Non-Thermal Einstein Relations
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Non-Thermal Einstein Relations
Robin Guichardaz , Alain Pumir and Michael Wilkinson , Univ Lyon, Ens de Lyon, Univ Claude Bernard, CNRS, Laboratoire de Physique, F-69342 Lyon, France Department of Mathematics and Statistics, The Open University, Walton Hall, Milton Keynes, MK7 6AA, England
PACS – Stochastic analysis methods (Fokker-Planck, Langevin, etc.)
PACS – Fluctuation phenomena, random processes, noise, and Brownian motion
Abstract – We consider a particle moving with equation of motion ˙ x = f ( t ), where f ( t ) is arandom function with statistics which are independent of x and t , with a finite drift velocity v = (cid:104) f (cid:105) and in the presence of a reflecting wall. Far away from the wall, translational invarianceimplies that the stationary probability distribution is P ( x ) ∼ exp( αx ). A classical example of aproblem of this type is sedimentation equilibrium, where α is determined by temperature. In thiswork we do not introduce a thermal reservoir and α is determined from the equation of motion.We consider a general approach to determining α which is not always in agreement with Einstein’srelation between the mean velocity and the diffusion coefficient. We illustrate our results with amodel inspired by the Boltzmann equation. Introduction. –
This Letter discusses a new perspec-tive on a classic problem of statistical physics. Considerthe motion of a particle with equation of motiond x d t = f ( t ) (1)where f ( t ) is a random function, with statistical proper-ties which are independent of x and of t . We might wishto characterise the probability distribution of the coordi-nate x ( t ). If we seek a probability distribution which isstationary in time, this distribution should respect, awayfrom the boundaries, the translational invariance of theproblem. The stationary probability density must havean exponential form P ( x ) = A exp( αx ) (2)where A is a normalisation constant. In this Letter wepresent a general formula determining the exponent α interms of the statistics of the function f ( t ).This problem is closely related to the classical treat-ment of sedimentation equilibrium [1] by Einstein [2] andSutherland [3], who used statistical mechanics to describethe particle motion, in terms of a diffusion process, andto relate α to temperature via the diffusion coefficient.In this work we treat equation (1) as a purely dynamicalprocess, and the exponential solution (2) is a consequenceof translation symmetry, rather than thermal equilibrium.We are concerned with the relation between α and dy-namical quantities. In a homogeneous system, we expect the motion at long time to resemble a biased random walkwith drift velocity v and diffusion coefficient D , given by v = (cid:104) f ( t ) (cid:105) , D = 12 (cid:90) ∞−∞ d t (cid:104) ( f ( t ) − v )( f (0) − v ) (cid:105) (3)where (cid:104) X (cid:105) denotes the expectation value of X through-out. We assume that both v and D are finite, and nonzero. In the case of a Brownian particle in a thermal bath,an appropriate description of the evolution of the proba-bility distribution function (PDF), P ( x, t ), is given by theFokker-Planck equation: ∂P∂t = − ∂∂x ( vP ) + D ∂ P∂x . (4)Seeking a stationary solution of Eq. (4) with an exponen-tial dependence on x , Eq. (2), of Eq. (4) leads to an explicitformula for α : α E = vD . (5)In the case of sedimentation equilibrium, the exponent ofthe exponential distribution is determined by the temper-ature, and equation (5) is the basis of the relation betweenmobility, diffusion coefficient and temperature which wasintroduced by Einstein [2] and Sutherland [3]. In the re-mainder of this text we refer to (5) as the classical Einsteinrelation, although we do not consider a coupling with athermal bath.In general, the evolution of the probability density forthe system is not always faithfully represented by (4). Al-though deriving the proper formulation is a challengingp-1 a r X i v : . [ n li n . C D ] F e b obin Guichardaz , Alain Pumir and Michael Wilkinson ,task (a variety of different approaches are discussed in[4–8]), our approach in this letter does not make explicituse of a generalisation of Eq. (4), but rather uses largedeviation theory [9, 10]. As a consequence of the fact thatEq. (4) is no more than an approximation, the status ofEq. (5) is uncertain.We analyse a simple model, where f ( t ) is telegraphnoise , and we determine a closed form for α , which dif-fers from Eq. (5). In the telegraph noise model, a particlemoves with one of two possible velocities, and the transi-tion between the two velocity states is completely random.Because the exponent α is a very fundamental charac-terisation of the simple dynamical process (1), we providea general analysis of this quantity. We show that largedeviation theory provides a powerful approach to derivinga generalised Einstein relation , in the form of an implicitequation for α in terms of cumulants of f ( t ). Equation (5)appears as an approximation of this general expression inthe case where the random process f is described by aGaussian process. The application of the general formuladerived from large deviation theory is illustrated here byusing the telegraph noise model as an example.Finally, we discuss how deviations from Eq. (5) couldaffect the sedimentation equilibrium. The telegraph noiseprocess can be viewed as a simplified model for the micro-scopic motion of molecules in gases, in which there are onlytwo possible velocities. The analysis is readily extendedto the Boltzmann equation, where atoms move ballisti-cally between collisions, which occur at random intervalsand result in an instantaneous change in the velocity of aparticle. In the general case the exponent α is not givencorrectly by (5) for a sedimentation equilibrium describedby the Boltzmann equation. This raises a question aboutthe validity of the classical Einstein relation for sedimen-tation equilibrium, and potentially for other physical pro-cesses. In the limit where the suspended particles are verymassive compared to the gas molecules, however, we no-tice that the collision term in the Boltzmann equation isreplaced by a diffusion term in the particle velocity. Weshow that for this model equation (5) is exact, so that theclassical Einstein relation is valid for the sedimentationequilibrium of macroscopic particles. Telegraph noise model. –
We first discuss the ex-ample where the velocity f ( t ) in Eq. (1) is a random tele-graph noise. Namely, we assume that f ( t ) can be either ofthe two values f + and f − . The system switches from f + to f − (respectively f − to f + ) with transition rates R + (re-spectively R − ). The probability in the steady state regimeof the velocity to be f + ( f − ), p + (respectively p − ), is sim-ply given by p + = R − / ( R − + R + ) ( p − = R + / ( R − + R + )).As a consequence, the mean velocity (cid:104) f (cid:105) is given by (cid:104) f (cid:105) = R + f − + R − f + R + + R − . (6)We assume the presence of an impervious wall, say at x = 0, and require that the two velocities f + and f − to be of opposite signs, which is required to impose zeroflux boundary condition at the wall. In fact, in order toreach a stationary state, the zero flux condition is neededeverywhere. Without any loss of generality, we assumethat f + > f − <
0, and that the averaged velocity (cid:104) f (cid:105) is negative.We introduce the probability P + ( x, t ) ( P − ( x, t )) thatthe position is x at time t , the velocity of the system being f + ( f − ). The evolution equation for P + , P − is simply: ∂∂t (cid:18) P + P − (cid:19) = − ∂∂x (cid:18) f + P + f − P − (cid:19) + (cid:18) − R + R − R + − R − (cid:19) (cid:18) P + P − (cid:19) . (7)Steady-state solutions of the form P + , − ( x ) ∝ A + , − exp( αx ), consistent with Eq. (2), can be read-ily found by imposing that the matrix M ( − α ), definedby M ( − α ) ≡ (cid:18) − αf + − R + R − R + − αf − − R − (cid:19) (8)has a zero determinant: det( M ( α )) = 0. This conditionleads to a simple algebraic equation, with only one non-zero root: α = − f + R − + f − R + f + f − . (9)With our assumptions for the signs, the exponent α isnegative. More generally, the product (cid:104) f (cid:105) × α >
0. Thisguarantees that away from the reflecting wall, the solu-tion decays exponentially, similar to what happens in thesedimentation problem of Brownian particles [2, 3].The value of α given by Eq. (9), however, differs fromthe prediction given by Eq. (5). From the solution ofEq. (7), in the homogeneous case ( ∂/∂ x → R − + R + , and by computing the variance of f we obtain the correlation function (cid:104) ( f ( t ) − (cid:104) f (cid:105) )( f (0) − (cid:104) f (cid:105) ) (cid:105) = R + R − ( f + − f − ) ( R + + R − ) exp[ − ( R + + R − ) t ] (10)so the diffusion coefficient D is equal to D = R + R − ( R + + R − ) ( f + − f − ) . (11)The resulting ratio v/D clearly differs from the expressionfor α , Eq. (9), thus calling for a revisiting of the Einstein-Sutherland relations. We find αα E = − R − R + ( R − + R + ) ( f + − f − ) f + f − . (12)In general, this ratio may be either very large or verysmall. After some algebra, it can be shown that the ratioapproaches unity whenever the dimensionless parameter µ = (cid:104) f (cid:105) D f + − f − R + + R − (13)p-2on-Thermal Einstein Relationsbecomes very small. The quantity µ can be rewritten as µ = α E (cid:96) , where α E is given by Eq. (5), and (cid:96) is effectivelythe mean free path of the particle. The length (cid:96) is theproduct of 1 / ( R + + R − ), which provides an estimate ofhow long the particle stays with either velocity f + or f − ,and of f + − f − = ( f + − (cid:104) f (cid:105) ) − ( f − − (cid:104) f (cid:105) ), which is the sizeof the difference between the mean and the instantaneousvelocity. Thus, (cid:96) is of the order of the size travelled by aparticle between two collisions, hence the mean free pathinterpretation. Thus, the condition µ → (cid:96) , is much smaller than the typicaldecay length predicted by Einstein theory.Note that the solution α = 0 is formally always valid. Itcorresponds to the homogeneous case, where the densityof probability is uniform, and thus, non-normalisable. A general form for the Einstein relation. –
Toproceed, we now consider the general problem describedby Eq. (1). We consider the integral of equation (1) x ( t ) = x (0) + ∆ x ( t ) , ∆ x ( t ) = (cid:90) t d t (cid:48) f ( t (cid:48) ) . (14)Let π (∆ x, t ) be the probability density of ∆ x at time t .We express the condition that the distribution P ( x ) isstationary in the form: P ( x ) = (cid:90) d∆ x P ( x − ∆ x ) π (∆ x, t ) . (15)Using explicitly the exponential form of the PDF P ( x ),Eq. (2), one obtains the expression: (cid:90) d∆ x exp( − α ∆ x ) π (∆ x ) = 1 . (16)Eq. (16) can be interpreted as the average of exp( − α ∆ x ),the variable ∆ x ( t ) being characterized by its PDF, π (∆ x, t ). It is valid provided t is much larger than thecorrelation time of the original process f ( t ) so that wecan assume that ∆ x ( t ) is independent of x . This gives (cid:28) exp (cid:18) − α (cid:90) t d t (cid:48) f ( t (cid:48) ) (cid:19)(cid:29) = 1 . (17)In the t → ∞ limit, the large deviation principle [10] pro-vides an appropriate approach. We introduce here thescaled cumulant generating function [10], λ ( k ), defined by: λ ( k ) ≡ lim T →∞ T ln (cid:68) exp (cid:16) k (cid:90) T d t (cid:48) f ( t (cid:48) ) (cid:17)(cid:69) (18)which describes the exponential growth of the average (cid:68) exp( k (cid:82) T d t (cid:48) f ( t (cid:48) )) (cid:69) as a function of time T . The condi-tion Eq. (17) merely states that λ ( − α ) = 0 . (19)Thus, the determination of spatial distribution of particlesin a sedimentation equilibrium amounts to finding solu-tions of Eq. (19), which is a simple condition for α that can be simply applied if the cumulant generating functioncan be determined.We now illustrate the application of the large devia-tion theory approach by using (19) to determine α forthe telegraph noise model. To this end, we discretizetime, and consider f n = f ( n ∆ t ) and x n = x ( n ∆ t ), where∆ t is a very small time step. Following the large devi-ation approach, we consider the function λ ( k ), definedby Eq. (18). To evaluate λ ( k ), we adapt the general ap-proach described in [10] (see in particular Section 4.3) asfollows. With the telegraph noise process, f n can takeonly two values, f + and f − , so the integral in Eq. (18)reduces (up to an overall factor ∆ t ) to a sum of termsequal to f + and f − , depending on the state of the system.The expectation value is computed by summing over allsequences f , f , . . . , f n , . . . . Because the steps are sta-tistically independent, the probability density for a se-quences of steps may be expressed as a product of theform (cid:81) j P ( f j +1 , f j ), where P ( f j +1 , f j ) is the probabilityto reach f j +1 at t j + ∆ t , if the particle is in velocity state f j at time t j . The summation over all possible values of f j can be represented as a product of a string of matri-ces (which are 2 × (cid:104) exp( k ∆ t (cid:80) ni =0 f i ) (cid:105) grows exponentially as afunction of n as ξ ( k ) n , where ξ ( k ) is the largest eigenvalueof the ‘tilted’ transition matrix [10], given by:Π k = (cid:18) (1 − R + ∆ t ) e kf + ∆ t R − ∆ t e kf + ∆ t R + ∆ t e kf − ∆ t (1 − R − ∆ t ) e kf − ∆ t (cid:19) . (20)Thus, λ ( k ) reduces to the logarithm of the largest valueof Π k . In the limit ∆ t →
0, the matrix Π k reduces to asum of the identity matrix, Id, plus ∆ t times the matrix M ( k ), defined by Eq. (8). From this simple representationof the matrix Π k , it immediately follows that the values of α for which ξ ( − α ) = 1 in the limit ∆ t → α for which det( M ( − α )) = 0, thus establishingthat α can be in fact established using large deviationtheory. The function λ ( k ) for the telegraph noise model isillustrated in Fig. 1.Equation (19) provides a simple criterion to determine α if the cumulant generating function λ ( k ) can be deter-mined. In many cases, this will not be practicable, andit is desirable to have an alternative approach. To pro-ceed further, we notice that the expression Eq. (17) canbe simply written as a series in powers of α , in the form: λ ( − α ) = lim T →∞ T ln (cid:68) ∞ (cid:88) n =0 ( − α ) n n ! (cid:16)(cid:90) T d t f ( t ) (cid:17) n (cid:69) = ∞ (cid:88) n =0 ( − n n ! c n α n (21)where c n are defined as the integrals of the n th order cu-mulants of the distribution of f ( t ): c n = lim T →∞ T (cid:90) T d t · · · (cid:90) T d t n κ [ f ( t ) , . . . , f ( t n )] . (22)p-3obin Guichardaz , Alain Pumir and Michael Wilkinson , -2 -1 0 1 2 3 . . . . k λ ( k ) h f i f + f − − α Fig. 1: (Colour online). In blue, plot of λ ( k ). The parametersare R − = 0 . R + = 0 . f − = −
1, and f + = 0 .
5, leading to (cid:104) f (cid:105) (cid:39) − . <
0. We remark that λ (cid:48) (0) = (cid:104) f (cid:105) (green line),and that the slope of the asymptote in k → −∞ ( k → + ∞ ) is f − ( f + ) (red dotted lines). Moreover, one has α (cid:104) f (cid:105) > The first cumulants are simply κ [ f ( t )] = (cid:104) f ( t ) (cid:105) κ [ f ( t ) , f ( t )] = (cid:104) f ( t ) f ( t ) (cid:105) − (cid:104) f ( t ) (cid:105) κ [ f ( t ) , f ( t ) , f ( t )] = (cid:104) f ( t ) f ( t ) f ( t ) (cid:105)− (cid:104) f ( t ) (cid:105)(cid:104) f ( t ) f ( t ) (cid:105) − (cid:104) f ( t ) (cid:105)(cid:104) f ( t ) f ( t ) (cid:105)− (cid:104) f ( t ) (cid:105)(cid:104) f ( t ) f ( t ) (cid:105) + 2 (cid:104) f (cid:105) . (23)It is straightforward to check that the coefficients c and c , as defined by Eq. (22) coincide with (cid:104) f (cid:105) and D , asdefined by Eq. (3). This immediately shows that theEinstein-Sutherland relations are exact when the cumu-lants of order higher than 3 vanish, which is the case when f is given by a Gaussian process. This conclusion does notdepend on whether the process is Markovian or not.Finally, in the telegraph model case, the parameter µ ,defined in Eq. (13), effectively specifies how far the processis from being Gaussian. Specifically, the deviation from aGaussian distribution in Eq. (21) are due to the terms c n for n >
2. One therefore has to compare the relative im-portance of c n α n E /n ! for n > c α E (or, equivalently,with c α / n > c n α n E n ! c α E = µ n − G n (cid:18) R − R + (cid:19) (24)where the G n are bounded functions, which implies thatthe solution of λ ( α ) = 0 in the limit µ → α = α E , thus justifying the Einstein equation. More refined models of sedimentation. –
Ourobservation that the exponent for sedimentation equilib-rium in the case of a telegraph noise model does not agreewith the classical Einstein relation raises the question asto whether the discrepancy exists in more refined models.The telegraph noise model is close in structure to theBoltzmann model for the motion of atoms in a dilute gas, where the atoms move ballistically between collisions, andhave their velocities changed discontinuously at collisionevents which occur at random times. The difference isthat the Boltzmann equation has a continuum of allowedvelocities, so that the probability density is a function of acontinuous velocity v and the probability density P ( x, v, t )satisfies a version of the Boltzmann equation in the form ∂P∂t ( x, v, t ) = − v ∂P∂x ( x, v, t ) − Γ( v ) P ( x, v, t )+ (cid:90) ∞−∞ d v (cid:48) R ( v, v (cid:48) ) P ( x, v (cid:48) , t ) (25)where R ( v, v (cid:48) ) is the rate for scattering from velocity v (cid:48) to v , and Γ( v ) = (cid:82) ∞−∞ d v (cid:48) R ( v (cid:48) , v ). Eq. (25) manifestlyreduces to Eq. (7) when only two velocities are possible.Therefore, the analysis for Eq. (25) follows the same stepsas for Eq. (7), except that operations involving matrixmultiplication are replaced by integral transforms. Thekey stages in the argument are unchanged, and we con-clude that in the general case the Boltzmann equationwill predict that α E (cid:54) = α .In sedimentation problems, however, we are usually con-cerned with the equilibrium of colloidal particles, whichare much larger than the size of the atoms. Because themass ratio is very large, the changes in the velocity of thecolloidal particle with each collision are small. This canbe described by replacing the general collision term in theBoltzmann equation (25) with a diffusion term. Specifi-cally the velocity of the particle undergoes diffusive fluc-tuations, with diffusion coefficient D , while relaxing to adrift velocity v with rate constant γ , so that v obeys thestochastic differential equationd v = − γ ( v − v )d t + √ D d η (26)where (cid:104) d η (cid:105) = 0 and (cid:104) d η (cid:105) = d t . The correspondingFokker-Planck equation is ∂P∂t = − v ∂P∂x + γ ∂∂v [( v − v ) P ] + D ∂ P∂v (27)where the collision kernel in (25) has been replaced by adiffusion term. This is a variant of the Ornstein-Uhlenbeckprocess [11], which is often an accurate description ofthe velocity of a Brownian particle (one then speaks ofa Rayleigh particle [12]). The normalisable steady-statesolution of the Fokker-Planck equation (27) is P ( x, v ) ∝ exp (cid:16) − γ D v (cid:17) exp (cid:18) v γ D y (cid:19) . (28)Determining the spatial diffusion coefficient for the pro-cess described by (26) gives D = D /γ , so that (28) agreeswith (5). In fact, as the process described by (27) is Gaus-sian, the expansion in (21) reduces to its two first terms,and thus one has α = α E . We conclude that the classicalEinstein relation for sedimentation equilibrium is valid formacroscopic colloidal particles, while it may fail for micro-scopic particles with a mass which is comparable to thatof the gas.p-4on-Thermal Einstein Relations Conclusions . –
In this letter, we have investigateda class of stochastic problems, with a mean drift, and areflecting wall. This corresponding to the classical andfundamental problem of sedimentation equilibrium [1–3].Very general considerations lead to the conclusion that thedistribution, far away from the wall, decays exponentially.We have shown that the decay rate, α , can be determinedquite simply from large deviation theory using equation(19) (where the cumulant generating function is available)or equation (21) (when the cumulants of f ( t ) are known).Whereas the classical Einstein relation can be derived froma Fokker-Planck description of the evolution of the PDF,our approach does not rest on any Fokker-Planck descrip-tion.In the case of the telegraph noise model we show explic-itly that α (cid:54) = α E . This raises the question as to whetherthere is a reason to doubt the validity of the Einstein re-lation for sedimentation equilibrium properties. We haveargued that while α need not equal α E for the Boltzmannequation, in the limit where the ratio of the mass of thesuspended particles is very large, the Boltzmann equationshould be replaced by a variant of the Ornstein-Uhlenbeckmodel. An explicit solution shows that α = α E for thiscase.Lastly, it is of interest to note that in some cases thescaled cumulant generating function λ ( k ) does not exist,for example then the process is discrete in time x n +1 = x n + f n and the velocities are independent and follow thedensity of probability p ( f n ) ∝ (1+ | f n − v | ) − β with β > (cid:104) f (cid:105) = v and D are finite, but λ ( k ) is nowheredefined, except in k = 0. The large tails in the distributionof f n avoid to properly define a region in space wherethe dynamic is considered being far from the wall, as theparticles are likely to do large jumps. It is then related tothe mean free path interpretation of the telegraph model;in this case not only the Einstein relation, but also theexponential sedimentation are no longer valid, and longrange corrections must be added.Acknowledgements: MW thanks the Kavli Institute forTheoretical Physics, Santa Barbara, where work on thispaper was supported in part by the National Science Foun-dation under Grant No. NSF PHY11-25915. REFERENCES[1]
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