Lyapunov and free energy functionals of generalized Fokker–Planck equations

5 November 2001

Physics Letters A 290 (2001) 93–100 www.elsevier.com/locate/pla

Lyapunov and free energy functionals of generalized Fokker–Planck equations T.D. Frank Faculty of Human Movement Sciences, Vrije Universiteit, Van der Boechorststraat 9, 1081 BT Amsterdam, The Netherlands Received 2 July 2001; accepted 28 September 2001 Communicated by J. Flouquet

Abstract We present the derivation of Lyapunov and free energy functionals for generalized Fokker–Planck equations including those proposed by Desai and Zwanzig, Kuramoto, Kaniadakis and Quarati, and Plastino and Plastino. Thus, we demonstrate a common feature of processes described by generalized Fokker–Planck equations: the decrease of an appropriately defined free energy.  2001 Elsevier Science B.V. All rights reserved. PACS: 05.20.-y; 05.40.+j

1. Introduction Fluctuations can be observed in various disciplines ranging from laser physics [1,2] to human movement sciences [3–5] and from the physics of superconductors [6] to population dynamics [7]. They have been modeled by conventional, linear Fokker–Planck equations with great success, e.g., [8–13]. Equally successful has been the application of generalized Fokker– Planck equations. Generalized Fokker–Planck equations with coefficients depending on mean fields can describe the dynamics of cross bridges during muscular contractions [14], the formation and breakup of swarms [15], entrainment of mutually coupled oscillators and synchronized oscillatory brain activity [16–22], charge density waves and spin-glasses [23], order–disorder transitions resembling phase transitions in ferromagnets [24–26], human group behav-

E-mail address: [email protected]. (T.D. Frank).

ior [27,28], and noise-induced shifts of phase transition points in spatially extended systems [29]. Generalized Fokker–Planck equations with nonlinear drift and diffusion terms have been discussed in the context of the nonextensive thermostatistics [30–43] proposed by Tsallis [44,45] and in the context of particles obeying exclusion or inclusion principles (e.g., fermions and bosons) [46,47] and can describe various nonlinear diffusion phenomena [48–50]. A salient feature that all the aforementioned phenomena have in common is irreversibility. Irreversibility of processes described by Fokker–Planck equations can be illustrated by means of Lyapunov functionals. For conventional linear Fokker–Planck equations Lyapunov functionals have been proposed on the basis of the Kullback distance measure and generalizations of it [51–54]. For a generalized Fokker–Planck equation of the Desai–Zwanzig type a Lyapunov functional was derived by Shiino [25,26]. Most recently, for other generalized Fokker–Planck equations Lyapunov functionals have been discussed [22,28,37,47].

0375-9601/01/$ – see front matter  2001 Elsevier Science B.V. All rights reserved. PII: S 0 3 7 5 - 9 6 0 1 ( 0 1 ) 0 0 6 3 8 - 7

94

T.D. Frank / Physics Letters A 290 (2001) 93–100

Although these studies hint at the existence of a unique Lyapunov functional for generalized Fokker–Planck equations, such a functional has not yet been derived. In this Letter, we will derive a Lyapunov-like functional for a generalized Fokker–Planck equation that includes as special cases the examples described above. Furthermore, we will discuss this functional in the context of free energy measures.

2. Irreversibility and generalized Fokker–Planck equations Let P(x, t) denote the probability density at time t of a stochastic process defined on a one-dimensional phase space Ω, that is, x ∈ Ω. We consider natural boundary conditions (i.e., Ω = R, P(x → ±∞) = 0) or periodic boundary conditions (i.e., Ω = [−a, a] with a > 0 and P(−a, t) = P(a, t)). We further assume that the stochastic process satisfies the generalized Fokker–Planck equation  ∂
∂ P(x, t) = − M P(x, t) ∂t ∂x  
  ∂
× h(x) + k[x, P] P(x, t) − D P(x, t) . ∂x    J

(1)

That is, this process is characterized by a probabilitydependent diffusion coefficient D(P) (related to generalized thermostatistics [42], exclusion–inclusion principles [46,47], or nonlinear diffusion phenomena [48–50]), a mean field interaction term k[x, P] [17, 24], and a probability-independent drift force h(x). Note that D(z) is a function of z, whereas k[x, P] is a functional of P and may also depend explicitly on the state variable x. In Eq. (1) we have introduced the probability-dependent coefficient M(z). In analogy to conventional linear Fokker–Planck equations, M may be interpreted as a probability-dependent mobility of the process under consideration and, consequently, we require M(z) > 0. In general, Fokker–Planck equations such as given by Eq. (1) have multiple stationary solutions [21,24]. Here, we require the existence of at least one stationary solution Pst and that in the stationary case the mobility-independent probability current

J vanishes. That is, Pst satisfies
 ∂ h(x) + k[x, Pst] Pst (x) = (2) D(Pst ). ∂x By definition, Eq. (1) unifies several special cases of generalized Fokker–Planck equations ranging from mean field Fokker–Planck equations of the Desai– Zwanzig type [24] and Kuramoto type [17] to the thermostatistical generalized Fokker–Planck equations proposed by Kaniadakis and Quarati [46] and by Plas˜ tino and Plastino [42]. For M(z) = 1, k[x, P] = k(x), and D(P) = QP(x, t) with Q > 0 Eq. (1) recovers the ordinary Fokker–Planck equation  ∂
∂ ˜ P(x, t) = − h(x) + k(x) P(x, t) ∂t ∂x ∂2 + Q 2 P(x, t). (3) ∂x One of the most essential principles in the theory of generalized Fokker–Planck equations is that these equations indeed generalize the conventional linear case. Consequently, generalized Fokker–Planck equations usually include linear Fokker–Planck equations as special cases. In what follows, we assume that there exists a limiting case in which Eq. (1) reduces to a linear equation of form (3). We denote this limiting case as “lin-lim”. For example, Fokker–Planck equations related to the nonextensive one-parametric entropy proposed by Tsallis become linear when the parameter converges to unity [42]. We can then define k˜ and the fluctuation strength Q in Eq. (2) by ˜ k(x) := lin-lim k[x, P], 1 Q := lin-lim D(P). (4) P We transform now Eq. (1) into a nonlinear diffusion equation. By means of the functional x



k x , P(x , t) P(x , t) dx
 + D P(x, t) ,

N[x, P] := −

(5)

Eq. (1) can equivalently be expressed as ∂ ∂ P(x, t) = − M(P) ∂t ∂x    ∂

× h(x)P(x, t) − N P(x, t) . ∂x

(6)

T.D. Frank / Physics Letters A 290 (2001) 93–100

Introducing the functional

 f u(x) := u(x0 ) x

1 1 dN[x , u(x )] × exp dx , Q u(x ) dx

2  f [P] ∂ dx
0. ln × ∂x f [Pst ] 

(7)

x0

∂ ln f [P] 1 ∂N[x, P] (8) = , ∂x QP ∂x d ln f [Pst ] 1 dN[x, Pst] h(x) (9) = = , dx QPst dx Q and    ∂ ∂ ∂ f [P] P(x, t)M(P) ln , P(x, t) = ∂t ∂x ∂x f [Pst ] (10) which is the desired nonlinear diffusion equation. Let us dwell on the functional f for a moment. From Eq. (9) it follows

x 1 h(x ) dx , f [Pst] = C exp (11) Q where C > 0 is an integration  x constant. By means of the potential V (x) := − h(x ) dx and the Boltzmann distribution W of V described by exp{−V (x)/Q} , exp{−V (x )/Q} dx Ω

(12)

˜ Eq. CW(x) with C˜ := C ×  (11) reads f [Pst ] = −1 [ Ω exp{−V (x )/Q} dx ] . Consequently, the functional f establishes a link between the stationary probability density Pst of the generalized Fokker–Planck equation (1) and the Boltzmann distribution W. Moreover, if f is invertible we can obtain Pst according to ˜ Therefore, we may interpret f −1 as Pst = f −1 [CW]. a distortion functional and f as the corresponding inverse operator: the inverse distortion functional [33, 36,37]. The form of Eq. (10) suggests that the functional   f [P] ∂P dx DI[P, Pst](t) := ln (13) f [Pst ] ∂t Ω

is smaller than or equal to zero. In fact, substituting Eq. (10) into Eq. (13) and integrating by parts yields DI[P, Pst](t) = − P(x, t)M(P) Ω

(14)

Consequently, on the basis of Eq. (13) we can construct a monotonically decreasing functional given by t I := DI(t ) dt . In more detail, we get    f [P] ∂P I [P, Pst](t) = dt (15) dx ln f [Pst ] ∂t Ω   d f [P] ∂P ⇒ I [P, Pst](t) = ln dx
0. dt f [Pst] ∂t t

that acts on a probability density u(x), we obtain

W(x) := 

95



Ω

(16) In the limiting case in which the generalized Fokker– Planck equation (1) becomes linear (i.e., M(z) → 1, ˜ k[x, P] → k(x), and D(z) → Qz) the inverse distortion functional (7) reduces to lin-lim f [u] = u(x) × x ˜ ) dx } and Eq. (16) yields exp{−Q−1 x0 k(x   P ∂P d dx lin-lim I [P, Pst ](t) = ln dt Pst ∂t Ω   d P = P ln dx dt Pst Ω   P dx, ⇒ lin-lim I [P, Pst ](t) = I0 + P ln Pst Ω

(17) where I0 denotes an arbitrary integration constant and the normalization condition was taken into account  (i.e., d Ω P dx/dt = 0). Hence, we realize that the functional I recovers the Kullback distance measure [8] in the linear limit. In sum, functional (15) may be regarded as (i) a generalization of the conventional Lyapunov functional for linear Fokker–Planck equations (i.e., the Kullback distance measure), and (ii) a measure indicating the irreversibility of stochastic processes described by Eq. (1).

3. Lyapunov functionals and free energy measures A functional L[P, Pst ] that satisfies the relations L  L0 > −∞, ∂ P =0 ∂t

⇔

d L
0, dt d L=0 dt

(18)

96

T.D. Frank / Physics Letters A 290 (2001) 93–100

is often called a Lyapunov functional [9]. Here, L0 is a constant. If there exists a Lyapunov functional L for the generalized Fokker–Planck equation (1), we can conclude that any transient solution P(x, t) of Eq. (1) converges to a stationary one [8,9,26]. On account of inequality (16), when searching for a Lyapunov functional of the generalized Fokker–Planck equation (1), the functional I defined by Eq. (15) is a promising candidate. In addition, I is constant in the stationary case, that is, ∂P/∂t = 0 ⇒ dI /dt = 0. In order to verify the boundedness and the implication dI /dt = 0 ⇒ ∂P/∂t = 0, we need to eliminate the factor ∂P/∂t in Eq. (15). To this end, we write Eq. (16) as ∂P d I [P, Pst ](t) = ln f [Pst ] dx dt ∂t Ω ∂P − ln f [P] (19) dx, ∂t Ω

integrate Eqs. (8) and (9) with respect to x and substitute the results into Eq. (19). Thus, we obtain  1 V (x) P (x,t ) + INL [P, Pst ](t), (20) Q 1 ∂P d INL [P, Pst](t) = dx dt Q ∂t

I [P, Pst ](t) =

 x × x0

Ω

  ∂ 1

dx N x , P(x , t) , P(x , t) ∂x

(21)

where . . .P (x,t ) denotes the average with respect to the probability density P(x, t). In the following, we will discuss several cases in which explicit analytical expressions can be derived for the nonlinear part INL . In addition, the functionals I thus obtained will be interpreted as free energy measures. 3.1. Systems with mean field interactions In order to study systems with mean field interactions, we set M(P) = 1 and D(P) = QP which x implies N[x, P] = − k[x , P(x , t)]P(x , t) dx + QP(x, t). Then, Eqs. (20) and (21) read I [P, Pst ](t) =



 1 V (x) P (x,t ) − SBGS P(x, t) Q + IMF [P, Pst ](t),

(22)

1 d IMF [P, Pst](t) = − dt Q

dx

∂P ∂t

Ω

 x ×





dx k x, P(x , t)



 , (23)

x0

where SBGS denotes the Boltzmann–Gibbs–Shannon  entropy, that is, SBGS [P] := − Ω P ln P dx [1]. 3.1.1. Desai–Zwanzig model For mean field systems of the Desai–Zwanzig type [24,28] the state variable x is defined on the real line and the mean field is usually established as the mean value of the process, that is, we have k = kDZ[x, P] = hDZ (xP ). Introducing the mean field potential z VDZ (z) := − hDZ (z ) dz , we can rewrite the righthand side of Eq. (23) as RHS = −

hDZ (xP ) Q

∞ (x − x0 ) −∞

∂P dx ∂t


 1 d = VDZ xP . Q dt

(24)

Then, the functional I reads I [P](t) =

V (x)P + VDZ (xP ) − SBGS [P]. Q

(25)

Following Shiino, the expression V (x)P + VDZ (xP ) may be interpreted as the averaged total energy U  of the system under consideration [25,26]. Then, F := QI = U  − QSBGS can be viewed as a measure of the system’s free energy. For globally attractive potentials V and VDZ it can be shown that the implication dI /dt = 0 ⇒ ∂P/∂t = 0 holds and that I is bounded from below, that is, I is a Lyapunov functional [34]. 3.1.2. Kuramoto model We consider now stochastic processes that are subjected to periodic boundary conditions (i.e., x ∈ Ω = [−a, a] and a > 0) and depend on a mean field k = a kK [x, P] = −a Γ (x − y)P(y, t) dy as proposed by Kuramoto [17]. By means of the potential VK (z) := z − Γ (z ) dz we can cast Eq. (23) into the form d 1 IMF [P, Pst](t) = dt Q

a a P(y, t) −a −a

∂P(x, t) ∂t

T.D. Frank / Physics Letters A 290 (2001) 93–100

× VK (x − y) dx dy.

(26)

For antisymmetric functions Γ or symmetric potentials VK (i.e., Γ (z) = −Γ (−z), VK (z) = VK (−z)) we obtain  a a  1  1 I [P](t) = P(y, t)P(x, t) V (x) P + Q 2 −a −a  × VK (x − y) dx dy − SBGS [P].

(27)

An interpretation of the mean field interaction term in Eq. (27) becomes apparent when we bear in mind that according to the mean field theory the joint probability density P(x, y, t) factorizes for large systems like P(x, y, t) ≈ P(x, t)P(y, t) [5,24]. Consequently, the mean field interaction term in Eq. (27) may be seen as the mean field approximationof the exact averaged P(x, y, t)VK(x − interaction energy Exy  = 0.5 y) dx dy. In this case, the functional  a a   1 F := QI = V (x) P + P(y, t)P(x, t) 2 −a −a  × VK (x − y) dx dy − QSBGS [P]   ≈ V (x) P + Exy  − QSBGS [P]

97

coincide with equilibrium distributions that maximize entropy functionals. This phenomenological approach establishes a link between transient properties determined by generalized Fokker–Planck operators and stationary properties related to generalized thermo ˜ statistics and entropies. Let S[P] = S(P) dx denote an entropy with a concave entropy kernel S˜ (i.e., 2 < 0). Furthermore, let us define the oper˜ d 2 S(z)/dz ˆ ator L[y(x)] := y − x dy/dx that is reminiscent of the Legendre transformation. Then, the stochastic process described by the generalized Fokker–Planck equation  ∂
∂ P(x, t) = − M P(x, t) ∂t  ∂x   ∂
× h(x)P(x, t) − Q Lˆ S˜ P(x, t) (29) ∂x has a stationary distribution that maximizes the entropyS under the constraints of a canonical ensemble (i.e., Pst dx = 1 and V (x)Pst = V0 ) [33,36]. Note that for M = 1 and S = SBGS (Boltzmann–Gibbs– Shannon entropy) Eq. (29) becomes linear. Eq. (29) was first studied by Plastino and Plastino [42] for the special case of the generalized nonextensive entropy proposed by Tsallis [44,45]. By comparison with ˆ Eq. (1), we obtain k[x, P] = 0 and D(P) = QL[P], ˆ which implies N[P] = QL[P], see Eq. (5). In this case, on account of the identity x

Q (28)

can be regarded again as a free energy measure. Furthermore, it can be shown that the functional I given by Eq. (27) is indeed a Lyapunov functional for the Kuramoto model with antisymmetric coupling function Γ [22]. 3.2. Nonequilibrium systems related to generalized nonextensive thermostatistics We discuss now two closely allied approaches to generalized Fokker–Planck equations with stationary probability densities that differ from Boltzmann distributions. In the following, we consider natural boundary conditions only. 3.2.1. Phenomenological approach Generalized Fokker–Planck equations can be derived such that their stationary probability densities

 1 ∂ ˆ

L P(x , t) dx P(x , t) ∂x x0   dS(z)  dS(z)  = −Q +Q , dz P (x,t ) dz P (x0 ,t )

(30)

Eq. (21) reads d INL [P, Pst ](t) = dt

∞

−∞

 ∂P dS(z)  dx ∂t dz P (x,t )

 ∞ dS(z)  ∂P dx − dz P (x0 ,t ) ∂t −∞    ⇒

INL [P] = S[P] + S0 ,

=0

(31)

where S0 is an arbitrary integration constant. From Eq. (20) it then follows  1 I [P] = − V (x) P (x,t ) − S[P] − S0 . (32) Q

98

T.D. Frank / Physics Letters A 290 (2001) 93–100

For S0 = 0, again, the expression F := QI = V (x)P (x,t ) − QS[P] can be interpreted as a free energy measure. By means of Eq. (9) we may express Eq. (32) in an alternative fashion which reads I [P] = ln Pst (x)P (x,t ) − S[P] − S0 . In this second form functional (32) was studied in [37]. There, it was shown that I satisfies relations (18) and therefore can be used as a Lyapunov functional for the processes defined by the generalized Fokker–Planck equation (29). In [37] it was also shown that the boundedness of I ˜ results from the concavity of S. 3.2.2. Kinetical interaction principle Kaniadakis and Quarati [46] and, recently, Kaniadakis [47] studied many particle systems subjected to probability-dependent transition probabilities. According to the kinetical interaction principle, in general, the probability of a transition between an initial and a target position depends on the probabilities to find the system in these initial and target positions. In this approach, the focus is on the kinetics of many particle systems and, in particular, on the impacts of heat baths and the interactions between the systems’ particles [47]. In this context inclusion and exclusion principles can be easily addressed. In line with these studies, we consider a stochastic process described by the generalized Fokker–Planck equation
 ∂ ∂ P(x, t) = − P(x, t)M P(x, t) ∂t  ∂x 
 ∂ × h(x) − Q ln κ P(x, t) , ∂x

(33)

where κ(z) is a monotonically increasing function. Eq. (33) may be considered as the Smoluchowski limit [8] of the generalized Fokker–Planck equation derived by Kaniadakis [47, Eq. (19)]. In this case, the function κ describes the dependency of transition probabilities on the process probability density P. In the absence of such a dependency we obtain κ(z) = z. Then, for M = 1 Eq. (33) becomes linear with respect to P. Comparing Eq. (33) with the Fokker–Planck equation (1), we can make the following identificax tions: k[x, P] = 0 and D(P) = Q dx P(x , t) × ∂ ln κ(P(x , t))/∂x . From Eqs. (5) and (7) we can compute the inverse distortion functional f and obtain f [u] = κ0 κ(u(x)) with κ0 = u(x0 )/κ(u(x0)). That is, apart from a proportional constant, the inverse distortion functional f employed in [33,36,37] agrees with

the deformation function κ discussed in [47]. Inserting this result into Eq. (15) and changing the order of the integrations, we obtain  t    κ(P) ∂P

dt ln . (34) I [P, Pst](t) = dx κ(Pst ) ∂t Ω

Since κ(z) is a function of z, Eq. (34) can be written as I [P, Pst](t) =

 P (x,t )  dx dZ ln

Ω

κ(Z) κ(Pst(x))

 .

(35) The functional K = −I has previously been derived [47]. In [47] it has been shown that I (or −K) is a Lyapunov functional which can be expressed in terms  P dZ × of the entropy measure Sκ [P] := − Ω dx −1 ln κ(Z)  x as I = −Sκ [P] + Q V (x)P with V (x) = − h(x ) dx . Consequently, we arrive once again at Shiino’s interpretation of generalized Lyapunov functionals as generalized free energy measures: F := QI = V P − QSκ [P]. 4. Conclusions Although the functional I defined by Eq. (15) or by Eqs. (20) and (21) does not look appealing, as argued in the following it can be regarded as a Lyapunovlike functional associated with generalized Fokker– Planck equations given by Eq. (1). First, the functional I decreases monotonically for transient solutions of Eq. (1). Second, the functional recovers the Kullback distance measure in the linear limit. Finally, in several special cases the functional reduces to Lyapunov functionals discussed in the literature (see Section 3). In addition, the functional I prompts the study of further special cases of Eq. (1) in which the factor ∂P/∂t occurring in I and INL can be eliminated and I and INL can be expressed in terms of the probability density only. For example, we may consider mean field interactions similar to the Desai–Zwanzig type r with k = kr [x, P] = rx r−1 r (x P ) and r  0. Using h z the potential Vr (z) := − hr (z ) dz , from Eqs. (22) and (23) we then obtain IMF [P, Pst ](t) =

Vr (x r P ) Q

T.D. Frank / Physics Letters A 290 (2001) 93–100

V (x)P + Vr (x r P ) − SBGS [P]. Q (36) Moreover, the functional I facilitates the study of systems with mean field couplings in the context of generalized nonextensive thermostatistics and the kinetical interaction principle. That is, it provides a tool to examine the interaction between mean fields and nonextensivity or inclusion and exclusion principles. For example, we may consider a stochastic process characterized by the above mentioned mean field interaction kr and a generalized entropy functional S[P] = ˜ Ω S(P) dx. Then, the evolution equation (1) is given by ⇒

I [P](t) =

∂ ∂ P(x, t) = − M(P) ∂t ∂x 

  × h(x) + rx r−1hr x r P(x, t)   ∂ ˆ ˜ − Q L S(P) , (37) ∂x and by analogy with Eq. (36) the corresponding functional I reads I [P](t) =

V (x)P + Vr (x r P ) − S[P]. Q

(38)

Acknowledgements I am indebted to Giorgio Kaniadakis for providing me with his manuscript [47] prior to publication. I wish to thank Andreas Daffertshofer for helpful discussions and Prof. Peter Beek for supporting my studies on generalized Fokker–Planck equations.

References [1] H. Haken, Synergetics. An Introduction, Springer, Berlin, 1977. [2] H. Haken, Light—Laser Light Dynamics, Vols. 1, 2, NorthHolland, Amsterdam, 1985. [3] H. Haken, Principles of Brain Functioning, Springer, Berlin, 1996. [4] R. Friedrich, S. Siegert, J. Peinke, S. Lück, M. Seifert, M. Lindemann, J. Raethjen, G. Deuschl, G. Pfister, Phys. Lett. A 271 (2000) 217. [5] T.D. Frank, A. Daffertshofer, P.J. Beek, Phys. Rev. E 63 (2001) 011905.

99

[6] F. Servant, S. Raymond, B. Fak, P. Lejay, J. Flouquet, Solid State Commun. 116 (2000) 489. [7] T.D. Frank, P.J. Beek, Phys. Rev. E (2001), in press. [8] H. Risken, The Fokker–Planck Equation—Methods of Solution and Applications, Springer, Berlin, 1989. [9] C.W. Gardiner, Handbook of Stochastic Methods, 2nd ed., Springer, Berlin, 1997. [10] R. Friedrich, J. Peinke, Phys. Rev. Lett. 78 (1997) 863. [11] J. Gradisek, S. Siegert, R. Friedrich, I. Grabec, Phys. Rev. E 62 (2000) 3146. [12] P. Hänggi, H. Thomas, Phys. Rep. 88 (1982) 207. [13] I. Santamaria-Holek, D. Reguera, J.M. Rubi, Phys. Rev. E 63 (2001) 051106. [14] H. Shimizu, Prog. Theor. Phys. 52 (1974) 329. [15] A.S. Mikhailov, D.H. Zanette, Phys. Rev. E 60 (1999) 4571. [16] S. Kim, S.H. Park, C.R. Doering, C.S. Ryu, Phys. Lett. A 224 (1997) 147. [17] Y. Kuramoto, Chemical Oscillations, Waves, and Turbulence, Springer, Berlin, 1984. [18] Y. Kuramoto, Int. J. Bifurc. Chaos 7 (1997) 789. [19] S.H. Strogatz, I. Stewart, Sci. Am. 269 (1993) 68. [20] P.A. Tass, Phase Resetting in Medicine and Biology—Stochastic Modelling and Data Analysis, Springer, Berlin, 1999. [21] T.D. Frank, A. Daffertshofer, C.E. Peper, P.J. Beek, H. Haken, Physica D 144 (2000) 62. [22] T.D. Frank, A. Daffertshofer, C.E. Peper, P.J. Beek, H. Haken, Physica D 150 (2001) 219. [23] L.L. Bonilla, J. Stat. Phys. 46 (1987) 659. [24] R.C. Desai, R. Zwanzig, J. Stat. Phys. 19 (1978) 1. [25] M. Shiino, Phys. Lett. A 112 (1985) 302. [26] M. Shiino, Phys. Rev. A 36 (1987) 2393. [27] M. Takatsuji, Biol. Cybern. 17 (1975) 207. [28] T.D. Frank, Phys. Lett. A 280 (2001) 91. [29] C. van den Broeck, J.M.R. Parrondo, J. Armero, A. Hernandez-Machado, Phys. Rev. E 49 (1994) 2639. [30] L. Borland, Phys. Rev. E 57 (1998) 6634. [31] L. Borland, F. Pennini, A.R. Plastino, A. Plastino, Eur. Phys. J. B 12 (1999) 285. [32] A. Compte, D. Jou, J. Phys. A: Math. Gen. 29 (1996) 4321. [33] T.D. Frank, A. Daffertshofer, Physica A 272 (1999) 497. [34] T.D. Frank, Phys. Lett. A 267 (2000) 298. [35] T.D. Frank, A. Daffertshofer, Physica A 285 (2000) 351. [36] T.D. Frank, A. Daffertshofer, Physica A 292 (2001) 392. [37] T.D. Frank, A. Daffertshofer, Physica A 295 (2001) 455. [38] T.D. Frank, Physica A (2001), in press. [39] E.K. Lenzi, C. Anteneodo, L. Borland, Phys. Rev. E 63 (2001) 051109. [40] L.C. Malacarne, R.S. Mendes, I.T. Pedron, E.K. Lenzi, Phys. Rev. E 63 (2001) 030101. [41] S. Martinez, A.R. Plastino, A. Plastino, Physica A 259 (1998) 183. [42] A.R. Plastino, A. Plastino, Physica A 222 (1995) 347. [43] C. Tsallis, D.J. Bukman, Phys. Rev. E 54 (1996) R2197. [44] C. Tsallis, J. Stat. Phys. 52 (1988) 479. [45] C. Tsallis, Braz. J. Phys. 29 (1999) 1. [46] G. Kaniadakis, P. Quarati, Phys. Rev. E 48 (1993) 4263. [47] G. Kaniadakis, Physica A 296 (2001) 405.

100

T.D. Frank / Physics Letters A 290 (2001) 93–100

[48] G.I. Barenblatt, V.M. Entov, V.M. Ryzhik, Theory of Fluid Flows through Natural Rocks, Kluwer Academic, Dordrecht, 1990. [49] Y.B. Zel’dovich, Y.P. Raizer, Physics of Shock Waves and High-Temperature Hydrodynamic Phenomena, Vol. 2, Academic Press, New York, 1967.

[50] [51] [52] [53]

H. Spohn, J. Phys. I France 3 (1993) 69. M.S. Green, J. Chem. Phys. 20 (1952) 1281. R. Graham, H. Haken, Z. Phys. 245 (1971) 141. L. Borland, A.R. Plastino, C. Tsallis, J. Math. Phys. 39 (1998) 6490, Erratum: 40 (1999) 2196. [54] M. Shiino, J. Phys. Soc. Jpn. 67 (1998) 3658.