluate h ( ∞ ) by integrating around C r and taking the limit r → + . Points on C r are given by p = re iθ . Therefore dp = ip dθ , and we can changevariables from p to θ . Since we are interested in the r → + limit, we can replace factors of p in the integrand by factors of r (there is no θ -dependence at the origin), giving h ( ∞ ) = lim t →∞ lim r → + πi (cid:73) C r dp e pt ˜ h ( p ) = lim t →∞ lim r → + π (cid:90) π dθ r e rt ˜ h ( r ) (6.17)= lim r → + π · π · r · · ˜ h ( r ) , (6.18)where the factor of unity comes from e rt → r → + (note that t is fixed while the r -limitis taken). Consequently, we find the elegant and compact result h ( ∞ ) = lim p → + p ˜ h ( p ) , (6.19)where we have changed variables from r back to p in the limit. Upon using (6.16) for theLaplace transform ˜ h ( p ), we can write the asymptotic form as h ( ∞ ) = lim p → + ( p + V + V ) − S . (6.20)53ote that the initial condition ˜ h (0) does not appear in the asymptotic form.Recall that the Laplace transform of e − at is ( p + a ) − , and we can therefore write( p + V + V ) − = (cid:90) ∞ dt e − ( p + V + V ) t (6.21)= 1(2 πi ) (cid:90) ∞ dt e − pt (cid:90) C dp e p t p + V (cid:90) C dp e p t p + V , (6.22)where we have expressed e − V t and e − V t as inverse Laplace transforms defined by contours C and C , respectively. These two contours are suitable inverse Laplace transform con-tours parallel to the imaginary axis, with all poles lying to their left. Upon performing the t -integral, we find( p + V + V ) − = 1(2 πi ) (cid:90) C dp (cid:90) C dp p − p − p p + V p + V , (6.23)where Re p > Re( p + p ) for the t -integral convergence at large t . The asymptotic form of h can now be written h ( ∞ ) = lim p → + πi ) (cid:90) C dp (cid:90) C dp p − p − p p + V p + V S [ f ] . (6.24)The source S [ f ] is defined by (4.106). The problem now reduces to an exercise in complexanalysis, albeit a rather involved exercise. The next step involves calculating the action ofthe operators ( p + V ) − and ( p + V ) − on the source S . B. Preliminary Example
As a prelude to finding the inverse operators above, let us consider a simpler problem inwhich h is a function of only one phase-space variable X (rather than X and X ), so that h ( X, t ) = h ( x , p , t ). Suppose now that h satisfies the simplified equation ∂h∂t + V h = 0 , (6.25)where the operator is defined by V h ( x , p ) = v · ∂h∂ x + (cid:90) dX h ( x , p ) F (3) x · ∂f ( p ) ∂ p . (6.26)As usual, F (3) x = e E ( x − x ) is the Coulomb force at x from a point charge at x , and theintegration variable is X = ( x , p ). We shall express the operator V in the more suggestiveform V h = v · ∂h∂ x + e E [ h ] · ∂f ( p ) ∂ p , (6.27)54here we define the electric field functional by E [ h ]( x ) = (cid:90) dX h ( x , p ) E ( x − x ) . (6.28)The quantity E [ h ] is analogous to the self-consistent electric field E [ f ], although it does notvanish in a uniform plasma. To solve (6.25) for h , let us take the spatial Fourier transformand the temporal Laplace transform of (6.25),( p + V ) ˜ h ( k , p , p ) = ˜ h ( k , p , , (6.29)which has the formal solution˜ h ( k , p , p ) = ( p + V ) − ˜ h ( k , p , . (6.30)The tilde over a function is used to denote both the Fourier and Laplace transforms. Thetransform of relevance should be clear from context (and from the presence of the variable x vs. k or t vs. p ). In other words, we are using a mixed notation in which ˜ h ( k , p , p ) is thespatial Fourier transform and the temporal Laplace transform of h ( x , p , t ), while ˜ h ( k , p , h ( x , p , t = 0). The momentum variable “just goes alongfor the ride,” so we will keep it implicit.To find an expression for ( p + V ) − that we can use in a calculation, let us repeat thesteps leading to the formal expression (6.30), except that now we shall employ the explicitform (6.27) for V . Note that we are using the Fourier conventions give by (2.60) and (2.61),or equivalently by (2.26) and (2.27). Clemmow and Dougherty [9] use a convention with theopposite sign of k and different factors of 2 π , so care must be taken when comparing theresults from these notes to Ref. [9]. Note that the spatial integral in (6.28) is a convolutionof h ( x ) and the Coulomb field E ( x ) of a point charge. We can therefore use the convolutiontheorem when taking the spatial Fourier transform of (6.27), giving p ˜ h + i k · v ˜ h + e ˜ E [ h ] · ∂f∂ p = ˜ h ( k , p , . (6.31)As stated above, the spatial Fourier transform ˜ E [ h ] of the self-induced field E [ h ] is performedby applying the convolution theorem, so that (being explicit with the arguments)˜ E [ h ]( k , p ) = (cid:90) d ν p (2 π (cid:126) ) ν ˜ h ( k , p , p ) ˜ E ( k ) , (6.32)where ˜ E ( k ) is the Fourier transform of the static Coulomb field,˜ E ( k ) = − i k ˜ φ ( k ) = i k ek . (6.33)55or notational simplicity, we drop the functional dependence on h from (6.32), and write˜ E ( k , p ). We can now solve (6.31) for ˜ h , giving( p + V ) − ˜ h ( k , p , ≡ ˜ h ( k , p , p ) (6.34)= 1 p + i k · v (cid:20) ˜ h ( k , p , − e ˜ E ( k , p ) · ∂f ( p ) ∂ p (cid:21) . As with the point charge in (6.33), the self-consistent electric field ˜ E ( k , p ) can be expressedin terms of a self-consistent potential ˜ φ ( k , p ) defined by˜ E ( k , p ) = − i k ˜ φ ( k , p ) . (6.35)In terms of this self-consistent potential, we have˜ h ( k , p , p ) = 1 p + i k · v (cid:20) ˜ h ( k , p ,
0) + e ˜ φ ( k , p )( i k ) · ∂f ( p ) ∂ p (cid:21) (6.36)˜ φ ( k , p ) = ek (cid:90) d ν p (cid:48) (2 π (cid:126) ) ν ˜ h ( k , p (cid:48) , p ) , (6.37)where we have changed integration variables from p to p (cid:48) .Let us now substitute (6.36) for ˜ h into (6.37) for the potential,˜ φ ( k , p ) = ek (cid:90) d ν p (cid:48) (2 π (cid:126) ) ν p + i k · v (cid:48) (cid:20) ˜ h ( k , p (cid:48) ,
0) + e ˜ φ ( k , p ) i k · ∂f ( p (cid:48) ) ∂ p (cid:48) (cid:21) , (6.38)where v (cid:48) = p (cid:48) /m . Note that ˜ φ ( k , p ) appears on both sides of this equation, and uponisolating the ˜ φ ( k , p ) term, we find (cid:34) − e k (cid:90) d ν p (cid:48) (2 π (cid:126) ) ν p + i k · v (cid:48) i k · ∂f ( p (cid:48) ) ∂ p (cid:48) (cid:35) ˜ φ ( k , p ) = ek (cid:90) d ν p (cid:48) (2 π (cid:126) ) ν ˜ h ( k , p (cid:48) , p + i k · v (cid:48) . (6.39)Solving (6.39) for the self-consistent potential therefore gives˜ φ ( k , p ) = e ¯ (cid:15) ( k , p ) k (cid:90) d ν p (cid:48) (2 π (cid:126) ) ν ˜ h ( k , p (cid:48) , p + i k · v (cid:48) , (6.40)where the “dielectric function” in Laplace space is defined by¯ (cid:15) ( k , p ) = 1 − (cid:90) d ν p (cid:48) (2 π (cid:126) ) ν e k p + i k · v (cid:48) i k · ∂f ( p (cid:48) ) ∂ p (cid:48) , (6.41)with p lying on the contour C . For future reference, we record the following identities: (cid:90) d ν p (cid:48) (2 π (cid:126) ) ν e k i k · ∂f ( p (cid:48) ) /∂ p (cid:48) p + i k · v (cid:48) = 1 − ¯ (cid:15) ( k , p ) (6.42) (cid:90) d ν p (cid:48) (2 π (cid:126) ) ν e k i k · ∂f ( p (cid:48) ) /∂ p (cid:48) p − i k · v (cid:48) = ¯ (cid:15) ( − k , p ) − . (6.43)56e will use these expressions throughout. Now, upon substituting (6.40) back into (6.36),we find( p + V ) − ˜ h ( k , p , ≡ ˜ h ( k , p , p )= 1 p + i k · v (cid:34) ˜ h ( k , p ,
0) + e ¯ (cid:15) ( k , p ) k i k · ∂f ( p ) ∂ p (cid:90) d ν p (cid:48) (2 π (cid:126) ) ν ˜ h ( k , p (cid:48) , p + i k · v (cid:48) (cid:35) . (6.44)We will generalize this result to the operators V and V shortly. To identify the quantity¯ (cid:15) ( k , p ) physically, we can analytically continue (6.41), allowing p to lie anywhere in thecomplex plane. When p = − iω , we note that¯ (cid:15) ( k , − iω ) = (cid:15) ( k , ω ) (6.45)Thus, the analytically continued dielectric function in Laplace space is just the ordinarydielectric function in temporal Fourier space. We also note that (cid:15) ( − k , − ω ) = (cid:15) ∗ ( k , ω ) , (6.46)and therefore ¯ (cid:15) ( − k , iω ) = (cid:15) ( − k , − ω ) = (cid:15) ∗ ( k , ω ) . (6.47)These complex conjugation properties will be useful in the forthcoming calculation. C. The Lenard-Balescu Equation
We now return to the Lenard-Balescu formalism in X - X space, and to the 2-pointcorrelation h ( X , X , t ). Bogoliubov’s hypothesis means that the time scale of h ( X , X , t )is much shorter than the time scale of f ( X, t ), so we can replace h by its t → ∞ limitrelative to f . Therefore, we shall assume that h ( X , X , t ) relaxes to its asymptotic value h ( X , X , ∞ ), and that the LBE kernel is L [ h ] ≡ − (cid:90) dX e E (2)1 · ∂h ( X , X , ∞ ) ∂ p = − ∂∂ p · J (6.48) J ( X ) ≡ (cid:90) dX e E (2)1 h ( X , X , ∞ ) . (6.49)In the single-particle distribution f ( p , t ), the time t is treated as a parameter. It is note-worthy that the approximation of replacing h ( t ) by its asymptotic value of h ( ∞ ) is wherethe non-reversibility in time enters the LBE kinetic equation. Because the distribution f is uniform, the 2-point correlation function h ( x , x , , p , p , ∞ ) reduces to a function of57nly a single spatial coordinate, h ( x , p , p , ∞ ). The Fourier transform of this function is˜ h ( k , p , p , ∞ ), and we see that h ( x − x , p , p , ∞ ) = (cid:90) d ν k (2 π ) ν e i ( x − x ) · k ˜ h ( k , p , p , ∞ ) . (6.50)In expression (6.49), let us write the inter-molecular as F (2)1 = e E ( x − x ), where E ( x ) isthe electric field of a point charge at the origin. Recall that the Fourier transforms of E ( x )and the corresponding potential φ ( x ) are e ˜ E ( k ) = − i k ˜ φ ( k ) (6.51)˜ φ ( k ) = e k , (6.52)where I have temporarily changed conventions by including an extra factor of e into theelectric potential φ . This is to avoid factors of eφ , as the stray electric charge is cumbersometo track. For a multi-species plasma, we would change e in (6.52) to e a e b . The Lenard-Balescu current (6.49) now becomes J = (cid:90) d ν p (2 π (cid:126) ) ν (cid:90) d ν x e E ( x − x ) h ( x − x , p , p , ∞ ) (6.53)= (cid:90) d ν p (2 π (cid:126) ) ν (cid:90) d ν x e E ( x ) h ( x , p , p , ∞ ) , (6.54)where we have made the change of variables x = x − x in (6.54), thereby illustrating that J is constant in space (independent of x ). We can express the x -integration in (6.54) as (cid:90) d ν x e E ( x ) h ( x ) = (cid:90) d ν x (cid:90) d ν k (2 π ) ν e i k · x e ˜ E ( k ) (cid:90) d ν k (2 π ) ν e i k · x ˜ h ( k ) (6.55)= (cid:90) d ν k (2 π ) ν (cid:90) d ν k (2 π ) ν (2 π ) ν δ ( ν ) ( k + k ) ( − i k ) ˜ φ ( k ) ˜ h ( k ) (6.56)= (cid:90) d ν k (2 π ) ν i k ˜ φ ( − k ) ˜ h ( k ) , (6.57)and since ˜ φ ( k ) is even in k , the current (6.54) becomes J ( p ) = (cid:90) d ν k (2 π ) ν d ν p (2 π (cid:126) ) ν i k ˜ φ ( k ) ˜ h ( k , p , p , ∞ ) (6.58)= − (cid:90) d ν k (2 π ) ν k ˜ φ ( k ) Im I ( k , p , p , ∞ ) , (6.59)where I ( k , p ) ≡ (cid:90) d ν p (2 π (cid:126) ) ν ˜ h ( k , p , p , ∞ ) . (6.60)58n (6.59) we have used the fact that J must be real, although we will continue to employ theform (6.58) until the end of the calculation. Note that we only need to find a momentumintegral of the correlation function, I ( k , p ), and not the correlation function itself. This isquite fortunate, since the integral (6.60) turns out to simplify considerably relative to thefull perturbation ˜ h ( k , p , p , ∞ ). Using expression (6.24) relating the perturbation to thesource term, the spatial Fourier transform of the correlation function becomes˜ h ( k , p , p , ∞ ) = (cid:90) d ν x e − i k · x h ( x , p , p , ∞ ) (6.61)= lim p → + πi ) (cid:90) C dp (cid:90) C dp p − p − p )( p + V )( p + V ) S ( k , p , p ) , (6.62)where the source term is S ( k , p , p ) = ˜ φ ( k ) i k · (cid:20) ∂f∂ p f ( p ) (cid:124) (cid:123)(cid:122) (cid:125) termB − ∂f∂ p f ( p ) (cid:124) (cid:123)(cid:122) (cid:125) termA (cid:21) . (6.63)For future reference, I have labeled the two terms of S by the names termA and termB, and(6.60) becomes I ( k , p )= 1(2 πi ) lim p → + (cid:90) d ν p (2 π (cid:126) ) ν (cid:90) C dp (cid:90) C dp p − p − p )( p + V )( p + V ) S ( k , p , p ) . (6.64)We must now calculate ( p + V ) − and ( p + V ) − on S . Expression (6.44) is the solutionto the inverse problem in the simpler context of a single space-variable, and with this resultin hand, let us return to the full equation involving V and V . Since x and x appear withopposite signs in (6.50), the value of k in V must be of the opposite sign as the correspondingvalue in V , and by referring back to (6.44) we can express( p + V ) − S ( p , p ) = 1 p + i k · v (cid:34) S ( p , p ) + ˜ φ ( k )¯ (cid:15) ( k , p ) i k · ∂f ( p ) ∂ p (cid:90) d p (cid:48) (2 π (cid:126) ) ν S ( p (cid:48) , p ) p + i k · v (cid:48) (cid:35) (6.65)( p + V ) − S ( p , p ) = 1 p − i k · v (cid:34) S ( p , p ) − ˜ φ ( k )¯ (cid:15) ( − k , p ) i k · ∂f ( p ) ∂ p (cid:90) d p (cid:48) (2 π (cid:126) ) ν S ( p , p (cid:48) ) p − i k · v (cid:48) (cid:35) . (6.66)59eturning to (6.64), we find (cid:90) d ν p (2 π (cid:126) ) ν ( p + V ) − S ( p , p ) = (cid:90) d ν p (2 π (cid:126) ) ν S ( p , p ) p − i k · v − (6.67)1¯ (cid:15) ( − k , p ) (cid:90) d ν p (2 π (cid:126) ) ν ˜ φ ( k ) p − i k · v i k · ∂f ( p ) ∂ p (cid:124) (cid:123)(cid:122) (cid:125) ¯ (cid:15) ( − k ,p ) − (cid:90) d ν p (cid:48) (2 π (cid:126) ) ν S ( p , p (cid:48) ) p − i k · v (cid:48) = 1¯ (cid:15) ( − k , p ) (cid:90) d ν p (cid:48) (2 π (cid:126) ) ν S ( p , p (cid:48) ) p − i k · v (cid:48) . (6.68)Thus, upon changing variables in the last integral from p (cid:48) to p , we can write (cid:90) d ν p (2 π (cid:126) ) ν ( p + V ) − S ( p , p ) = 1¯ (cid:15) ( − k , p ) (cid:90) d ν p (2 π (cid:126) ) ν S ( p , p ) p − i k · v , (6.69)and (6.64) becomes I ( k , p )= − πi ) lim p → + (cid:90) d ν p (2 π (cid:126) ) ν (cid:90) C dp (cid:90) C dp p + p − p (cid:15) ( − k , p ) 1 p − i k · v × ( p + V ) − S ( k , p , p ) . (6.70)We now perform the integral over p . There are poles at p = i k · v and p = p − p , andthe zeros of ¯ (cid:15) ( k , p ). Recall that the contour C runs parallel to the imaginary axis with allsingularities of ( p + V ) − lying to the left of C . As illustrated in Fig. 5, we can completethe contour C to include a large semicircle at infinity (as the integrand vanishes there). Thecontour now encloses the pole p = p − p and is clock-wise oriented, and so the p integralmay be performed using the residue theorem, I ( k , p ) = 12 πi lim p → + (cid:90) d ν p (2 π (cid:126) ) ν (cid:90) C dp (cid:15) ( − k , p − p ) 1 p − p − i k · v ( p + V ) − S ( k , p , p ) . (6.71)Using (6.65) to express ( p + V ) − S , we can now write (6.71) in the form I ( k , p )= 12 πi lim p → + (cid:90) d ν p (2 π (cid:126) ) ν (cid:90) C dp (cid:15) ( − k , p − p ) 1 p − p − i k · v p + i k · v (6.72) (cid:34) S ( p , p ) (cid:124) (cid:123)(cid:122) (cid:125) term1 + ˜ φ ( k )¯ (cid:15) ( k , p ) i k · ∂f ( p ) ∂ p (cid:90) d p (cid:48) (2 π (cid:126) ) ν S ( p (cid:48) , p ) p + i k · v (cid:48) (cid:124) (cid:123)(cid:122) (cid:125) term2 (cid:35) , IG. 5: Contour C in the p -plane. The contour can be closed in the right half-plane, oriented inthe clockwise direction and enclosing the simple pole p = p − p . where I have labeled the two terms of (6.72) by term1 and term2. The source S given by(6.63) contains two terms, termA and termB, and we must therefore examine a total of fourterms: I ( k , p ) = I + I + I + I , (6.73)where I = − πi lim p → + (cid:90) C dp (cid:90) d ν p (2 π (cid:126) ) ν (cid:15) ( − k , p − p ) 1 p − p − i k · v p + i k · v (cid:20) ˜ φ ( k ) i k · ∂f ( p ) /∂ p f ( p ) (cid:21) , (6.74) I = 12 πi lim p → + (cid:90) C dp (cid:90) d ν p (2 π (cid:126) ) ν (cid:15) ( − k , p − p ) 1 p − p − i k · v p + i k · v (cid:20) ˜ φ ( k ) i k · ∂f ( p ) /∂ p f ( p ) (cid:21) , (6.75) I = − πi lim p → + (cid:90) C dp (cid:90) d ν p (2 π (cid:126) ) ν (cid:15) ( − k , p − p ) 1 p − p − i k · v p + i k · v (cid:20) ˜ φ ( k )( i k ) · ∂f /∂ p ¯ (cid:15) ( k , p ) (cid:90) d ν p (cid:48) (2 π (cid:126) ) ν ˜ φ ( k )( i k ) · ∂f /∂ p f ( p (cid:48) ) p + i k · v (cid:48) (cid:21) , (6.76)and I = 12 πi lim p → + (cid:90) C dp (cid:90) d ν p (2 π (cid:126) ) ν (cid:15) ( − k , p − p ) 1 p − p − i k · v p + i k · v (cid:20) ˜ φ ( k )( i k ) · ∂f /∂ p ¯ (cid:15) ( k , p ) (cid:90) d ν p (cid:48) (2 π (cid:126) ) ν ˜ φ ( k )( i k ) · ∂f /∂ p (cid:48) f ( p ) p + i k · v (cid:48) (cid:21) . (6.77)61e combine the A -terms together and the B -terms together. Using (6.43) in (6.74), we canperform the p -integral to give I = − πi lim p → + (cid:90) C dp f ( p ) p + i k · v (cid:15) ( − k , p − p ) × (cid:20) ¯ (cid:15) ( − k , p − p ) − (cid:21) = 12 πi lim p → + (cid:90) C dp p + i k · v (cid:20) (cid:15) ( − k , p − p ) − (cid:21) f ( p ) , (6.78)Similarly, we use (6.43) in (6.76) to perform the p -integral, and after rearranging terms andchanging the remaining integration variable from p (cid:48) to p , we find I = 12 πi lim p → + (cid:90) C dp p + i k · v (cid:20) (cid:15) ( − k , p − p ) − (cid:21) (cid:15) ( k , p ) (cid:20) ˜ φ ( k ) i k · ∂f∂ p (cid:90) d ν p (2 π (cid:126) ) ν f ( p ) p + i k · v (cid:21) . (6.79)Upon reorganizing the terms in (6.75) we can write I = 12 πi lim p → + (cid:90) C dp p + i k · v (cid:15) ( − k , p − p ) (cid:20) ˜ φ ( k ) i k · ∂f∂ p (cid:90) d ν p (2 π (cid:126) ) ν f ( p ) p − p − i k · v (cid:21) . (6.80)Finally, upon using (6.42) to perform the p (cid:48) -integral, expression (6.77) becomes I = 12 πi lim p → + (cid:90) C dp p + i k · v (cid:15) ( − k , p − p ) (cid:20) (cid:15) ( k , p ) − (cid:21)(cid:20) ˜ φ ( k ) i k · ∂f∂ p (cid:90) d ν p (2 π (cid:126) ) ν f ( p ) p − p − i k · v (cid:21) . (6.81)Note that the k · v term in the p integrals of (6.79) and (6.81) have opposite signs, a factthat will be critical as the calculation proceeds. Note that (6.78) and (6.79) give I + I = 12 πi lim p → + (cid:90) C dp p + i k · v (cid:20) (cid:15) ( − k , p − p ) − (cid:21)(cid:20) f ( p ) +˜ φ ( k ) i k · ∂f∂ p (cid:15) ( k , p ) (cid:90) d ν p (2 π (cid:126) ) ν f ( p ) p + i k · v (cid:21) , (6.82)while the final two terms become I + I B = 12 πi lim p → + (cid:90) C dp p + i k · v (cid:15) ( − k , p − p ) ¯ (cid:15) ( k , p ) (cid:20) ˜ φ ( k ) i k · ∂f∂ p (cid:90) d ν p (2 π (cid:126) ) ν f ( p ) p − p − i k · v (cid:21) . (6.83)62 IG. 6: The Laplace contour C and associated poles p = − i k · v , p = − i k · v , and p = p − i k · v ,along with the singularities arising from the zeros of ¯ (cid:15) ( k , p ) and ¯ (cid:15) ( − k , p − p ). Not every pole orsingularity in the Figure is associated with every term in (6.84). Since Re ( p − p ) >
0, for real p > < η < p . This allows us take the limit η → + before taking p → + (theorder of limits cannot be reversed). Upon adding (6.82) and (6.83) we can write I ( k , p ) in the form I = 12 πi lim p → + (cid:90) C dp p + i k · v (cid:40) ˜ φ ( k ) i k · ∂f /∂ p ¯ (cid:15) ( − k , p − p ) ¯ (cid:15) ( k , p ) (cid:90) d ν p (2 π (cid:126) ) ν f ( p ) p − p − i k · v (cid:124) (cid:123)(cid:122) (cid:125) (e) + (cid:20) (cid:15) ( − k , p − p ) (cid:124) (cid:123)(cid:122) (cid:125) (a) − (cid:124)(cid:123)(cid:122)(cid:125) (b) (cid:21)(cid:20) f ( p ) (cid:124) (cid:123)(cid:122) (cid:125) (c) + ˜ φ ( k ) ( i k ) · ∂f /∂ p ¯ (cid:15) ( k , p ) (cid:90) d ν p (2 π (cid:126) ) ν f ( p ) p + i k · v (cid:124) (cid:123)(cid:122) (cid:125) (d) (cid:21) (cid:41) , (6.84)where I have labeled the terms as in Ref. [9]. From (6.59), the Lenard-Balescu current canbe expressed as J ( p ) = − (cid:90) d ν k (2 π ) ν k ˜ φ ( k ) (cid:20) Im I [a+b] × c + Im I b × d + Im I [a × d]+e (cid:21) , (6.85)where I [a+b] × c is the result of the (a)+(b) term times the (c) term in (6.84), I b × d is (b) times(d), and I [a × d]+e is (a) times (d) plus term (e).The contour C lies parallel to the imaginary axis in complex p -plane such that thesingularities from ( p + V ) − lie to the left of C . There are simple poles at p = − i k · v , p = − i k · v , p = p − i k · v . The latter pole, however, is not associated with ( p + V ) − , asit is arose from the term 1 / ( p − p − p ) in (6.64). There are also singularities arising fromthe zeros of ¯ (cid:15) ( k , p ) (the singularities and corresponding branch cut lie in the left half-planefor plasma stability) and the zeros of ¯ (cid:15) ( − k , p − p ) (whose singularities and branch cut liein the right half-plane). As shown in Fig. 6, we can offset C by a small amount η in the63eal direction, with 0 < η < p . We must therefore take the η → + limit before taking p → + . Not every term in (6.85) will involve every singularity, so care must be taken whenevaluating (6.84). Reference [9] emphasizes that the guiding principle in closing the contour C is to avoid enclosing a singularity arising from the zeros of the dielectric function ¯ (cid:15) . Inthis way, we avoid crossing a branch-cut when closing the contour at infinity. The first termwe consider is I [a+b] × c = 12 πi lim p → + (cid:90) C dp p + i k · v (cid:20) (cid:15) ( − k , p − p ) − (cid:21) f ( p ) . (6.86)There is a simple pole at p = − i k · v , so we complete the C contour by a large semi-circle in the left-hand p -plane to form a closed contour C L , as illustrated in the left panelof Fig. 7. This closed contour has a counter-clockwise orientation and encircles the pole p = − i k · v . The path C L avoids the singularity in the right half-plan arising from thezeros of the dielectric function ¯ (cid:15) ( − k , p − p ), and there are no such singularities in the left-halfplane from ¯ (cid:15) ( k , p ), so upon applying the residue theorem we find I [a+b] × c = 12 πi (cid:90) C L dp p + i k · v (cid:20) (cid:15) ( − k , − p ) − (cid:21) f ( p ) (6.87) (cid:20) (cid:15) ( − k , i k · v ) − (cid:21) f ( p ) = (cid:20) (cid:15) ( − k , − k · v ) − (cid:21) f ( p ) . (6.88)In the last equality we have used (6.45) to express the result in terms of the ordinary dielectricfunction (cid:15) ( k , ω ) in temporal Fourier space. We only need the imaginary component,Im I [a+b]c = Im (cid:15) ( k , k · v ) | (cid:15) ( k , k · v ) | f ( p ) , (6.89)where we have used the fact that (cid:15) ( − k , − ω ) = (cid:15) ∗ ( k , ω ). Recall that the sing-componentdielectric function takes the form (cid:15) ( k , ω ) = 1 + (cid:90) d ν p (2 π (cid:126) ) ν ˜ φ ( k ) k · ∂f /∂ p ω − k · v + iη . (6.90) I ( k , p ) = 12 πi lim p → + (cid:90) d ν p (2 π (cid:126) ) ν (cid:90) C dp (cid:15) ( − k , p − p ) 1 p − p − i k · v ( p + V ) − S ( k , p , p ) . (6.91)To find the imaginary part, we use the functional relation1 ω − k · v + iη = P 1 ω − k · v − iπ δ ( ω − k · v ) , (6.92)where the first term in (6.92) gives the principle-part integral. We can therefore expressIm (cid:15) ( k , ω ) = − π (cid:90) d ν p (2 π (cid:126) ) ν δ ( ω − k · v ) ˜ φ ( k ) k · ∂f∂ p , (6.93)64 IG. 7: Closed contours C L and C R for the integrals I [a+b] × c of Eq. (6.86) and I b × d of Eq. (6.97),respectively. and hence Im I [a+b] × c = − π (cid:90) d ν p (2 π (cid:126) ) ν ˜ φ ( k ) δ ( k · v − k · v ) | (cid:15) ( k , k · v ) | k · ∂f ( p ) ∂ p f ( p ) . (6.94)This gives the corresponding contribution to the Lenard-Balescu current J [a+b] × c = − (cid:90) d ν k (2 π ) ν k ˜ φ ( k ) Im I [a+b]c ( k , p , p ) (6.95)= π (cid:90) d ν k (2 π ) ν d ν p (2 π (cid:126) ) ν [ ˜ φ ( k )] k δ ( k · v − k · v ) | (cid:15) ( k , k · v ) | k · ∂f ( p ) ∂ p f ( p ) . (6.96)We now evaluate b × d contribution. Interestingly, this term vanishes upon closing thecontour C to the right to form a closed contour C R lying in the right half-plane, I b × d ∝ (cid:90) C R dp p + i k · v p + i k · v (cid:15) ( k , p ) = 0 . (6.97)This is because the contour C R does not enclose the simple poles p = − i k · v and p = − i k · v on the imaginary axis, nor the zeros of ¯ (cid:15) ( k , p ) in the left half-plane. Theresidue theorem therefore gives a vanishing integral. The final term involves [a × d] + e. Wecan in fact take the limit p → + inside ¯ (cid:15) ( − k , p − p ) to give ¯ (cid:15) ( − k , − p ) ¯ (cid:15) ( k , p ) = | ¯ (cid:15) ( k , p ) | ,and therefore I [a × d]+e = 12 πi lim p → + (cid:90) d ν p (2 π (cid:126) ) ν f ( p ) ˜ φ ( k ) i k · ∂f∂ p (cid:90) C dp p + i k · v × | ¯ (cid:15) ( k , p ) | (cid:20) p + i k · v + 1 p − p − i k · v − p (cid:21) (6.98)= lim p → + (cid:90) d ν p (2 π (cid:126) ) ν ˜ φ ( k ) k · ∂f∂ p f ( p ) I C ( k , p , p ) , (6.99)65here we define the contour integral I C = 12 π (cid:90) C dp | ¯ (cid:15) ( k , p ) | p + i k · v (cid:20) p + i k · v − p + i k · v − p (cid:21) . (6.100)All factors of i from and all signs in (6.98) have been placed in the contour integral I C of(6.100). We can parameterize points p ∈ C by p = − iω + η for arbitrary real ω and fixedreal η , with 0 < η < p . We must therefore take the η → + limit before the p → + limit.The contour integral over C in (6.98) can now be expressed an integral over real ω , I C = − i π (cid:90) ∞−∞ dω | (cid:15) ( k , ω ) | ω − k · v + iη (cid:20) ω − k · v + iη − ω − k · v − ip (cid:21) , (6.101)where we taken the η → + limit inside the dielectric function, and then used (6.45) to set¯ (cid:15) ( k , − iω ) = (cid:15) ( k , ω ). Furthermore, since the p -limit is taken at the end of the calculation, inperforming the η -limit we have p (cid:54) = 0. Therefore, in the last term of (6.101), we have taken η → + , leaving the small imaginary piece − ip in the denominator. Since η and p are bothinfinitesimal, we have 1 ω − k · v + iη − ω − k · v − ip = − πi δ ( ω − k · v ) , (6.102)where the principles parts cancel. Similarly, the second term in (6.101) becomes1 ω − k · v + iη = P 1 ω − k · v − πi δ ( ω − k · v ) . (6.103)The principle part is real and does not contribute to the imaginary piece of I [a × d]+e (it alsointegrates to zero when performing the k -integration), and therefore I C = πi (cid:90) ∞−∞ dω | (cid:15) ( k , ω ) | δ ( ω − k · v ) δ ( ω − k · v ) = πi δ ( k · v − k · v ) | (cid:15) ( k , k · v ) | . (6.104)We therefore arrive atIm I [a × d]+e = π (cid:90) d ν p (2 π (cid:126) ) ν ˜ φ ( k ) k · ∂f∂ p f ( p ) δ ( k · v − k · v ) | (cid:15) ( k , k · v ) | , (6.105)which gives a contribution to the current J [a × d]+e = − (cid:90) d ν k (2 π ) ν k ˜ φ ( k ) Im I [a × d]+e ( k , p , p ) (6.106)= − π (cid:90) d ν k (2 π ) ν d ν p (2 π (cid:126) ) ν [ ˜ φ ( k )] k δ ( k · v − k · v ) | (cid:15) ( k , k · v ) | k · ∂f ( p ) ∂ p f ( p ) . (6.107)Upon adding (6.96) and (6.107) we find the total Lenard-Balescu current, J = π (cid:90) d ν k (2 π ) ν d ν p (2 π (cid:126) ) ν [ ˜ φ ( k )] (cid:124) (cid:123)(cid:122) (cid:125) ( e /k ) k δ ( k · v − k · v ) | (cid:15) ( k , k · v ) | (cid:20) k · ∂∂ p − k · ∂∂ p (cid:21) f ( p ) f ( p ) , (6.108)and the proof is complete for a single-component plasma.66 . Generalization to a Multi-species Plasma It is easy to generalize the previous result to a multi-species plasma. The variables X i andthe distribution functions must contain species indices, e.g. f ( a )1 ( X a , t ) and f ( ab )2 ( X a , X b , t ).The latter gives the joint probability of finding species a at X a and species b at X b . Thefirst order correction becomes f ( ab )2 ( X a , X b , t ) = f a ( X a , t ) f b ( X b , t ) + h ab ( X a , X b , t ) . (6.109)The Lenard-Balescu kernel becomes (cid:88) b L ab [ f ] = − (cid:88) b ∂∂ p a · J ab [ f ]( p a ) , (6.110)where J ab [ f ]( p a ) = π (cid:90) d ν k (2 π ) ν d ν p b (2 π (cid:126) ) ν (cid:16) e a e b k (cid:17) k δ [ k · ( v b − v a )] | (cid:15) ( k , k · v a ) | (cid:20) k · ∂∂ p b − k · ∂∂ p a (cid:21) f a ( p a ) f b ( p b ) , (6.111)and (cid:15) ( k , ω ) is the multi-component dielectric function.67 II. CONCLUSIONS
Calculating the rate of Coulomb energy exchange in a plasma is notoriously difficult, evenfor the case of a fully ionized weakly coupled plasma. Two examples of experimental relevanceare the charged particle stopping power and the temperature equilibration rate betweenelectrons and ions in a non-equilibrium plasma. Naive calculations of these processes sufferfrom logarithmic divergences at both long- and short-distance scales, and we must thereforeresort to more sophisticated methods of calculation. Corresponding to these divergences aretwo broad classes of kinetic equations, applicable in complementary regimes, represented bythe Boltzmann equation (BE) and the Lenard-Balescu equation (LBE). The BE describes theshort-distance effects of 2-body scattering, including large angle scattering, while the LBEmodels 2-point long-distance correlations. It is well known that the BE suffers a long-distancelogarithmic divergence for Coulomb scattering (in three spatial dimensions), confirming thatit is indeed missing long-distance physics (correlations are being ignored). Conversely, theLBE suffers from a short-distance logarithmic divergence for Coulomb interactions (in threedimensions), another indication that relevant physics is being overlooked (the short-distancescattering physics).The fact that the BE and the LBE are relevant in complementary regimes allows usto capitalize on the lessons physicists have learned from quantum field theory, a formalismdeveloped by particle physicists for understanding the fundamental interactions of nature. Inquantum field theory, an array of divergences are encountered, from logarithmic to quadraticand higher, and the so called renormalization program was developed to form meaningfuland finite predictions from these divergent results. The first ingredient is to temporarily regularize the theory by rendering the integrals finite. At the end of the calculation, theregularization will be removed, but in the interim, the finite expressions can be algebraicallymanipulated in a meaningful fashion. After regularization has been performed, one then renormalizes the theory by reinterpreting physical properties like the electric charge and massin such a way as to give finite predictions as the regularization scheme is removed. Thereare many regularization schemes in use, each with their own strengths and weaknesses. Thesimplest one is to choose arbitrary large- and small-distance cutoffs in the integrals, suchas the Debye wavelength and the classical distance of closest approach for the electrons orions in the plasma. This is the regularization scheme first adopted by Landau and Spitzer,and it produces a scaling factor called the
Coulomb logarithm , which is defined to be thenatural logarithm of the ratio of the large- to small-distance scales. Much effort has beendevoted to determining the precise value of the Coulomb logarithm. However, such a cruderegularization method is inherently uncertain in determining the exact value of the Coulomblogarithm, and this exercise is doomed to failure from the start. For example, one couldjust as correctly take twice the Debye length as the long distance cutoff, thereby leaving theconstant inside the logarithm undetermined by this regularization method. It is interesting68o note that if the divergence had been higher order rather than logarithmic, this crudecutoff method would not have been acceptable to plasma physicists. It is only because thedivergence in Coulomb exchange processes is logarithmic that one can get away with such anaive regularization scheme for so long.The most pertinent feature of relativistic quantum field theory is that it is a many-bodytheory. The non-relativistic limit of these theories provides a rigorous treatment of plasmaphysics from which the framework of a non-relativistic many-body field theory [14, 15]. Oneof the subtleties of the renormalization program is that the regularization scheme often breaksthe symmetries of the system, thereby changing the structure of the theory. For example,the cut-off method breaks Lorentz invariance, which is essential for electrodynamics. Whilethe symmetries must return when the regularization is removed, the system becomes morecomplex and when its symmetries are broken, and the restoration of the symmetries as theregularization is removed can often be nontrivial. Indeed, some symmetries remain broken.For example, gauge symmetry and particle number conservation cannot both be preserved inthe standard model of particle physics. As gauge invariance is essential for defining the the-ory, it turns out that matter is not stable and decays by so-called nonperturbative sphaleron processes (albeit the proton is quite long lived, with a decay rate many times the age ofthe universe). Since a plasma is a many-body system, it is not surprising that we encounterdivergences similar to those in quantum field theory. Furthermore, since the renormalizationprogram makes experimental predictions, we must take it seriously, and it is not surprisingthat techniques developed in field theory are applicable to plasma theory. Regularizationmethods are often chosen in such a way as to preserve as many symmetries as possible. Themethod of dimensional regularization is one such method that stands apart from most othersin that it preserves the essential symmetries, such as Lorentz invariance and gauge invari-ance. The dimensional continuation formalism of Brown-Preston-Singleton (BPS) relies ona technique adopted from quantum field theory called dimensional regularization. Processesthat are divergent in ν = 3 spatial dimensions can often be regularized by looking at thesystem in a general number of dimensions ν . The three-dimensional divergences show up assimple poles of the form 1 / ( ν − ν space. The generalrule is that long distance fluctuations are greater in lower dimensions, while short distancephysics is more important in higher dimensions. In fact, in ν = 1, it has been shown thatthe quantum fluctuations are so large that spontaneous symmetry breaking cannot occur,even if it is permitted classically [16]. Another interesting result is that in ν = 1 dimensions ,the photon acquires a mass via quantum loop corrections [17]. Other phenomena are uniqueto ν = 2 dimensions, such as high temperature superconductivity. BPS does not requirethe introduction of a Coulomb logarithm, as the regularization is performed by changing thedimension ν of space. The BPS method uses dimensional continuation to find the Coulombenergy exchange at the integer values ν = 1 , , , , · · · (except for ν = 3). By applying69arlson’s theorem [18], we can define an analytically continued quantity for complex ν , in asimilar way that the factorial function on the positive integers can be analytically continuedto the gamma function over the complex plane. For Coulomb energy exchange processes,the continuation to complex ν allows us to take the ν → finite resultvalid in ν = 3 dimensions. In this way, the BPS formalism regularizes the traditional ν = 3divergences, and allows us to define the theory in three dimensions. In this third installmentof the BPS Explained lecture series, we have proven a pivotal result of process of dimensionalcontinuation upon which the BPS formalism resides. Namely, that to leading order in theplasma coupling g , the BBGKY hierarchy of kinetic equations reduces to (i) the Boltzmannequation for spatial dimensions ν >
3, and (ii) the Lenard-Balescu equation for ν <
3. Thebulk of these notes were devoted to proving the latter.
Acknowledgments
I would like to thank Jean-Etienne Sauvestre, Lowell Brown, and Don Shirk for readingthe manuscript and providing critical feed-back. I would also like to thank Lowell Brown fora number of very useful technical discussions.70 ppendix A: The Cross Section and Hyperspherical Coordinates
To make these notes self contained, and to establish some notation, I shall give a quickreview of the material required from Lectures I and II.
1. Hyperspherical Coordinates
Kinematic quantities such as ν -dimensional momentum or position vectors are elementsof Euclidean space R ν . We can decompose any element x ∈ R ν in terms of a rectilinearorthonormal basis ˆ e (cid:96) , so that x = (cid:80) ν(cid:96) =1 x (cid:96) ˆ e (cid:96) , or in component notation x = ( x , · · · , x ν ).Each component is given by x (cid:96) = ˆ e (cid:96) · x , and a change d x in the vector x corresponds to achange dx (cid:96) = ˆ e (cid:96) · d x in the rectilinear coordinate x (cid:96) . Letting x vary successively along eachindependent direction ˆ e (cid:96) , we can trace out a small ν -dimensional hypercube with sides oflength dx (cid:96) ; therefore, the rectilinear volume element is given by the simple form d ν x = ν (cid:89) (cid:96) =1 dx (cid:96) = dx dx · · · dx ν . (A1)Similar considerations hold for momentum volume element d ν p . In performing integralsover the kinematic variables, however, symmetry usually dictates the use of hypersphericalcoordinates rather than rectilinear coordinates. I will therefore review the hypersphericalcoordinate system in this section, deriving the measure for a ν -dimensional volume element d ν x in terms of hyperspherical coordinates. It should be emphasized again that this formalismalso holds in momentum space for the momentum volume element. For our purposes, theprimary utility of hyperspherical coordinates is that the volume element d ν x can be writtenas a product of certain conveniently chosen dimensionless angles, which I will collectivelyrefer to as d Ω ν − , and an overall dimensionfull radial factor r ν − dr , so that d ν x = d Ω ν − r ν − dr . (A2)To prove this, let us recall why the three dimensional volume element takes the form d x =sin θ dθ dφ r dr (with 0 ≤ θ ≤ π , 0 ≤ φ < π , and 0 ≤ r < ∞ ). As depicted in Fig. 8,the three dimensional vector x has length r , and subtends a polar angle θ relative to the z -axis, while its projection onto the x - y plane subtends an azimuthal angle φ relative tothe x -axis. The two angles θ and φ specify completely the direction of the unit vector ˆ x ,while an additional coordinate r determines the total vector x = r ˆ x . As we increase thepolar angle θ by a small amount dθ , the vector x sweeps out an arc of length dR = rdθ .Similarly, a change dφ in the azimuthal angle will cause x to sweep out an arc in the x - y plane of length dR = r sin θ dφ , where the factor of sin θ in dR arises from the projectionof x onto the x - y plane. Moving along the radial direction gives the final independent71 IG. 8:
Spherical coordinates r, θ, φ of a point x in three dimensional space: radial distance r , polar angle θ , and azimuthal angle φ . The angles range over the values 0 ≤ θ ≤ π and 0 ≤ φ < π . displacement dR = dr . For small displacements in dθ , dφ , and dr , the vector x sweepsout a small cubic volume element with sides of length dR , dR , and dR , and therefore d x = dR dR dR = rdθ · r sin θdφ · dr .Let us now consider the volume element d x in four dimensional space, and denote thecoordinate axes by x, y, z, w . Since we cannot visualize four dimensional space, let us examinethis problem in two steps, each of which can be visualized in three dimensions. As shownin Fig. 9a, let θ be the angle between the w -axis and the four dimensional vector x . The w -axis and the vector x lie in a plane, and θ can therefore be visualized. Let us nowproject x onto the w = 0 hyperplane (a three dimensional slice of four-space), denoting theprojected three-vector by x w . Since this vector lies in three-space, it too can be visualized.Since the three-plane w = 0 lies perpendicular to each of the axes x , y , and z , the vector x w lies in the three dimensional space shown in Fig. 9b, and its length is | x w | = r sin θ .Let the angle θ be the polar angle between the z -axis and the vector x w , while θ is theusual azimuthal angle φ . The last angle θ runs between 0 and 2 π , while all previous anglesrun between 0 and π . As we vary the three angles and the radial coordinate, we sweepout a four-dimensional cube (or an approximate cube) with sides of length dR = r dθ , dR = r sin θ dθ , dR = r sin θ sin θ dθ , and dR = dr . This gives a four dimensionalvolume element d x ≡ dR dR dR dR = sin θ dθ sin θ dθ dθ r dr , (A3)where 0 ≤ θ (cid:96) ≤ π for (cid:96) = 1 , ≤ θ < π . It is amusing to calculate the four-volume ofa four-dimensional ball of radius r by integrating the volume element over the appropriateangles, B = (cid:90) π dθ sin θ (cid:90) π dθ sin θ (cid:90) π dθ (cid:90) r dr (cid:48) r (cid:48) = 12 π r . (A4)The derivative of B with respect to r gives the hypersurface area of the enclosingthree-sphere, S = dB dr = 2 π r . (A5)72 IG. 9:
Hyperspherical coordinates r, θ , θ , θ of a point x in four dimensional space. As before, r = | x | isthe radial distance. The angles are defined as follows. (a) First, let θ be the angle between x and the w -axis.Let us now project x onto the orthogonal three dimensional space, so that x = ( w, x, y, z ) → x w = (0 , x, y, z ).The length of the projection x w is r w = r sin θ . (b) The vector x w can be viewed as a three dimensionalvector x w = ( x, y, z ), which then defines the usual polar and azimuthal angles of Fig. 8, denoted here by θ and θ respectively. These are well known results, analogous to a three dimensional ball of radius r and volume B = 4 π r /
3, which is of course bounded by the two-sphere of area S = 4 πr .We can readily generalize this procedure to an arbitrary number of dimensions. Considera point x ∈ R ν given by the rectilinear coordinates x = ( x , x , · · · , x ν ). Let θ be the anglebetween the vector x and the x -axis, in a manner similar to that of Figs. 8 and 9a. Note that dR = rdθ is the arc length swept out by x as the angle θ is incremented by dθ . Let us nowproject x onto the hyperplane x = 0, the ( ν − x -axis and passingthrough the origin. Denote this projection by x , that is to say, let x → x = (0 , x , · · · , x ν ),and note that the length of x is r = r sin θ . We proceed to the next step and define θ as the angle between the x -axis and the projection x . Note that as the angle θ isincremented by dθ , the vector x sweeps out an arc of length dR = r dθ = r sin θ dθ .In a similar fashion, project x onto the x -plane, that is, the plane described by x = 0and x = 0. This projection is given by x → x = (0 , , x , · · · , x ν ), and the length of x is r = r sin θ = r sin θ sin θ . For the general (cid:96) th iteration, let θ (cid:96) be the angle between the x (cid:96) -axis and x (cid:96) − , so that dR (cid:96) = r (cid:96) − dθ (cid:96) = r sin θ sin θ · · · sin θ (cid:96) − dθ (cid:96) , where we have usedthe fact that r (cid:96) − = r sin θ sin θ · · · sin θ (cid:96) − . This gives the ν -dimensional volume element d ν x = ν (cid:89) (cid:96) =1 dR (cid:96) = sin ν − θ dθ · sin ν − θ dθ · · · sin θ ν − dθ ν − · dθ ν − · r ν − dr . (A6)The angles θ , · · · , θ ν − run from 0 to π , while θ ν − runs from 0 to 2 π . For notationalconvenience, I will write the angular measure in (A6) as d Ω ν − = sin ν − θ dθ sin ν − θ dθ · · · sin θ ν − dθ ν − dθ ν − , (A7)73o that d ν x = d Ω ν − r ν − dr , which establishes (A2). Also note that d Ω ν − = d Ω ν − sin ν − θ dθ . (A8)It is easy to show that the integration over all angles gives the total solid angleΩ ν − ≡ (cid:90) d Ω ν − = 2 π ν/ Γ( ν/ . (A9)To prove this, first consider the one-dimensional Gaussian integral (cid:90) ∞−∞ dx e − x = √ π . (A10)If we multiply both sides together ν times, we find( √ π ) ν = (cid:90) ∞−∞ dx e − x (cid:90) ∞−∞ dx e − x · · · (cid:90) ∞−∞ dx ν e − x ν = (cid:90) d ν x e − x , (A11)where the vector x in the exponential of the last expression is the ν -dimensional vector x = ( x , x , · · · , x ν ), and x = x · x = (cid:80) ν(cid:96) =1 x (cid:96) . As in (A2), we can factor the angularintegrals out of the right-hand-side of (A11), and the remaining one-dimensional integralcan be converted to a Gamma function with the change of variables t = r , π ν/ = (cid:90) d Ω ν − · (cid:90) ∞ dr r ν − e − r = (cid:90) d Ω ν − ·
12 Γ( ν/ , (A12)and solving for (cid:82) d Ω ν − gives (A9).A few general remarks on calculating physical quantities in the BPS program are in order.When we calculate the temperature equilibration rate between plasma species or the chargedparticle stopping power, we encounter integrals of the form I ( ν ) ≡ (cid:90) d ν x F ( r ) = Ω ν − (cid:90) ∞ dr r ν − F ( r ) (A13) I ( ν ) ≡ (cid:90) d ν x F ( r, θ ) = Ω ν − (cid:90) ∞ dr r ν − (cid:90) π dθ sin ν − θ F ( r, θ ) , (A14)respectively. The exact expressions for F and F are not important here, except thattheir angular dependence is determined by the following considerations. The integral (A13)is spherically symmetric because the energy exchange between plasma species is isotropic,while in integral (A14), the motion of the charged particle defines a preferred directionaround which one must integrate, thereby leaving a single angular dependence. The integrals I ( ν ) and I ( ν ) can be viewed as functions defined on the integers ν ∈ N , and as discussedat length in Lecture I [3], Carlson’s Theorem [18] ensures that there are unique analyticcontinuations of I ( ν ) and I ( ν ) for ν ∈ C . This is similar to extending the factorial function n ! on the integers to the gamma function Γ( ν ) on the complex ν -plane. Let us examinemore closely how this analytic continuation to complex ν works in practice. First, the solid74 r(a) B (r) B ν S ν−1 (r) S (b)(r)(r)
FIG. 10:
A ( ν − S ν − of radius r bounds the ν -dimensional ball B ν ( r ) of radius r . Byintegrating over successive shells of area, we can find the volume by B ν ( r ) = (cid:82) r dr (cid:48) S ν − ( r (cid:48) ); or conversely S ν ( r ) = B (cid:48) ν ( r ). angles Ω ν − and Ω ν − are well defined for complex arguments ν , as they are composed ofsimple exponential factors like π ν/ and Gamma functions, whose analytic properties are wellknown. As for the integrals, simply treat ν as an arbitrary integer dimension, and performthe integral for general ν . The integral will of course depend upon the value of ν , and oncethe integral has been performed exactly (not approximately and not numerically), we arefree to set the value of ν to a complex number (presumably in a small neighborhood about ν = 3). This provides functions I ( ν ) and I ( ν ) with complex argument ν ∈ C .
2. The Hypervolume of Spheres, Disks, and Cylinders
We shall now calculate the hypervolume of several useful geometric objects. Let us firstconsider a ν -dimensional ball of radius r , defined by the set of points x ∈ R ν for which | x | ≤ r . We will denote this object by B ν ( r ), and in two and three dimensions this is a diskand a ball, both volume centered at the origin. We can find the ν -dimensional hypervolumeof the ball B ν ( r ) by simply integrating (A6) over all permissible values of the coordinates.It should cause no confusion to denote the hypervolume of the region B ν ( r ) by the samesymbol, and using (A9), together with x Γ( x ) = Γ( x + 1), we find B ν ( r ) = (cid:90) d Ω ν − (cid:90) r dr (cid:48) r (cid:48) ν − = π ν/ Γ( ν/ r ν . (A15)The boundary of B ν ( r ) is a ( ν − S ν − ( r ) defined by | x | = r , or (cid:80) ν(cid:96) =1 x (cid:96) = r . By differentiating (A15) with respect to the radius r , we can also find thehyperarea of a ( ν − S ν − ( r ) of radius r in R ν , S ν − ( r ) = dB ν ( r ) dr = 2 π ν/ Γ( ν/ r ν − = Ω ν − r ν − . (A16)75 −1 B (r)L r (r)S ν−2
FIG. 11:
The hyperarea of a hypercylinder C ν − ( r, L ) of length L and radius r is C ν − ( r, L ) = S ν − ( r ) · L ,and the hypervolume bounded by the cylinder is V ν ( r, L ) = B ν − ( r ) · L . For brevity, I have denoted the hyperarea by the same symbol S ν − ( r ) as the sphere itself,which is simply the ( ν − B ν ( r ). This is illustrated inFig. 10. The distinction I am making between “hypervolume” and “hyperarea” is somewhatarbitrary, since these are both terms involving regions in a higher dimensional space. WhenI wish to talk about a ν -dimensional subregion of the hyperspace R ν , such as B ν ( r ), I willuse the term hypervolume. On the other hand, when I wish to emphasize a boundary regionof a hypervolume, such as S ν − ( r ), I will use the term “hyperarea.” Regarding the usage ofthe term “solid angle,” suppose we keep the radius r fixed but vary the angles θ i over ranges dθ i . The region swept out by this procedure lies on the ( ν − S ν − ( r )with a hyperarea dS ν − = d Ω ν − r ν − . We are therefore justified in calling d Ω ν − the solidangle in ν dimensions.Finally, let us discuss the ( ν − C ν − ( r, L ) of radius r and length L . Again, it is easiest to argue from analogy in three dimensions. To form a two-cylinder C ( r, L ) in R , we let a two dimensional disk B ( r ) sweep out a volume as it moves a distance L in the orthogonal direction, as illustrated in Fig. 11. Similarly, a ( ν − ν − B ν − ( r ) sweep out a distance L along the orthogonal axis. Therefore, the hyperarea of the ( ν − C ν − ( r, L ) = S ν − ( r ) · L , (A17)and the ν -dimensional hypervolume enclosed by this cylinder is V ν ( r, L ) = B ν − ( r ) · L . (A18)Note that this is the natural geometry of a scattering experiment to measure the cross sectionin ν dimensions, which leads us to the next section.76 . The Cross Section Now that we have examined the Coulomb plasma in some detail, we should address two-body scattering and the cross section. This is necessary formalism, since the Boltzmannequation contains the differential cross section for two-body scattering. For continuity, wereview the notion of “cross section” in ν -dimensions. As illustrated in Fig. 12, we consider abeam of projectiles 1 with flux I striking a fixed target 2, although we can perform a similaranalysis in the lab frame in which the scattering centers are also moving. In ν dimensions,the spatial region normal to the beam axis is a ( ν − I is the number of particles per second per unit hyperarea passing through this plane. Theengineering units of I are therefore L − ν · T − . In other words, the number of particles ina time interval dt passing through a hyperarea dA normal to the beam is dN = I · dA · dt ;therefore, the differential rate through the normal area dA is dR = I · dA . Let us now placea particle counter along position ˆΩ some distance away from the scattering center, and let usmeasure the rate dR ( ˆΩ) at which the 1-particles enter a given solid angle centered aboutdirection ˆΩ. We can therefore define the differential cross section dσ in the usual way, dσ · I = dR . (A19)Note that the engineering of dσ are L ν − . The cross section is usually quite sensitive to thedetails of any given physical theory, thereby making it a good experimental probe. Indeed,in high energy physics, it is the primary diagnostic. FIG. 12:
Definition of the cross section in a general number of dimensions. The incident flux I of species 1is the rate of particles per unit hyperarea normal to the beam. The units of I are L − ν · T − , where L andT denote the units of space and time. By definition, the differential cross section dσ is related to the rate dR , each at angular position ˆΩ, by dR ( ˆΩ) = I dσ ( ˆΩ). The cross section per unit solid angle about thedirection ˆΩ is denoted by dσ /d Ω. The engineering units of dσ are L ν − . ν -dimensional Coulomb field. Then two-body particle motion is confined along a two-dimensional plane, and this holds true even in ν dimensions. Let b denote the impactparameter of the projectile relative to the scattering center. As the particle traverses itsplane of motion, its position is uniquely characterized by a function b = b ( θ ), where θ is theangle between the beam direction and the projectile (with the scattering center defining theorigin). The rate at which particles pass through the hyperannulus of width db and radius b is dR = Ω ν − b ν − db · I , and by particle number conservation, the same number of scatteredparticles reaches the hyperannulus around ˆΩ. This is the analog of dR = 2 πb db · I in threedimensions. It is actually better to consider a differential d Ω ν − rather than the total angularextent Ω ν − . Again, this corresponds to dR = dθ b db · I in three dimensions. The crosssection in a ν -dimensional central potential is therefore given by dσ = d Ω ν − b ν − db . (A20)This is the differential form of Eq. (8.31) of Ref. [2]. However, for include two-body quantumscattering effects, it is more convenient to replace the cross section dσ by the quantumscattering amplitude T (1 + 2 → (cid:48) + 2 (cid:48) ) ≡ T (cid:48) (cid:48) ; 12 ( W, q ) by using the relation | v − v | dσ = (cid:90) d ν p (2 π (cid:126) ) ν d ν p (2 π (cid:126) ) ν (cid:12)(cid:12) T (cid:48) (cid:48) ;12 ( W, q ) (cid:12)(cid:12) (2 π (cid:126) ) ν δ ν (cid:16) p (cid:48) + p (cid:48) − p − p (cid:17) × (2 π (cid:126) ) δ (cid:16) E (cid:48) + E (cid:48) − E − E (cid:17) . (A21)In the amplitude, W is the total center-of-mass energy and q is the square of the momentumexchange. This is just a rewriting of the expression I · dσ = dR , since | v − v | isproportional to the flux I , and the rate dR is proportional to the square of the scatteringamplitude | T (cid:48) (cid:48) ; 12 | . The integration is over all values of p and p consistent with energyand momentum conservation. In three dimensions, for example, there are six momentumintegrals and four δ -functions, leaving two differential degrees of freedom, namely, the crosssectional area. This is why I continue to use the differential cross section dσ on the left-hand-side of (A21), to imply that some of the angular coordinates have not been integratedover. 78 ppendix B: Center-of-Momentum Coordinates In calculating the convective terms in (5.9), it is useful to transform to center-of-momentum coordinates. We will generalize to two species for the scattering. We definethe total and relative moment, and the center-of-mass and the relative position by P = m v + m v = p + p (B1) p = m ( v − v ) = m p − m p M (B2) R = m x + m x M (B3) x = x − x , (B4)where M = m + m (B5) m = m m m + m . (B6)I am using the notation x for the relative position, rather than usual notation r , because inthe text, the beam-axis in two-body scattering has been called x . The inverse transformsare p = m M P − p p = m M P + p (B7) x = R − m M x x = R + m M x , (B8)and it is easy to check that the gradients transform as, ∂∂ p = ∂∂ P − m M ∂∂ p ∂∂ p = ∂∂ P + m M ∂∂ p (B9) ∂∂ x = m M ∂∂ R − ∂∂ x ∂∂ x = m M ∂∂ R + ∂∂ x , (B10)and ∂∂ p = ∂∂ p − ∂∂ p ∂∂ P = m M ∂∂ p + m M ∂∂ p (B11) ∂∂ x = m M ∂∂ x − m M ∂∂ x ∂∂ R = ∂∂ x + ∂∂ x . (B12)I have recorded these formulae here for convenience.79 ppendix C: The Multi-component Poisson-Vlasov Equation Let us restore the species index in this section, and consider a collisionless plasma, ∂f a ∂t + v · ∂f a ∂ x + e a E · ∂f a ∂ p = 0 (C1) E ( x , t ) = (cid:88) b (cid:90) dX b f b ( X b , t ) E ( b ) x , (C2)where E ( x , t ) is the self-consistent electric field associated with the f b . There is a charge e a at position x , and the quantity E ( b ) x inside the integral is the static Coulomb field at x originating from a point-charge of type b at position x b , so that E ( b ) x = E b ( x − x b ) = e b Γ( ν/ π ν/ x − x b | x − x b | ν . (C3)This means that the divergence of the self-consistent electric field is ∇ · E ( x , t ) = (cid:88) b (cid:90) d ν p (2 π (cid:126) ) ν e b f b ( x , p , t ) . (C4)To emphasize that E in (C1) is a functional of the distributions f b , we shall often write E [ f ]in place of E ( x , t ). It is actually more convenient to write (C1) and (C4) in terms of theelectric potential φ , where E = − ∂φ/∂ x , so that the kinetic equations become ∂f a ∂t + v · ∂f a ∂ x − e a ∂φ∂ x · ∂f a ∂ p = 0 (C5) ∇ φ ( x , t ) = − (cid:88) b (cid:90) d ν p (2 π (cid:126) ) ν e b f b ( X, t ) . (C6)This form of the kinetic equations is known as the Poisson-Vlasov equations, and we shalluse it interchangeably with (C1)–(C4). For visual clarity, I am using a mixed notation inwhich the Laplacian is denoted by ∂∂ x · ∂φ∂ x = ∇ φ . (C7)Let us now perform a perturbative analysis on the Vlasov equation (C1), or equivalently(C5). Rather than perturbing about an equilibrium configuration, let us take the 0-th orderstarting point as a solution to the Vlasov equation itself. In other words, suppose f a is asolution to ∂f a ( X, t ) ∂t + v · ∂f a ( X, t ) ∂ x + e a E · ∂f a ( X, t ) ∂ p = 0 , (C8)where the 0-th order self-consistence electric field at x is E ( x , t ) = (cid:88) b (cid:90) dX b f b ( x b , p b , t ) E b ( x − x b ) . (C9)80ow suppose that the perturbation f a ( x , p , t ) = f a ( x , p , t ) + h a ( x , p , t ) (C10)satisfies the Vlasov equation (C1), and let us find the corresponding equation satisfied by h a . Upon substituting (C10) for h a into the electric field (C2), the self-consistent electricfield receives a 0-th order contribution from f a and a 1-st order contribution from h a , E ( x , t ) = (cid:88) b (cid:90) dX b (cid:104) f b ( x b , p b , t ) + h b ( x b , p b , t ) (cid:105) E b ( x − x b )= E ( x , t ) + E ( x , t ) , (C11)where the first-order self-consistent field is E ( x , t ) = (cid:88) b (cid:90) dX b h b ( x b , p b , t ) E b ( x − x b ) . (C12)We also substitute the perturbation (C10) back into the kinetic equation (C1), using theelectric field (C11) and working only to first order in h a . The 0-th order terms vanish because f a is a solution to equation (C8), and after some algebra we find that the perturbationsatisfies ∂h a ∂t + v · ∂h a ∂ x + e a E [ f ] · ∂h a ∂ p + e a E [ h ] · ∂f a ∂ p = 0 . (C13)It should be reiterated that we have dropped the second-order term E [ h ] · ( ∂h a /∂ p a ). Wecan also write (C13) in the form ∂h a ∂t + V h a = 0 , (C14)where the operator V is defined by V h a = v · ∂h a ∂ x + e a E [ f ] · ∂h a ∂ p + e a E [ h ] · ∂f a ∂ p (C15)= v · ∂h a ∂ x + e a (cid:88) b (cid:20) (cid:90) dX b f b ( X b , t ) E ( b ) x · ∂h a ∂ p + (cid:90) dX b h b ( X b , t ) E ( b ) x · ∂f a ∂ p (cid:21) . (C16)From Bogoliubov’s hypothesis, the time dependence of f a is much slower than that of h a ,and we can regard the operator V as constant in time as far as its action on a perturbation h a is concerned. Note that equations (C15) and (C16) serve a definition of the operator V on any function h a ( X, t ), whether h a is the perturbation or not.As in Section VI, to solve (C14) we take the temporal Laplace transform and the spatialFourier transform, ( p + V )˜ h ( k , p , p ) = ˜ h ( k , p , , (C17)81iving the formal solution ˜ h ( k , p , p ) = ( p + V ) − ˜ h ( k , p , . (C18)We now preform the inversion (C18) for a specific example in which the unperturbed plasmais a function of momentum only, f a = f a ( p ). We then see that the 0-th order self-consistentfield vanishes, E ( x , t ) = (cid:88) b (cid:90) d ν p b (2 π (cid:126) ) ν f b ( p b ) (cid:90) d ν x b E b ( x a − x b ) = 0 , (C19)since the electric field E b ( x a − x b ) = − ∂φ ( x ) ∂ x (cid:12)(cid:12)(cid:12)(cid:12) x = x a − x b = ∂φ ( x a − x b ) ∂ x b (C20)is a total derivative of the potential φ ( x ). The operator V now take the form V h a = v · ∂h a ∂ x + e a E · ∂f a ∂ p (C21)= v · ∂h a ∂ x + e a (cid:88) b (cid:90) dX b h b ( x b , p b , t ) E b ( x − x b ) · ∂f a ( p ) ∂ p . (C22)Since E vanishes, the first-order field E defined in (C12) is the total self-consistent electricfield, and I have therefore dropped the 1-superscript. Upon taking the temporal-Laplacetransform and the spatial-Fourier transform of (C14) and (C21), we find p ˜ h a + i k · v a ˜ h a + e a ˜ E · ∂f a ∂ p = ˜ h a (0) , (C23)where the Fourier transform ˜ E can be calculated from (C22) by the convolution theorem,˜ E ( k , p ) = (cid:88) b (cid:90) d ν p b (2 π (cid:126) ) ν ˜ h b ( k , p b , p ) ˜ E b ( k ) . (C24)By way of notation, the tilde over a function is used to denote both Laplace and Fouriertransforms, so care must be taken when interpreting such terms. In other words, we take˜ h a = ˜ h a ( k , p , p ), ˜ E = ˜ E ( k , p ), and ˜ h a (0) = ˜ h b ( k , p , t = 0). That is to say, ˜ h a (0) is thespatial Fourier transform of h a ( x , p , t ) evaluated at t = 0, while ˜ h a and ˜ E are spatial Fouriertransforms and temporal Laplace transforms. We can now solve (C23) for the perturbation,giving ˜ h a ( k , p , p ) = ( p + V ) − ˜ h a ( k , p ,
0) (C25)= 1 p + i k · v a (cid:20) ˜ h a ( k , p , − e a ˜ E ( k , p ) · ∂f a ( p ) ∂ p (cid:21) . (C26)82ecall that ˜ E b ( k ) is the Fourier transform of the point Coulomb field E b ( x ), and can thereforebe written in terms of the Fourier transform of the potential ˜ φ b ( k ) as˜ E b ( k ) = − i k ˜ φ b ( k ) = − i k e b k . (C27)In like manner, the self-consistent electric field ˜ E ( k , p ) can be expressed in terms of a self-consistent potential ˜ φ ( k , p ) defined by˜ E ( k , p ) = − i k ˜ φ ( k , p ) . (C28)We can now write (C25) and (C24) in the Poisson-Vlasov form˜ h a ( k , p , p ) = 1 p + i k · v a (cid:20) ˜ h a ( k , p ,
0) + e a ˜ φ ( k , p )( i k ) · ∂f a ( p ) ∂ p (cid:21) (C29)˜ φ ( k , p ) = (cid:88) b e b k (cid:90) d ν p b (2 π (cid:126) ) ν ˜ h b ( k , p b , p ) . (C30)Let us substitute (C29) for the perturbation ˜ h a into (C30) for the potential, thereby giving˜ φ ( k , p ) = (cid:88) b e b k (cid:90) d ν p b (2 π (cid:126) ) ν p + i k · v b (cid:20) ˜ h b ( k , p b ,
0) + e b ˜ φ ( k , p ) i k · ∂f b ( p b ) ∂ p b (cid:21) . (C31)Note that ˜ φ ( k , p ) appears on both sides of this equation, and upon isolating the ˜ φ ( k , p ) term,we find (cid:34) − (cid:88) b e b k (cid:90) d ν p b (2 π (cid:126) ) ν p + i k · v b i k · ∂f b ( p b ) ∂ p b (cid:35) ˜ φ ( k , p ) = (cid:88) b e b k (cid:90) d ν p b (2 π (cid:126) ) ν ˜ h b ( k , p b , p + i k · v b . (C32)Solving for the self-consistent potential gives˜ φ ( k , p ) = (cid:88) b e b ¯ (cid:15) ( k , p ) k (cid:90) d ν p b (2 π (cid:126) ) ν ˜ h b ( k , p b , p + i k · v b , (C33)where the “dielectric function” in Laplace space is defined by¯ (cid:15) ( k , p ) = 1 − (cid:88) b (cid:90) d ν p b (2 π (cid:126) ) ν e b k p + i k · v b i k · ∂f b ( p b ) ∂ p b , (C34)with p lying on the contour C . In fact, we can analytically continue (C34), and allow p tolie anywhere in the complex plane to the right of C . For future reference, we record thefollowing identities: (cid:88) b (cid:90) d ν p b (2 π (cid:126) ) ν e b k i k · ∂f b /∂ p b p + i k · v b = 1 − ¯ (cid:15) ( k , p ) (C35) (cid:88) b (cid:90) d ν p b (2 π (cid:126) ) ν e b k i k · ∂f b /∂ p b p − i k · v b = ¯ (cid:15) ( − k , p ) − . (C36)83pon substituting (C33) back into (C29) we find the inverse( p + V ) − ˜ h a ( k , p , ≡ ˜ h a ( k , p , p ) (C37)= 1 p + i k · v (cid:34) ˜ h ( k , p ,
0) + (cid:88) b e a e b ¯ (cid:15) ( k , p ) k i k · ∂f a ( p ) ∂ p (cid:90) d ν p (cid:48) (2 π (cid:126) ) ν ˜ h b ( k , p (cid:48) , p + i k · v (cid:48) (cid:35) . We have use expressions (C35), (C36), and (C37) in Section VI C in their single-componentforms. This analysis shows that the results in Section VI C also hold for a multi-componentplasma.Let us pause now to understand these results physically. We see from (2.50) that thedielectric function (cid:15) ( k , ω ) can be analytically continued to complex values of ω , therebytaking the form (cid:15) ( k , ω ) = 1 + (cid:88) b (cid:90) d ν p (2 π (cid:126) ) ν e b k ω − k · v b k · ∂f b ∂ p , (C38)where Re ω >
0. The quantity ¯ (cid:15) ( k , p ) in (C34) is just the analytically continued dielectricfunction (cid:15) ( k , ω ) to a complex frequency ω = ip (for real p ), and we see that (cid:15) ( k , ω = ip ) = ¯ (cid:15) ( k , p ) , (C39)or equivalently, ¯ (cid:15) ( k , p = − iω ) = (cid:15) ( k , ω ). We can also analytically continue the self-consistentpotential (C33) to Fourier space by setting p = − iω + η , where ω is real and η >
0. Upontaking the limit η → + , we find˜ φ ( k , ω ) = (cid:88) b e b (cid:15) ( k , ω ) k (cid:90) d ν p b (2 π (cid:126) ) ν i ˜ h b ( k , p b , ω − k · v b + iη . (C40)Note that the k term in the denominator of the self-consistent potential (C40) is accom-panied by a factor of (cid:15) ( k , ω ) relative to the static Coulomb potential of a point charge,˜ φ a ( k ) = e a /k . This means that the self-consistent field is accompanied by Landau screen-ing, and in fact we could write the equations using only the screened potential ,˜ φ landau a ( k , ω ) = e a (cid:15) ( k , ω ) k . (C41)We choose, however, to keep the factors of (cid:15) ( k , ω ) explicit. The Fourier components of theelectric field are determined by ˜ E ( k , ω ) = − i k ˜ φ ( k , ω ), so that We should actually set ω = ip + η with η >
0, so that Re ω >
0. We can then take the limit η → + ,which gives (C39). E ( k , ω ) = (cid:88) b k (cid:15) ( k , ω ) e b k (cid:90) d ν p b (2 π (cid:126) ) ν ˜ h b ( k , p b , ω − k · v b + iη , (C42)where v b = p b /m b . If we wanted to find the electric field E ( x , t ) as a function of space andtime, it is more convenient to revert back to Laplace space by setting p = − iω in (C42),˜ E ( k , p ) = (cid:88) b − i k ¯ (cid:15) ( k , p ) e b k (cid:90) d ν p b (2 π (cid:126) ) ν ˜ h b ( k , p b , p + i k · v b , (C43)and then taking the inverse Laplace transform. We will, however, not work through thisalgebra, and remain content to have found the electric field and the perturbation in Laplaceand Fourier space. Chapter 24 of Ref. [13] does a good job of finding E ( x , t ) in variousphysical cases of interest. 85
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