On the Manifestly Covariant Juttner Distribution and Equipartition Theorem
Guillermo Chacon-Acosta, Leonardo Dagdug, Hugo A. Morales-Tecotl
OOn the Manifestly Covariant J¨uttner Distribution andEquipartition Theorem
Guillermo Chac´on-Acosta ∗ and Leonardo Dagdug † Departamento de F´ısica, Universidad Aut´onomaMetropolitana-Iztapalapa, M´exico D. F. 09340, M´exico
Hugo A. Morales-T´ecotl ‡ Departamento de F´ısica, Universidad Aut´onomaMetropolitana-Iztapalapa, M´exico D. F. 09340, M´exico andInstituto de Ciencias Nucleares, Universidad Nacional Aut´onoma de M´exico,A. P. 70-543, M´exico D. F. 04510, M´exico a r X i v : . [ c ond - m a t . s t a t - m ec h ] O c t bstract The relativistic equilibrium velocity distribution plays a key role in describing several high-energyand astrophysical effects. Recently, computer simulations favored J¨uttner’s as the relativistic gen-eralization of Maxwell’s distribution for d = 1 , , d +1 momentum space, with d spatial components. The use ofthe multiplication theorem of Bessel functions turns crucial to regain the known invariant form ofJ¨uttner’s distribution. Since equilibrium kinetic theory results should agree with thermodynamicsin the comoving frame to the gas the covariant pseudo-norm of a vector entering the distributioncan be identified with the reciprocal of temperature in such comoving frame. Then by combiningthe covariant statistical moments of J¨uttner’s distribution a novel form of the Equipartition The-orem is advanced which also accommodates the invariant comoving temperature and it contains,as a particular case, a previous not manifestly covariant form. PACS numbers: 05.70.-a,03.30.+p,51.10.+y ∗ Electronic address: [email protected] † Electronic address: [email protected] ‡ Electronic address: [email protected] . INTRODUCTION Incorporating the relativity principles in kinetic theory is crucial not only to understandthe theoretical grounds in the description of relativistic many particles systems [1, 2] but tointerpret relativistic high-energy experiments like those involving heavy-ion collisions [3] aswell as phenomena in the astrophysical [4] and cosmological realms [5]. These include, forinstance, the use of the relativistic Bolztmann equation to understand the thermal historyof the Universe [6, 7] and the structure of the Cosmic Microwave Background Radiationspectrum associated to its interaction with hot electrons in galaxy clusters [8].In the case of equilibrium the history of the relativistic analogue of Maxwell’s velocitydistribution goes back to F. J¨uttner, who in 1911 turned to relativity to consistently get ridof the contribution of particles with speeds exceeding that of light in vacuum, denoted by c , and which are contained in Maxwell’s distribution. This was achieved upon replacementof the non-relativistic energy of the free particles by its relativistic form and a maximumentropy principle thus yielding the so called one-particle J¨uttner’s distribution [9] f = n πkT m cK ( mc /kT ) e − √ p c m c kT , (1)which is written here in momentum rather than velocity space. m is the rest mass ofany of the particles forming the gas, k is Boltzmann’s constant and K is the modifiedBessel function of order two [10]. The remaining quantities: particle number density n ,temperature T and the relativistic spatial momentum of the particle p , should be hereunderstood as determined by an observer comoving with the gas as a whole. The mass shellcondition, ( E/c ) − p = m c , is assumed along the sequel. In the non-relativistic regime | p | (cid:28) mc, mc (cid:28) kT , we have f ≈ f Maxwell , where f Maxwell stands for Maxwell’s velocitydistribution. Thus particles’s speeds beyond c are just an artifact of the non-relativisticapproximation.J¨uttner’s distribution in the form (1) can be disadvantageous in that it is not manifestlycovariant, namely it is not expressed in terms of Lorentz tensors which in turn explicitlyshow its behavior under Lorentz transformations to investigate its description for frames inrelative motion. To make it manifestly covariant two key information pieces are needed: thetransformation under change of frame of f and that of T . Both of them have been consideredin the literature at large [11, 12] and [1, 13, 14, 15, 17], for temperature and distributionfunction, respectively. 3pon multiplying (1) by the characteristic function for the box confining the gas so that χ Box ( x ) = 1 for x within the box, and zero otherwise, the resulting distribution ¯ f ( x , p ) := χ Box ( x ) f ( p ) is defined on the single-particle phase space. An observer would thus determine¯ f particles of the gas in a volume d x located at x and having momentum p within range d p . Moreover ¯ f must be Lorentz invariant [13, 16, 17]. This is readily seen in the case ofequilibrium [18]: for a simple gas the number of particles N is invariant and so N = (cid:82) dµ ¯ f ,with dµ = d xd p , must be. Since dµ is a Lorentz invariant measure due to a compensationbetween the transformations of the (mass-shell) spatial momentum and space measure, ¯ f must be so too. Now since χ Box is invariant [44] then f is. Hence a manifestly covariantform of f amounts to manifest invariance and, in particular, it should involve the behavior oftemperature under Lorentz transformations. The manifestly invariant J¨uttner distributionwas given long ago [2, 14, 15, 19]. It was determined by introducing spherical coordinateson the pseudosphere associated to the mass shell in relativistic four-momentum space. Thisprovided a rather elegant and fruitful treatment which however does not seem to have beenfully appreciated in particular regarding temperature.Alternatives to (1) have been proposed recently for which Lorentz covariance is incorpo-rated in a different manner. In [20] a quantum mechanical approach including an invariant“historical time” was considered to derive a manifestly covariant Boltzmann equation withstationary solution implying a different ultra-relativistic mean energy-temperature relationwhereas in [21, 22] a maximum relative entropy principle was combined with an invariantgroup theoretical measure approach to obtain an equilibrium distribution. Both alternativeshave been recently criticized in [23]. Moreover recent Molecular Dynamics Simulations of atwo component one dimensional simple relativistic gas showed agreement with (1) and tem-perature defined through an equipartition theorem was shown to be invariant under changeof frame [24]. The study of the two-dimensional case along these lines has been reported in[25] and [26]. For three spatial dimensions Monte Carlo simulations have been consideredfavoring also J¨uttner’s distribution [27]. Amusingly, as for temperature, such kind of analy-sis take us back to the long standing discussion of whether a moving object appears cooler[11, 12, 24, 25].In this work we shall obtain the manifestly invariant J¨uttner distribution by adopting“cartesian” rather than spherical coordinates in relativistic ( d +1)-momentum space [14,19]. Our approach can be considered as an alternative to such previous derivations of the4¨uttner’s distribution and to others [1, 28] in the sense that we avoid adopting any framealong the sequel which otherwise would raise the question on the Lorentz transformation oftemperature. We only allude to the frame comoving with the gas at the end of the analysisto relate the kinetic description with thermodynamics in particular to identify temperaturewith the Lorentz invariant norm of a thermal vector. We once more obtain in this waythe thermal four vector introduced long ago [2, 14, 15, 19] which is formed by the productof the inverse comoving temperature with the four velocity of the gas as a whole. Alsothe lower dimensional cases recently studied [24, 25, 26, 27] are contained in our results.Hence comoving temperature is seen to play a role analogue to rest mass of a particle. Thecompatibility of such an interpretation is further investigated in relation with a manifestlycovariant form of the equipartition theorem.The paper is organized as follows. For the sake of completeness section II briefly re-views the derivation of the J¨uttner distribution as an equilibrium solution of the relativisticBoltzmann equation. This includes a brief description of the two types of approaches: themanifestly covariant one adopting spherical coordinates on the mass shell pseudo-spherein momentum space and the one adopting the comoving frame and cartesian components.Section III provides the details of our analysis in which we use “cartesian” coordinates inmomentum space to get the manifestly invariant distribution. In particular the thermalLorentz vector is here characterized. Its role in the relativistic covariant equipartition theo-rem is the subject of section IV. Finally section V summarizes our results. We also describethe behavior of a Planckian distribution when use is made of invariant temperature. The dif-ficulties to define a non-comoving temperature for a gas of massive particles is also touchedupon to further stress that invariant comoving temperature seems to be the only consis-tent possibility to define temperature according to the standard relativistic kinetic theoryframework.Our conventions are the following. Spacetime components of tensors are denoted by greekindices: µ, ν, · · · = 0 , , ,
3, the zero component referring to time whereas spatial componentsare denoted by latin indices i, j, · · · = 1 , ,
3. Einstein sum convention is assumed for bothtypes of indices throughout. The Minkowski metric has components η µν = diag { + , − , − , −} .5 I. J ¨UTTNER DISTRIBUTION AND THE RELATIVISTIC BOLTZMANNEQUATION
Consider a simple relativistic gas composed by identical particles of mass m . To eachparticle correspond space-time and four-momentum coordinates, respectively given by x µ and p µ . The evolution of the one-particle distribution function of the gas is governed by therelativistic Boltzmann equation [1, 28, 29] p µ ∂ ¯ f∂x µ + m ∂ ¯ f K µ ∂p µ = C [ ¯ f ∗ , ¯ f ] C [ ¯ f ∗ , ¯ f ] := (cid:90) (cid:0) ¯ f ∗ ¯ f ∗ − ¯ f ¯ f (cid:1) F σ d Ω d p p . (2)Here C [ ¯ f ∗ , ¯ f ] is the so called collision term and ¯ f ∗ ≡ ¯ f ( x , p ∗ , t ), ∗ implying a quantityis evaluated after the collision. σ is the differential cross section of the scattering processwhereas Ω is the solid angle. K µ denotes an external 4-force while F is the so called invariantflux, F = (cid:112) ( p µ p µ ) − m c , and d p p the invariant measure on mass shell. We shall beinterested in the case K µ = 0.The equilibrium state can be defined so that the entropy production vanishes [1, 14, 15,28]. In covariant form this means ∂S µ ∂x µ = 0, with the entropy flux given by S µ = − kc (cid:90) d pp p µ ¯ f ln( h ¯ f ) , (3)and h is a constant with dimensions to make the argument of the logarithm dimensionless.Zero entropy production requires ¯ f ∗ ¯ f ∗ = ¯ f ¯ f and so the collision term C [ ¯ f ∗ , ¯ f ] becomeszero too. Such condition can be written asln f ∗ + ln f ∗ = ln f + ln f , (4)where χ Box canceled out. Eq. (4) is fulfilled by the collisional invariants [28] which for thepresent case become the four-momentum up to a constant (independent of p µ ), so that an f solving (4) takes on the formln f = − (cid:16) Λ( x ) + ˜Θ µ ( x ) p µ (cid:17) ⇔ f = α ( x ) exp ( − Θ µ ( x ) p µ ) . (5)Here the negative sign has been introduced for later convenience, p µ is the 4-momentum ofa particle of the gas and α = e − Λ . The equilibrium distribution function is fully obtained6y requiring consistency between (5) and the left hand side of (2) equated to zero. Suchsubstitution yields Λ independent of x µ and ∂ ν ˜Θ µ ( x ) + ∂ µ ˜Θ ν ( x ) = 0, whose solution is˜Θ µ ( x ) = ω µν x ν + Θ µ and ω µν = − ω νµ . These correspond to the ten Killing vectors (6for ω µν x ν and 4 for Θ µ ) under which Minkowski spacetime metric is invariant. They areassociated with Lorentz transformations and spacetime translations, respectively [30]. Aswe will see below Θ µ can be identified with the four velocity of the fluid as a whole and thusit inherits spacetime symmetries in the form of rigid motions: world lines of neighbor fluidelements would keep their separation whenever they lie along Killing vectors [14]. We shallrestrict such motions to translations in the present work and so only Θ µ is considered.To determine α and Θ µ one assumes that typical physical quantities are related to thestatistical moments of the distribution. For instance the particle number density flux andthe energy-momentum tensor N µ = c (cid:90) d pp p µ ¯ f , (6) T µν = c (cid:90) p µ p ν ¯ f d d pp , (7)can be respectively written as N µ = − αc ∂ I ∂ Θ µ , (8) T µν = αc ∂ I ∂ Θ µ ∂ Θ ν , (9) I := (cid:90) d pp e − Θ α p α , (10)from which the denomination of I as generating functional suggests itself [14]. Havingexplicitly defined I will allow to determine α . On the other hand to obtain Θ µ one relatesthe kinetic theory form for thermodynamical quantities with equilibrium thermodynamicequations assumed to hold in a frame comoving with the gas as a whole. This will relate Θ µ with a comoving temperature.To evaluate (10) one can express it in spherical coordinates on the mass shell in momen-tum space [14, 15, 19], namely, since p = (cid:112) p + m c , one has for the components of theparticle’s momentum p µ = ( mc cosh χ, mc sinh χ sin θ cos φ,mc sinh χ sin θ sin φ, mc sinh χ cos θ ) , (11)0 ≤ χ ≤ ∞ , ≤ θ ≤ π, ≤ φ ≤ π ,
7n the pseudosphere p µ p µ = m c . Hence (10) becomes I = (cid:90) d Ω (3) dχ sinh χ e − mc Θ cosh χ = 4 π K ( mc Θ) mc Θ , Θ µ Θ µ = Θ , (12)where d Ω (3) is the element of solid angle in three spatial dimensions, K is the modifiedBessel function or order one [10] and Θ µ has been chosen to lie along the χ = 0 axis. Uponuse of the identity between Bessel functions ddu [ u − n K n ( u )] = − u − n K n +1 ( u ) to evaluate thecomponents of the statistical moments (8) and (9) allows to relate their components in thecomoving frame to get the equation of state for the gas P = ρc Θ, with P the pressure and ρ the density of energy, both in the comoving frame. This led Israel [14] to identify Θ = ckT ,with T the invariant comoving temperature.A different way to deal with (10) is to consider that, being invariant, it can be calculatedin a convenient frame, say one in which Θ µ = (Θ , Θ = ). Here cartesian coordinates havebeen used in momentum space [1, 28]. Noticing that (i) Θ µ in (6) should be timelike to makethe integral to converge and (ii) the only available timelike vector for the gas as a whole isits velocity U µ , one is led to propose Θ µ = U µ kT , so that T is identified with a quantity in aframe in which U , the spatial part of U µ , is zero. Such frame is the comoving frame to thegas. This approach in which one picks the comoving frame from scratch begs the questionabout which is the Lorentz transformation of the temperature.It would be desirable to be able to combine the power of the spherical components of themanifestly covariant approach mentioned afore with the more intuitive and easier to handlecartesian ones and still be able to investigate the behavior of temperature under Lorentztransformations. This is indeed possible and will be discussed in the next section. III. MANIFESTLY COVARIANT GENERATING FUNCTIONAL WITH CARTE-SIAN COORDINATES
Since our analysis remains the same independently of the number of spatial dimensionswe consider such general case at once. Hence unless otherwise stated, tensor indices run asfollows: µ, ν · · · = 0 , , , . . . , d , for d spatial dimensions.8 . Determination of α Let us consider a frame non-comoving with the gas so that the spatial components ofΘ µ , Θ , are non-zero. In d -dimensional space the integral I , Eq. (10), just requires tochange the measure from d p to d d p . Now we adopt spherical coordinates only for the spa-tial part of the momentum and assume that Θ · p = | Θ || p | cos ϑ . We then have d d p = | p | d − d | p | d Ω ( d ) , where d Ω ( d ) = (sin ϑ ) d − (sin ϑ ) d − . . . (sin ϑ d − ) dϑ dϑ . . . dϑ d − dϕ = (cid:81) d − i =1 (sin ϑ i ) d − i − dϑ i dϕ , where 0 < ϕ < π y 0 < ϑ i < π [31]. Thus I can be writtenas I = S d − (cid:90) d | p || p | d − p (sin ϑ ) d − dϑ e − Θ p e | Θ || p | cos ϑ . (13)Here S d − = π d − Γ ( d − ) is the hyper-surface of the d − ϕ and d Ω ( d − , which excludes ϑ . To integrate over ϑ it is better to usethe series form of the spatial exponential e | Θ || p | cos ϑ = (cid:80) ∞ k =0 ( | Θ || p | cos ϑ ) k k ! . In this way (13)becomes I = S d − ∞ (cid:88) k =0 | Θ | k k ! (cid:90) | p | d − k d | p | p e − Θ p ×× (cid:90) π sin ϑ d − cos ϑ k dϑ . (14)The remaining angular integral is non-zero only for k = 2 n, n = 0 , , , . . . and, moreover,it can be related to the beta functions B (cid:0) n +12 , d − (cid:1) [10] so that I = S d − ∞ (cid:88) n =0 | Θ | n (2 n )! B (cid:18) n + 12 , d − (cid:19) × (cid:90) e − Θ p (cid:104) p − m c (cid:105) n + d − dp . (15)Where we have exchanged the independent variable | p | by p by using the mass shell con-dition. The further change y = p /mc and z = mc Θ together with the integral form (cid:82) ∞ dx e − ax ( x − m − = (cid:0) a (cid:1) m Γ( m + )Γ( ) K m ( a ) for the modified Bessel function [10] leads to I = S d − ∞ (cid:88) n =0 | Θ | n (2 n )! B (cid:18) n + 12 , d − (cid:19) ( mc ) n + d − ×× (cid:18) z (cid:19) n + d − Γ (cid:0) n + d (cid:1) Γ (cid:0) (cid:1) K n + d − ( z ) . (16)9t this point we make the following considerations: (i) We use the known expression ofthe beta function in terms of the gamma functions [10] and then (ii) introduce z µ = mc Θ µ together with β := | z | /z , 0 ≤ β <
1, the latter following from the fact that Θ µ is timelike.Thereby Eq. (16) may be rewritten as I = 2 d +12 ( mc ) d − (cid:18) πz (cid:19) d − ∞ (cid:88) n =0 β n z n K n + d − ( z )2 n n ! . (17)Here we arrive at a critical point from the technical perspective. The sum in (17) can befurther reduced to a modified Bessel function upon using the multiplication theorem of theBessel functions [32] K ν ( λx ) = λ ν ∞ (cid:88) m =0 ( − m ( λ − m ( z ) m m ! K ν + m ( x ) , (18)with | λ − | <
1. This finally produces I = 2( mc ) d − (cid:16) πz (cid:17) d − K d − ( z ) . (19)Formula (18) is essential to trade the Bessel series in (17) by a single modified Bessel functionin (19). This becomes (12) when d = 3. We see that I is only function of the invariant z .Now we can obtain the relativistic particle number density flux from (8) producing N µ = 2 m d c d +1 α (cid:16) πz (cid:17) d − K d +12 ( z ) z µ z . (20)Such equation can be solved for α giving α := N c ( mc ) d K d +12 ( mc Θ) (cid:18) mc Θ2 π (cid:19) d − . (21)It is clear from (20) that z µ and N µ point in the same direction and since N µ = N U µ /c wehave that Θ µ = Θ c U µ . (22)The physical significance of Θ within our approach is discussed in the next subsection. B. Θ µ and the invariant Temperature Now we seek to identify Θ µ with a thermodynamical quantity. Let us take as our startingpoint the Gibbs form of the second law of thermodynamics for a closed system [15, 33],10hich we shall assume to hold in the comoving frame δU = T δ
S − P δV. (23)Clearly we must relate the relativistic kinetic expressions like the energy-momentum tensorand entropy flux with internal energy, U , entropy, S , pressure P and volume V appearingin (23). To begin with let us introduce the distribution function (5) and Eq. (21), in theexpression for the energy-momentum tensor (9). This leads to T µν = − N Θ (cid:32) η µν − K d +32 ( z ) K d +12 ( z ) Θ µ Θ ν Θ mc (cid:33) . (24)The comoving pressure can be obtained from the energy-momentum tensor in d -dimensionsas P ≡ − d ∆ µν T µν = N Θ , (25)where the projector is ∆ µν ≡ η µν − U µ U ν c − . The corresponding d -dimensional entropy flow(3) can be conveniently reexpressed as S µ = − k (cid:2) ln (cid:0) αh d (cid:1) N µ − T µν Θ ν (cid:3) . (26)It is worth stressing that the quantities N µ , T µν and S µ are densities, and therefore donot depend on the size of the system. It is however rewarding to include the fact that thegas is inside a box and to do so we make use of the characteristic function χ Box defined inthe introduction. In particular integrating over d spatial dimensions N = c − (cid:90) χ Box N µ dσ µ , (27) S = c − (cid:90) χ Box S µ dσ µ . (28) G µ = c − (cid:90) χ Box T µν dσ ν , (29)In the previous expressions (27)-(29), dσ µ = n µ d d S , with n µ a unit normal to the spatialhypersurface, and therefore a timelike vector; in particular d d S = d d x when n µ = δ µ .Obviously N and S are Lorentz invariants while G µ is a vector. The latter is just the totalrelativistic momentum of system.Combining (26) and (28) yields S = − k (cid:2) N ln (cid:0) αh d (cid:1) − Θ µ G µ (cid:3) , (30)11ow we can make contact with (23) by going to the comoving frame. Consider (30) forwhich such process means, in light of (22), Θ µ G µ → Θ G = Θ G . The differential of theentropy, (30), in thermodynamic space (Θ constant) becomes hereby δ S = k Θ (cid:20) δ G − N P c δNN (cid:21) , = k Θ c (cid:2) δ ( c G ) + P δV (cid:3) , (31)where use has been made of (21), (25) and the relation δV = − N δNN . Comparison of (23)with (31) gives rise to the identification Θ = ckT . (32)with T the comoving temperature by interpreting c G as the relativistic generalization ofinternal energy U ; this clearly holds in the low speed regime in which c G ≈ N mc + U non − rel ,with U non − rel the usual non-relativistic internal energy of the gas. Thus (21), (22) and (32)complete our analysis of the manifestly covariant distribution function which is expressed interms of invariant quantities, N, Θ , m, c, Θ µ p µ , in the fashion f ( p ) = N c ( mc ) d K d +12 ( mc Θ) (cid:18) mc Θ2 π (cid:19) d − exp ( − Θ µ p µ ) . (33)Eq. (1) is obtained from (33) by considering d = 3 and using the comoving frame to the gas.It should be stressed however that in deriving (33) no assumption was made in regard to theLorentz transformation of temperature. The latter came from the requirement to reproduceequilibrium thermodynamics in the comoving frame. IV. COVARIANT EQUIPARTITION THEOREM
Equipartition theorems are usually considered to connect temperature with averages ofsay kinetic energy in the non-relativistic case and certain functions of momenta in therelativistic case [12, 34]. Now, as we have shown in the previous section a manifestly covariantapproach to relativistic equilibrium distribution function unveils the convenience of using aninvariant temperature which is the one associated with the comoving frame of the gas. Thusit is of interest to investigate the role an invariant temperature plays within a manifestlycovariant equipartition theorem. 12he non manifestly covariant but relativistic equipartition theorem seems to have beenfirst considered by Tolman [34] and later by others [35]. Its use as a criterion to determine theLorentz transformation of temperature was criticized by Landsberg [12] who stressed that itcan accommodate both an invariant as well as a transforming temperature under a changeof frame. Recently Cubero et al . performed numerical simulations indicating the existenceof an invariant temperature [24] on the basis of the relativistic equipartition theorem of [12]thus hinting at the invariant temperature included in Landsberg’s analysis. In this sectionwe shall provide a manifestly covariant form of the equipartition theorem which not onlycontains an invariant temperature but includes Landsberg’s version. Moreover our form ofthe theorem is neatly expressed in terms of the total momentum of the gas.Let us start by reexpressing the generating functional I (10) in ( d + 1)-dimensionalmomentum space [1] I = 2 (cid:90) e − Θ µ p µ δ (cid:0) p σ p σ − m c (cid:1) H ( p ) d d +1 p, (34)where H ( p ) is the Heaviside function that restricts (34) to positive energies. The mass shellcondition is accounted for by the Dirac’s delta.By covariantly extending the argument in [34], we next integrate by parts d + 1 times-one for each p µ - and discarding the boundary terms at infinity we obtain I = − d + 1 (cid:90) p ν ∂∂p ν (cid:2) e − Θ µ p µ δ (cid:0) p σ p σ − m c (cid:1)(cid:3) H ( p ) d d +1 p . (35)To integrate over p we apply the properties of the Dirac delta which finally yields I = 1 d (cid:90) e − Θ µ p µ (cid:34)(cid:18) | p | p (cid:19) + Θ µ ∂p µ ∂p i p i (cid:35) d d pp . (36)Remarkably, notwithstanding the second term in (36) containing the spatial components p i it is Lorentz invariant overall.To connect with the physical framework Eq. (36) is plugged back into (8) and (9) toyield N α = cd (cid:90) d d pp f (cid:40) p α (cid:34)(cid:18) | p | p (cid:19) + p i ∂p µ ∂p i Θ µ (cid:35) − p i ∂p α ∂p i (cid:41) , (37) T αβ = cd (cid:90) d d pp f (cid:40) p α p β (cid:34)(cid:18) | p | p (cid:19) + p i ∂p µ ∂p i Θ µ (cid:35) − p i ∂p α p β ∂p i (cid:41) . (38)13 further spatial integration of (37) and (38) according to (27) and (29) gives rise to d = Θ µ (cid:28)(cid:28) p i ∂p µ ∂p i (cid:29)(cid:29) (39) G α = N d (cid:20) Θ µ (cid:28)(cid:28) p i ∂p µ ∂p i p α (cid:29)(cid:29) − (cid:28)(cid:28) p i ∂p α ∂p i (cid:29)(cid:29)(cid:21) . (40)where (cid:104)(cid:104)·(cid:105)(cid:105) := N (cid:82) · d d µ has been used. It is worth stressing that Eq. (39) is just themanifestly covariant form of the equipartition theorem corresponding to that advanced byTolman and Landsberg [12, 34], respectively, expressed using the invariant comoving tem-perature T = c/k Θ, Eq. (32). A comment regarding covariance of (39) is here in order.Although it contains the spatial sum p i ∂∂p i , we arrive to it from the manifestly covariantequation (34). This is analogous to recognizing the invariance of d p p . Clearly the argumentholds also for both terms in the r.h.s. of Eq. (40).Observing that (39) and (40) contain a common term suggests that the relativistic mo-mentum can be made to enter the covariant equipartition theorem. To proceed further weneed to determine the first term in (40). First we combine (29) and (24) to obtain theexplicit form for the momentum G α = N Θ α Θ (cid:34) mc Θ K d +32 ( mc Θ) K d +12 ( mc Θ) − (cid:35) . (41)By equating the projections Θ α G α of (40) and (41) we identifyΘ α Θ µ d (cid:28)(cid:28) p i ∂p µ ∂p i p α (cid:29)(cid:29) = K d +32 ( mc Θ) K d +12 ( mc Θ) mc Θ . (42)From here we finally arrive atΘ µ G µ N − z = F d ( z ) , z = mc Θ (43) F d ( z ) := z K d +32 ( z ) K d +12 ( z ) − − z . (44)Amusingly just in the same way as the energy of a particle having momentum p µ particle isdetermined by an observer having velocity U ν obs is given by E obs = p ν particle U ν obs we interpret Θ µ G µ as the energy of the gas determined by the comoving observer. F d ( z ) reduces to the usual values of the equipartition theorem in the limiting cases (SeeFig. 1). F d ( z ) = d , z (cid:29) d, z (cid:28)
1. (45)14 igure 1: The graph shows F d ( z ) vs. z = mc /kT for d = 3. F ( z ) corresponds to the quotient ofthe average of the relativistic kinetic energy per particle and the comoving temperature. We seethat for z (cid:29) z (cid:28) z , theratio is a fraction between 3 < F ( z ) < / Notice that although a relativistic equipartition theorem had been considered before byother authors [19, 36] their approach was not manifestly covariant. Their work and ourscoincide in terms of the function F d (44), in particular the behavior in the non-relativistic aswell as in the ultra-relativistic limits. Remarkably, recent numerical calculations adoptingMonte Carlo methods [27] confirm F d as giving the relativistic kinetic energy divided by kT . V. DISCUSSION
The interest in incorporating the relativity principles into kinetic theory goes beyond thetheoretical foundations: Actual observations and experiments like for instance in high-energyphysics [3], astrophysics [4] and cosmology [5] require a description of relativistic many-15articles systems in terms of, say, Boltzmann’s equation and the corresponding equilibriumdistribution [6, 7, 8]. Recently, Cubero et al . [24] (See also references there) have developednumerical simulations based on Molecular Dynamics pointing to J¨uttner’s as the equilibriumdistribution in agreement with their numerical analysis, as opposed to other proposals inthe literature, for one and possibly other number of spatial dimensions. This was furtherconfirmed for two [25, 26] and three [27] spatial dimensions, the latter adopting MonteCarlo simulations instead. In [24], the old problem regarding the relativistic transformationof temperature [11] was also considered in connection with a previous relativistic version ofthe Equipartition Theorem proposed by Landsberg [12]. Based upon such theorem Cubero et al . determined, within their approach, that temperature should possess an invariantcharacter. It is worth mentioning that such analysis were not formulated in a manifestlycovariant form and indeed some further insight was needed to interpret the contributionsentering such theorem as well as temperature.In this work we have obtained a new derivation of the manifestly invariant J¨uttner’srelativistic distribution function (33). This is based on cartesian coordinates in ( d +1)-momentum space ( d spatial dimensions) in contrast with the known results developed usingspherical coordinates. This was made possible by the use of the multiplication theorem forBessel’s functions (18) which simplified the treatment of a series involving Bessel functions[32]. In this approach no assumption is made a priori of any specific relativistic character fortemperature. The latter appears through the invariant norm, Θ, of a four-vector, Θ µ (22),and it is invariant just for the same reason a point particle’s rest mass is. Indeed becauseof the assumption that relativistic equilibrium kinetic theory should agree with standardthermodynamics for an observer comoving with the system under study the pseudo-normbecomes Θ = c/kT , Eq. (32), T being the comoving temperature of the gas. Finally we havedeveloped a manifestly covariant Equipartition Theorem, Eq. (43), in which the average ofthe energy-momentum of the gas as determined by the comoving observer is given by afunction F d , Eq. (44) of the invariant temperature. Indeed this is analogous to the caseof a point particle for which the energy is obtained by projecting momentum along thefour-velocity of the observer. Here we have the thermal vector Θ µ = ckT U µ , with U µ thefour-velocity of the gas as whole which defines a comoving observer reading the invarianttemperature T . A further comment is here in order regarding the difference between ourapproach and previous ones. While previous versions of the Equipartition theorem [12, 34]16elate temperature with a peculiar combination of relativistic quantities (See for exampleEq.(39)), here we interpret that combination as included in (43) which relates the invarianttemperature with the averaged relativistic momentum, G µ .In a nutshell the manifestly covariant form of J¨uttner’s distribution leads naturally toconsider the invariant comoving temperature to characterize the equilibrium regime. As anillustrative example of how this result applies to known cases let us consider the case of blackbody radiation. Let us recall the standard analysis of this problem [5, 7, 18]: The Lorentzinvariance of I ν ν , containing the specific intensity I ν and the frequency ν of the photons,implies the invariance of the Planckian distribution I ν hν = 1e hvkT − . (46)Indeed for two frames in relative motion one should have hνkT = hν (cid:48) k T (cid:48) , (47)where primed quantities refers to an observer that moves with respect to radiation and, inparticular, T (cid:48) is suggested as the non-comoving temperature. Since the photons suffer aDoppler shift: νν (cid:48) = γ (1 − β cos δ ) , (48)where δ is the angle between the momentum of the photon and the gas velocity whereas β is the ratio between the gas speed and c . The quantity T (cid:48) takes the anisotropic form T (cid:48) = Tγ (1 − β cos δ ) . (49)Notice that we could alternatively write a manifestly invariant form for (46) as (e Θ µ p µ − − .In this way one can focus in the invariant product Θ µ p µ . By evaluating it in the two framesmentioned above and using that for photons p (cid:48) = | p (cid:48) | hνkT = Θ (cid:48) p (cid:48) (1 − β cos δ ) . (50)Considering the invariant comoving temperature T the components Θ (cid:48) µ become γkT ( c, U ) andthus (50) reduces to (48). This shows the consistency of adopting such invariant comovingtemperature, trough Θ µ , in the description of the black body radiation without resorting to T (cid:48) , Eq. (49). The latter can be regarded only as an auxiliary quantity for the following rea-sons. Firstly, actual observations of the Cosmic Microwave Background Radiation (CMBR)1737] reveal there is a frame in which it presents the black body structure. Measurementshowever involve brightness depending on frequency rather than a non-comoving temperature(49). Secondly, in the case of massive particles one finds and obstacle to handle (49) (Seefor instance [38]). For massive particles we would have T (cid:48) = Tγ (1 − β p β cos δ ) (51) β p = | p | p , which makes no sense due to the dependence on the particles’s momentum. However adopt-ing an invariant comoving temperature is viable just for the same reason that the case forphotons described above works.Our work then adds to recent claims elaborated on the basis of an Unruh-DeWitt detector[39] pointing to the impossibility to have a relativistic transformation of temperature [40].In any case all these results reinforce the idea that temperature makes undisputable sensein the comoving frame.There are several possible extensions of the present work which could be of interest. Onepossibility is to extend the analysis in the present work to the case of relativistic Fokker-Planck equation, say along the lines of [41]. 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