H theorem for telegrapher type kinetic equations

Physics Letters A 171 (1992) 26-30 North-Holland

PHYSICS LETTERS A

H theorem for telegrapher type kinetic equations J. C a m a c h o a n d D. J o u

Departament de Fisica (FlsicaEstadistica), UniversitatAutbnoma de Barcelona, 08193Bellaterra, Catalonia, Spain Received 22 June 1992; revised manuscript received 21 September 1992; accepted for publication 28 September 1992 Communicated by J. Flouquet

We generalize the Boltzmann H function in order to be able to state a H theorem for kinetic equations of the telegraphertype. The generalizedH function reduces to the Boltzmann formula in equilibrium.

Given a kinetic equation, the general problem of finding a functional of the probability density whose temporal derivative has a definite sign is known in the literature as the H theorem. Its interest rests on the fact that it provides a criterion of irreversibility in the evolution of the system, and thus it constitutes a kinetic analog of the second law of thermodynamics. For some equations, as e.g. the master equations [ 1,2 ], there exist more than one functional (or even an infinite number of them) satisfying the latter itreversibility criterion. For others, however, like the Smoluchowski equation, describing the dispersion of non-interacting Brownian particles under an external potential U(x),

T 02f(x, - - qt) - -Of(x, - t) Ot2 Ot =V.(DokTVf(x, t)+ [ V U ( x ) I f ( x , t ) } ,

Of(x, t) 8t =V.Do{kTVf(x,t)+[VU(x)]f(x,t)},

(1)

where f ( x , t) denotes the probability distribution function, D f ~ the friction coefficient, T t h e absolute temperature, x the particle position and V the gradient operator, only a functional of the form

H= - k I d x f ( x , t) [lnf(x, t) - l ] ,

(2)

J

k being the Boltzmann constant, obeys a H theorem [3 ]. The same thing happens for the Boltzmann equation, for which the configuration space also exAlso at lnstitut d'Estudis Catalans. 26

tends to velocities, i.e. the so-called/~-space [4]. In this case, the functional (2) displays also the interesting feature of providing in equilibrium the entropy of an ideal gas. Both features, dH/dt>~O and H~q=S,q, suggest that the H theorem is the microscopic counterpart of the second law of thermodynamics. The aim of this Letter is to examine the possibility of stating a H theorem for kinetic equations of the telegrapher kind, the so-called generalized telegrapher equations (GTE), which have been used in several contexts (see below),

(3)

where z is obviously a time parameter. Equations of the GTE type arise, for instance, in the study of the one-dimensional random walk [ 6 ]; the persistent random walk [7 ] - which can also be seen as a random walk with a continuous distribution of pausing times [ 8,9 ] - ; differential stochastic equations with telegraphist noise [ 10]; adiabatic elimination of fast variables (for instance, in the study of the position of an overdamped anharmonic oscillator [ 1 1 ] ); as an extension of the Smoluchowski equation also valid at short times [ 12,13 ]; and generalized master equations with memory effects [ 11 ]. Recently, Hongler and Streit have studied the conditions under which an equation like (3) keeps f positive in its evolution, an obvious con-

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dition for f t o be considered as a probability distribution [ 15 ]. The hyperbolic character of eq. (3) leads to finite speeds of propagation for probability perturbations, in contrast to eq. ( 1 ), which has the form of a diffusion equation with an infinite speed of propagation. The inclusion of inertial effects in constitutive equations to avoid such an infinite speed is well known, for instance, in the context of heat conduction, where the relaxation terms account for experimentally observed phenomena such as thermal waves. This topic has been extensively reviewed in refs. [5,16]. However, the fundamental feature of GTE is that, being second order differential equations in time, their solutions require two initial conditions, such as f ( x, to) and [Of(x, to ) / Ot] (to). This implies that the microscopic state of the system is not completely determined by the value of the distribution function at a single point in time, since different values of Of(x, t)/Ot at that instant would lead to different evolutions. For this reason, it should not be surprising that the Boltzmann H function, which only containsf, is inadequate for kinetic equations of this type. The purpose of this Letter is thus to examine the H theorem in the context of (3). We take this equation, whose motivations have been examined just above, as a model equation to obtain a better understanding of the second law out of equilibrium, in conditions where the usual ideas of local equilibrium are no longer valid. This has also becn a source of inspiration in phenomenological nonequilibrium thermodynamics [ 5,17 ]. We first show that the Boltzmann H function does not lead to a positive rate for an equation of the GTE type and propose its generalization in order to obtain a monotonous functional; later on, we complete the H theorem by showing h o w f ( x , t) reaches equilibrium at long times. Since (3) stands for a constant temperature situation, we evaluate the destruction of dynamical free energy ~ = U- TH, with U= f dxf(x, t) U(x) the internal energy of the system, instead of the entropy production. The evolution of M is simply

d ~ t - f dx~t

.

a:f (kTln f+ U) dMdt--r dr-~ - f dx Do f [kTVf+ (VU)f]. [krVf+ (VU)f] (5) as can be easily seen after an integration by parts taking into account that there is no flux at the boundary. The second term is obviously negative, since the friction coefficient is positive, but the first one has no definite sign. It cannot be immediately seen whether (4) or ( 5 ) has always a well defined sign. In principle, one could argue that since eq. (3) is a second-order differential equation in time, the initial conditions for OflOt may be chosen arbitrarily. However, the condition that f must remain positive is not generally satisfied unless Of/Ot= 0 at the initial time [ 15,18]. In the appendix we present a simple example which shows that the sign in (4) may be either positive or negative under several circumstances. This particular example shows that the evolution equation (3) does not satisfy in general the H theorem. At this point, one can adopt several attitudes. One is to consider that eq. (3) is not a good equation for a distribution function; this does not seem to be the case since several systems are described for such an equation. Another point of view is to accept that this dynamics does not admit a H theorem, that is to say, to consider that it is a counterexample to the possibility of extending the second law to nonequilibrium situations in a general way. Finally, another position - the one which we adopt here - is to look for a more general version of the H theorem, in order to preserve the existence of a microscopic analog of the macroscopic entropy, with the same essential feature of having a positive production. In analogy with external irreversible thermodynamics, where the dissipative fluxes are independent variables in the generalized entropy [ 5 ], we conceive a H function that not only depends on the distribution function, f, but also on the probability flux j, which is considered as an independent variable, and propose

(4) H=

Substituting

Of/Ot from eq. (3) one obtains

30 November 1992

--k

fd dx [ f l n f + o t ( J ) j 2] ,

(6)

27

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with a ( f ) a scalar parameter which in general depends on f. Notice that in the equilibrium state j = 0 and H = - k f dxf~q lnf~q gives the equilibrium entropy. Furthermore, in the case a = 0 one recovers the Boltzmann expression. The generalized H function leads us to a generalized free energy

~¢= f dx [f(U+kTlnf)+kTctj2l

(7)

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which is the same as eq. (3) identifying r - 2 k T a D and Do-D/f. This identification allows us to obtain an explicit expression for a in terms of dynamical magnitudes, T

a = 2kTDof"

(13)

We can thus write the generalized H function (6) in the more explicit form

and a generalized free energy rate

H = _ k f dx(flnf+2--~TDj2). d ~ _ f dx [ (Of/Ot) U+kTlnf) dt + kTot~20f/ Ot+ 2kTaj.Oj/ dt ] ,

(8)

where we denote dot ( f ) / d f a s ct'. Using the balance equation

Of(x, t) + V . j ( x , t)=O Ot

(9)

and after an integration by parts, one easily obtains d~¢ I

dt

d x j . [V( U + kTln f + kTc~j 2)

+ 2kTaaj/Ot] .

(10)

To preserve the negative character o f d ~ / d t one finds that j cannot be independent of the term inside the brackets; the simplest relation is

j(x, t)= -D(x, t)[V(U+kTlnf+kTa'j 2) +2kTctOj/Ot],

OflOt. Up to now, it has been proved that the GTE provide a dynamics for f f o r which the ~¢[f, j ] functional decreases with increasing time. To complete the H theorem it must be shown that the distribution function reaches its equilibrium value for asymptotic long times. To do that we use the known method of eigenfunction expansion [ 3 ]. We rewrite (3) as

O2f(x, t) Ot 2

q-

Of(x, t)

O-"""~--

F(x)f(x,t),

(15)

(l l)

fast perturbations around equilibrium, and for small values of j, the term in j2 can be neglected as usual in front of the one in Oj/Ot since it is a correction of higher order [5 ]; obviously, this is the case when ot # ot (f). In a situation where the probability flux is high enough, other terms, such asj4,j6 .... should be taken into account both in the generalized H function and in eq. ( 11 ). For a D independent of x one gets, introducing ( l l ) into (9)

02f. Of 2kTaD~-fi . O--t

28

The new term in H accounts for some details of the dynamics of the distribution function. This H function has no longer a static magnitude: two states which at to have the same configuration (same f ) can give different values of H, since it also depends on the value of j, which is an independent variable. In this simple case, j not only depends on f, but on

T

D being a scalar function of x and t. In the study of

= V. { (D/f) [kTVf+ (VU)f] },

(14)

with F the linear operator

F(x)f=-V.Do[kTVf+ ( V U ) f ] .

The probability density f ( x , t) can be expanded in terms of the eigenfunctions of the adjoint operator F * , f ~ ( x ) , as [3]

f(x, t ) = ~ au(t)f~(x)f~q(X), p=O

(17)

where ap(t) = f dxf~(x)f(x, t) .

( 12 )

(16)

(18)

After introducing ( 17 ) into ( 15 ), one directly gets

Volume 171, number 1,2 d2 d ) t ~ - i + ~-~ +20

av(t)=O

PHYSICS LETTERSA

(19)

(Ao indicating the p th eigenvalue ), whose solution is a combination of exponentials,

ap(t)=Aexp(-t/r~)+Bexp(-t/z;) ,

(20)

with characteristic times t~=

2t . 1 -+x/l --4t20

(21)

which eq. (3) does not satisfy the usual H theorem, i.e., the sign of (4) has not a well-defined sign or, in other words, the free energy does not decrease monotonically. A system from x = 0 to x f L , with U(x) = 0 (and thus, f ~ = 1 / L ) suffers a small perturbation of the form 8./'--(Sf)ocos(2~x/L), in such a way that f dxf(x)= 1, with f=f~+Sf. We also assume that (Of/Ot)o=O, to ensure that fwiU always be positive [ 15,18 ]. Introduction of ~if in eq. (3) gives for the decay of the perturbation 8f,

8f(x, t) = (6f)o c o s ( 2 ~ x / L )

It can be shown also that all the eigenvalues 2p are positive - except the one related to f o = f ~ that is obviously zero, and for which normalization conditions require ao= 1 [3]. This implies that the real part of z~ is positive for any value of Jlo, and consequently ao--,0 for long enough times, that is to say

ftx, t)=f~tx)+ ~ aott)f*rtx)f~(x)

Xexp( - t / 2 r ) [cos(Bt) + (2tB) - t sin (Bt) ] ,

B=(2z)-1(4Az-l) t/2, where A=(2x/L) 2 ×DokT. This solution is valid provided z is large

with

enough to satisfy 4Az ~t, I. Then the evolution of the free energy according to (4) will be given, up to order (Sf) 2, by

dA

p>O

dt

'-~.fm(x)

30 November 1992

(22)

as we wanted to show. In summary, we have proved that is is possible to formulate a H theorem for the generalized telegrapher equation under the condition of generalizing the Boltzmann H function. The generalized expression reduces to the equilibrium entropy in equilibrium situations, but it differs from the Boltzmann H function out of equilibrium. Fruitful discussions with Professor Garcia-Coltn are acknowledged. J.C. wants to thank sincerely all the members of the Statistical Physics Group of the Universidad Aut6noma de Mexico for their warm welcome during a stay in this group, as well as for financial support. J.C. also acknowledges a scholarship in the program Formaci6n de Personal Investigador of the Spanish Ministry of Education. The present work has been supported in part by the Direcci6n General de Investigaci6n Cientifica y Tdcnica of the Spanish MEC under grant PB/90-0676.

Appendix In this appendix we present a simple situation in

-

k TLA 2n ( 4 A t - 1 )f~

(gf)o2 exp( -

t/z)

× [sin2(Bt) + 2 t B sin(Bt) cos(Bt) ] .

Then dA/dt will not decrease monotonically if 2Bt > I. This particularexample shows that the evolution equation (3) does not always satisfy the H theorem in its classicalform.

References [ 1] H. Risken, The Fokker-Planck equation (Springer, Berlin, 1984). [2] N.G. van Kampen, Stochastic processes in pysics and chemistry (North-Holland, Amsterdam, 1981). [31 M. Doi and S.F. Edwards, The theory of polymerdynamics (Oxford Univ. Press, Oxford, 1986). [4] S. Harris, An introduction to the theory of the Boltzmann equation (Holt, Rinehart and Winston, New York, 1971). [5] D. Jou, J. Casas-V/~.quezand G. Lebon, Rep. Prog. Phys. 51 (1988) 1105. 16] H.D. Weymann,Am. J. Phys. 35 (1965) 48g. [71S. Goldstein, Q. J. Mech. Appl. Math. 4 (195!) 129. [ 8 ] J. Maaoliver,K. Lindenbergand G.H. Weiss,PhysicaA 157 (1989) 891. [9] J. Masoliver, J.M. Porrb and G.H. Weiss, Physica A 182 (1992) 593. [10]W. Horsthemke and R. Lefever, Noise induced phase transitions (Springer, Berlin, 1983). [ 111 C.W. Gardiner, Handb(mkof stochastic methods (Springer, Berlin, 1983). 29

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[ 12 ] H.C. Brinkman, Physica 22 ( 1956 ) 29. [ 13] R.A. Sack, Physica 22 (1956) 917. [14]E.W. Montroll and B.J. West, Stochastic processes, in: Fluctuation phenomena, eds. E.W. Montroll and J. Lebowitz (North-Holland, Amsterdam, 1979). [ 15 ] M.O. Hongler and L. Streit,Physica A 165 (1990) 196.

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[16] D.D. Joseph and L. Preziosi, Rev. Mod. Phys. 61 (1989) 41; 62 (1990) 375. [ 17 ] C. Truesdell, Rat/onal thermodynamics (McGraw-Hill, New York, 1985). [ 18 ] P.C. Hemmer, Physica A 271 ( 1961 ) 79.