A General Nonlinear Fokker-Planck Equation and Its Associated Entropy

Eur. Phys. J. B 58, 159–165 (2007) DOI: 10.1140/epjb/e2007-00217-1 THE EUROPEAN PHYSICAL JOURNAL B

A general nonlinear Fokker-Planck equation and its associated entropy

V. Schw¨ammle, E.M.F. Curado, and F.D. Nobrea Centro Brasileiro de Pesquisas F´ısicas, Rua Xavier Sigaud 150, Rio de Janeiro, RJ 22290-180, Brazil

Received 3 April 2007 Published online 10 August 2007 – c EDP Sciences, Societ`a Italiana di Fisica, Springer-Verlag 2007

Abstract. A recently introduced nonlinear Fokker-Planck equation, derived directly from a master equation, comes out as a very general tool to describe phenomenologically systems presenting complex behavior, like anomalous diﬀusion, in the presence of external forces. Such an equation is characterized by a nonlinear diﬀusion term that may present, in general, two distinct powers of the probability distribution. Herein, we calculate the stationary-state distributions of this equation in some special cases, and introduce associated classes of generalized entropies in order to satisfy the H-theorem. Within this approach, the parameters associated with the transition rates of the original master-equation are related to such generalized entropies, and are shown to obey some restrictions. Some particular cases are discussed.

PACS. 05.40.Fb Random walks and Levy flights – 05.20.-y Classical statistical mechanics – 05.40.Jc Brownian motion – 66.10.Cb Diffusion and thermal diffusion

1 Introduction Nevertheless, restrictions to the applicability of the BG statistical-mechanics formalism have been found in many The standard statistical-mechanics formalism, as pro- systems, including for instance, those characterized by posed originally by Boltzmann and Gibbs (BG), is con- nonlinearities, long-range interactions and/or long-time sidered as one of the most successful theories of physics, memories, which may present several types of anomalous and it has enabled physicists to propose theoretical models behavior, e.g., stationary states far from equilibrium [6–8]. in order to derive thermodynamical properties for real sys- These anomalous behaviors suggest that a more general tems, by approaching the problem from the microscopic theory is required; as a consequence of that, many at- scale. Such a prescription has led to an adequate descrip- tempts have been made, essentially by proposing general- tion of a large diversity of physical systems, essentially izations of the BG entropy [9–19]. Among these, the most those represented by linear equations and characterized successful proposal, so far, appears to be the one suggested by short-range interactions and/or short-time memories. by Tsallis [9], through the introduction of a generalized en- Although BG statistical mechanics is well formulated (un- tropy, characterized by an index q, in such a way that the der certain restrictions) for systems at equilibrium, the BG entropy is recovered in the limit q → 1. The usual same is not true for out-of-equilibrium systems, in such extensivity property of some thermodynamic quantities a way that most of this theory applies only near equilib- holds only for q =1,andifq = 1 such quantities do not rium [1–3]. One of the most important phenomenological increase linearly with the size of the system; this has led to equations of nonequilibrium statistical mechanics is the the so-called nonextensive statistical-mechanics formalism linear Fokker-Planck equation (FPE), that rules the time [6–8]. evolution of the probability distribution associated with a given physical system, in the presence of an external force Among many systems that present unusual behavior, field [4], provided that the states of the system can be ex- one should mention those characterized by anomalous dif- pressed by a continuum. This equation deals satisfactorily fusion, e.g., particle transport in disordered media. A pos- with many physical situations, e.g., those associated with sible alternative for describing anomalous-transport pro- normal diffusion, and is essentially associated with the BG cesses consists in introducing modifications in the stan- formalism, in the sense that the Boltzmann distribution, dard FPE. Within the most common procedure, one con- which is usually obtained through the maximization of the siders nonlinear FPEs [20], that in most of the cases BG entropy under certain constraints (the so-called Max- come out as simple phenomenological generalizations of Ent principle), also appears as the stationary solution of the usual linear FPE [21–32]. In these nonlinear sys- the linear FPE [4,5]. tems, interesting new aspects appear, leading to a wide range of open problems, in such a way that their in- a e-mail: [email protected] vestigation has led to a new research area in physics, 160 The European Physical Journal B with a lot of applications in natural systems. It is very and so, common to find the power-law-like probability distri- ∞ ∞ bution that maximizes Tsallis’ entropy as solutions of dx P (x, t)= dx P (x, t0)=1 (∀t). (4) some nonlinear FPEs [20–22,24,25,31–33]. It seems that −∞ −∞ the nonextensive statistical mechanics formalism appears to be intimately related to nonlinear FPEs, motivating In the present work we investigate further properties of an investigation for a better understanding of possible the nonlinear FPE of equation (1), finding stationary so- connections between generalized entropies and nonlinear lutions in several particular cases, and discussing its asso- FPEs [20–22,26,27,31,32,34–37]. ciated entropies, that were introduced in order to satisfy Recently, a general nonlinear FPE has been derived the H-theorem. In the next section we present station- directly from a standard master equation, by introducing ary solutions of this equation; in Section 3 we prove the nonlinear contributions in the associated transition prob- H-theorem by using equation (1), and show that the valid- abilities, leading to [38,39] ity of this theorem can be directly related to the definition of a general entropic form associated with this nonlinear ∂P(x, t) ∂(A(x)P (x, t)) ∂ ∂P(x, t) FPE. In Section 4 we discuss particular cases of this gen- = − + Ω[P (x, t)] ; eral entropic form and their associated nonlinear FPEs. ∂t ∂x ∂x ∂x Finally, in Section 5 we present our conclusions. Ω[P ]=aµP µ−1 + b(2 − ν)P ν−1, (1) where a and b are constants, whereas µ and ν are real exponents1. The system is in the presence of an exter- 2 Stationary state nal potential φ(x), associated with a dimensionless force x The nonlinear FPE of equation (1) is very general and cov- A(x)=−dφ(x)/dx [φ(x)=− A(x )dx ]; herein, we ers several particular cases, e.g., the one related to Tsallis’ −∞ thermostatistics [21,22,33]. In this section we will restrict assume analyticity of the potential φ(x) and integrability ourselves to a stationary state, and will derive the corre- of the force A(x) in all space. sponding solutions for particular values of the parameters In what concerns the functional Ω[P (x, t)], we are as- associated with this equation. suming its differentiability and integrability with respect Let us then rewrite equation (1) in the form of a con- to the probability distribution P (x, t), in such a way that tinuity equation, at least its first derivative exists, i.e., that it should be at least Ω[P ] ∈ C1. Furthermore, this functional should be a ∂P(x, t) ∂j(x, t) positive finite quantity, as expected for a proper diffusion- + =0; like term; this property will be verified later on, as a direct ∂t ∂x ∂P(x, t) consequence of the H-theorem. j(x, t)=A(x)P (x, t) − Ω[P (x, t)] , (5) As usual, we assume that the probability distribution, ∂x together with its first derivative, as well as the product A(x)P (x, t), should all be zero at infinity, in such a way that a stationary solution of equation (1), Pst(x), is associated with a stationary probability flux, jst(x) = constant, which becomes jst(x) = 0, when one ∂P(x, t) P (x, t)|x→±∞ =0; =0; uses equation (2). Therefore, using the functional Ω[P ]of ∂x x→±∞ equation (1), the stationary-state solution satisfies, A(x)P (x, t)|x→±∞ =0, (∀t). (2) µ−2 ν−2 ∂Pst(x) A(x)= aµPst (x)+b(2 − ν)Pst (x) , (6) The conditions above guarantee the preservation of the ∂x normalization for the probability distribution, i.e., if for ∞ which, after integration, becomes agiventimet0 one has that −∞ dx P (x, t0)=1,then a simple integration of equation (1) with respect to the µ µ−1 2 − ν ν−1 φ0 − φ(x)=a P (x)+b P (x), (7) variable x yields, µ − 1 st ν − 1 st ∞ ∂ ∞ where φ0 represents a constant. The equation above may dx P (x, t)=− [A(x)P (x, t)]−∞ ∂t −∞ be solved easily in some particular cases, e.g., ν = µ, ν = ∞ 2, and µ =0, ∂P(x, t) + Ω[P (x, t)] =0, (3) ∂x −∞ 1 α−1 1 − φ(x) 1 Pst(x)= (1) 1 ; Even though one could use a simpler notation for the func- Z φ0 + µ ν tional Ω[P ], e.g., Ω[P ]=a P +b P , herein we will keep the ∞ 1 φ(x) α−1 notation of equation (1), as appeared naturally in the deriva- Z(1) = dx 1 − , (8) tion of the above FPE, for a consistency with previous publi- φ0 + cations [38,39]. −∞ V. Schwämmle et al.: A general nonlinear Fokker-Planck equation and its associated entropy 161 where α = µ,inthecasesν = µ and ν =2,whereas corresponds to a well-defined sign for the time derivative of α = ν,ifµ = 0. In the equation above, [y]+ = y,fory>0, the free-energy functional defined in equation (12). Using and zero otherwise. Another type of solution applies for the definitions above, µ =2ν − 1, or ν =2µ − 1, ⎛ ⎞ ∞ ∞ 1 ∂F ∂ 1 1 α−1 = ⎝ dx φ(x)P (x, t) − dx g[P ]⎠ Pst(x)= 1 ± 1+K (φ(x) − φ0) ; (2) + ∂t ∂t β Z −∞ −∞ ∞ 1 (2) α−1 ∞ Z = dx 1 ± 1+K (φ(x) − φ0) , (9) 1 ∂g[P ] ∂P + = dx φ(x) − . (13) −∞ β ∂P ∂t −∞ where ⎧ − − Now, one may use the FPE of equation (1) for the time ⎪ 2a (2ν 1)(ν 1) − ⎨⎪ 2 − 2 , if µ=2ν 1(α=ν), derivative of the probability distribution; carrying out an b (2 ν) integration by parts, and using the conditions of equa- K = ⎪ tion (2), one obtains, ⎩⎪ 2b (3 − 2µ)(µ − 1) 2 2 , if ν =2µ−1(α=µ), a µ ∞ (10) ∂F dφ(x) − ≥ = − dx P (x, t) and we are assuming that [1 + K(φ(x) φ0)] 0. Some ∂t dx well-known particular cases of the stationary solutions −∞ presented above come out easily, e.g., from equation (8) 2 ∂P dφ(x) − 1 ∂ g[P ] ∂P one obtains, in all three situations that yielded this equa- +Ω[P ] 2 . (14) tion, the exponential solution associated with the linear ∂x dx β ∂P ∂x → ∝ − −1 FPE in the limit α 1[withφ0 (α 1) ], as well Usually, one is interested in verifying the H-theorem as the generalized exponential solution related to Tsallis − from a well-defined FPE, together with a particular en- thermostatistics, for α =2 q,whereq denotes Tsallis’ tropic form, in such a way that the quantities Ω[P ]and entropic index. ∂2g[P ]/∂P 2 are previously defined (see, e.g., Refs. [20, 40]). Herein, we follow a different approach, by assuming that the general equations (1) and (11) should be satisfied; 3 H-theorem and the associated entropy then, we impose the condition,

In this section we will demonstrate the H-theorem by mak- ∂2g[P ] Ω[P ] ing use of equation (1), and for that purpose, an entropic = −β , (15) ∂P2 P (x, t) form related to this equation will be introduced. Let us therefore suppose a general entropic form satisfying the in such way that, following conditions, ∞ ∞ 2 2 ∂F dφ(x) Ω[P ] ∂P d g ≤ = − dx P (x, t) + ≤ 0. (16) S = dx g[P (x)]; g[0] = 0; g[1] = 0; 2 0, (11) ∂t dx P (x, t) ∂x dP −∞ −∞ where one should have g[P (x, t)] at least as g[P (x, t)] ∈ It should be noticed that equation (15), introduced in such C2; in addition to that, let us also define the free-energy a way to provide a well-defined sign for the time deriva- functional, tive of the free-energy functional, yields two important conditions, as described next. ∞ 1 F = U − S; U = dx φ(x)P (x, t), (12) (i) Ω[P ] ≥ 0 [cf. Eq. (11)], which is expected for an ap- β propriate diffusion-like term. −∞ (ii) It expresses a relation involving the FPE of equa- where β represents a Lagrange multiplier, restricted to tion (1) and an associated entropic form, allowing for β ≥ 0. Furthermore, we will show that this free-energy the calculation of such an entropic form, given the functional is bounded from below; this condition, together FPE, and vice-versa. Since the FPE is a phenomeno- with the H-theorem [(∂F/∂t) ≤ 0),leads,afteralong logical equation that specifies the dynamical evolution time, the system towards a stationary state. associated with a given physical system, equation (15) may be useful in the identification of the entropic form associated with such a system. In particular, the 3.1 H-theorem present approach makes it possible to identify entropic forms associated with some anomalous systems, ex- The H-theorem for a system that exchanges energy with hibiting unusual behavior, that are appropriately de- its surrounding, herein represented by the potential φ(x), scribed by nonlinear FPEs, like the one of equation (1). 162 The European Physical Journal B

As an illustration of this point, let us consider the sim- the entropy. A similar result may also be obtained by prov- ple case of a linear FPE, that describes the dynamical ing the H-theorem using the master equation from which evolution of many physical systems, essentially those equation (1) was derived, with the transition probabilities characterized by normal diffusion. This equation may introduced in references [38,39] (see the Appendix). be recovered from equation (1) by choosing µ = ν =1, in such a way that Ω[P ]=a + b = D,whereD represents a positive constant diffusion coefficient with 3.2 Boundness from below units (time)−1. One may now set the Lagrange multiplier β = kB/D,wherekB represents the Boltzmann Above, we have proven that the free-energy functional de- constant; integrating equation (15), and using the con- creases in time, and so, for the existence of a stationary ditions of equation (11), one gets the well-known BG state at long times of an evolution process, characterized entropic form, by a probability distribution Pst(x), one should have that − g[P ]= kBP ln P. (17) F (P (x, t)) ≥ F (Pst(x)) (∀t). (22) In the next section we will explore further the relation of In what follows, we will show this inequality and find equation (15), by analyzing other particular cases. the conditions for its validity. Therefore, using equa- The simplest situation for which condition (i) above is tions (7, 11), and (12), we can write, satisfied may be obtained by imposing both terms of the functional Ω[P ] to be positive, which leads to ∞ µ µ−1 a ≥ 0; µ ≥ 0, or a<0; µ<0, (18) F (P (x, t)) = P (x, t) φ0 − a Pst (x) µ − 1 −∞ and ∞ b ≥ 0; ν ≤ 2, or b<0; ν>2. (19) 2 − ν ν−1 1 −b P (x) dx − g[P ]dx, (23) ν − 1 st β It should be stressed that it is possible to have Ω[P ] ≥ 0 −∞ with less restrictive ranges for the parameters above. How- ever, an additional property for the free-energy functional and so, of equation (12) to be discussed next, namely, the boundness from below, requires the conditions of equations (18) ∞ µ µ−1 and (19), with the additional restriction ν>0. F (Pst) − F (P )= (P − Pst) a Pst Now, integrating equation (15) for the general func- µ − 1 −∞ tional Ω[P ] of equation (1), and using the standard con- ∞ ditions for g[P ] defined in equation (11), one gets that 2 − ν ν−1 1 +b P dx + (g[P ] − g[Pst]) dx, (24) ν − 1 st β a 2 − ν −∞ g[P ]=−β P µ + b P ν µ − 1 ν(ν − 1) where we have used the normalization condition for the aν(1 − ν)+b(2 − ν)(1 − µ) + P . (20) probabilities. Now, we insert the entropic form of equa- (1 − µ)(1 − ν)ν tion (20) in the equation above to obtain,

This entropic form recovers, as particular cases, the BG ∞ entropy (e.g., when µ, ν → 1) and several generalized en- a µ F (Pst) − F (P )= Pst Γµ[P/Pst] tropies defined previously in the literature, like those in- µ − 1 troduced by Tsallis [9], Abe [10], Borges-Roditi [12], and −∞ Kaniadakis [17–19]. Such particular cases, as well as their b(2 − ν) ν + Pst Γν [P/Pst] dx, (25) associated FPEs, will be discussed in the next section. ν(ν − 1) For the simpler situation of an isolated system, i.e., φ(x) = constant, the H-theorem should be expressed in where, terms of the time derivative of the entropy, in such a way α that equation (14) should be replaced by Γα[z]=1− α + αz − z (α = µ, ν). (26) ∞ 2 By analyzing the extrema of the functional Γα[P/Pst], ∂S[P ] − ∂P ∂ g[P ] ∂P = dx Ω[P ] 2 one can see that Γα[P/Pst] ≥ 0for0<α<1and ∂t −∞ ∂x ∂P ∂x Γα[P/Pst] ≤ 0forα>1andα<0. Therefore, the in- ∞ 2 2 − ∂ g[P ] ∂P ≥ equality of equation (22) is satisfied for the range of pa- = dx Ω[P ] 2 0. (21) −∞ ∂P ∂x rameters specified by equations (18) and (19), if one con- siders additionally, ν>0. In this case all that one needs is the standard condition For the case of an isolated system, the stationary solu- associated with the FPE [same condition (i) above], i.e., tion turns out to be the equilibrium state, which is the one Ω[P ] ≥ 0, and the general restrictions of equation (11) for that maximizes the entropy. Therefore, one may use the V. Schwämmle et al.: A general nonlinear Fokker-Planck equation and its associated entropy 163 concavity property of the entropy [cf. Eq. (11)] in order to (d) Kaniadakis entropies [17–19]: the entropy proposed get, in references [17,18] is also a two-exponent entropic form, but slightly different from those presented in examples (b) ∞ and (c) above; it may be reproduced from equation (20) by choosing {a = b = D/[2(1 + q)],µ=1+q, ν =1− q}, S[Peq(x)]−S[P (x, t)] = (g[Peq] − g[P ]) dx ≥ 0(∀t). in such a way that −∞ (27) k 1 1 g[P ]=− P 1+q − P 1−q ; 2q 1+q 1 − q D 4Someparticularcases Ω[P ]= (P q + P −q). (31) 2 In this section we analyze some particular cases of the entropic form of equation (20) and using equation (15), However, the most recent entropy introduced by Ka- we find for each of them, the corresponding functional niadakis [19], characterized by three free parameters Ω[P ] of the associated FPE. In the examples that follow, {q, r, ζ}, we will set the Lagrange multiplier β = k/D,withk and k D representing, respectively, a constant with dimensions g[P ]=− ζqP r+q+1 − ζ−qP r−q+1 +(ζ−q − ζq)P ; of entropy and a constant diffusion coefficient. B q −q (a) Tsallis entropy [9]: this represents the most well-known B =(q + r)ζ +(q − r)ζ , (32) generalization of the BG entropy, which has led to the development of the area of nonextensive statistical me- may be identified with the general entropic form of equa- chanics [6–8]. One may find easily that equation (20) tion (20), for recovers Tsallis entropy in several particular cases, e.g., {b =0,a = D, µ = q}, {a =0,b = Dν/(2 − ν),ν = q}, µ = r + q +1; ν = r − q +1; { − } q −q and a = D/2,b = Dν/[2(2 ν)],µ = ν = q . For all a ζ b(2 − ν) ζ these cases one may use equation (15), in order to get the = D ; = −D , (33) µ − 1 B ν(ν − 1) B corresponding functional Ω[P ], q or yet, P − P q−1 g[P ]=k − ; Ω[P ]=qDP . (28) 1 q µ − ν µ + ν − 2 q = ; r = ; With the functional Ω[P ] above, one identifies the non- 2 2 linear FPE that presents the well-known q-exponential, 2 − ν 2q a ν(1 − ν) → − D = a + b ; ζ = . (34) or Tsallis distribution (replacing q 2 q), as a time- ν b (µ − 1)(2 − ν) dependent solution [21,22,26]. (b) Abe entropy [10]: this proposal was inspired in the In this case, considering the identifications above, the as- area of quantum groups, where certain quantities, usu- sociated FPE is given by equation (1). ally called q-deformed quantities, are submitted to defor- Except for the well-known example (a), i.e., Tsallis en- mations and are often required to possess the invariance tropy and its corresponding FPE, the other three particu- −1 q ↔ q . The Abe entropy may be obtained from equa- lar cases presented herein were much less explored in the −1 tion (20) in the particular case {a = D(q−1)/(q−q ),b= literature. Their associated FPEs, defined in terms of their −1 −Dq(q +1)/[(q − q )(q +2)],µ= −ν = q},forwhich respective functionals Ω[P ] above are, to our knowledge, P q − P −q presented herein for the first time. These equations, whose − nonlinear terms depend essentially in two different pow- g[P ]= k − −1 ; q q ers of the probability distribution, may be appropriated q(q − 1) q−1 q(q +1) −q−1 for anomalous-diffusion phenomena where a crossover be- Ω[P ]=D P − P . (29) q − q−1 q − q−1 tween two different diffusion regimes occurs [25]. (c) Borges-Roditi entropy [12]: this consists in another generalization of Tsallis entropy, where now one has two distinct entropic indices, q and q,withamoregen- 5Conclusion eral invariance q ↔ q; this case may be obtained from We have analyzed important aspects associated with a re- equation (20) by choosing {a = D(q − 1)/(q − q),b = cently introduced nonlinear Fokker-Planck equation, that Dq (q − 1)/[(q − q )(2 − q )],µ= q, ν = q }.Onegets, was derived directly from a master equation by setting P q − P q nonlinear effects on its transition rates. Such equation g[P ]=−k ; q − q is characterized by a nonlinear diffusion term that may present two distinct powers of the probability distribu- 1 − q−1 − − q−1 Ω[P ]=D q(q 1)P q (q 1)P . tion; for this reason, it may reproduce, as particular cases, q − q a large range of nonlinear FPEs of the literature. We have (30) obtained stationary solutions for this equation in several 164 The European Physical Journal B cases, and some of them recover the well-known Tsallis dis- equation one gets, tribution. We have proven the H-theorem, and for that, an ∂P1 µ µ ν−1 ν−1 important relation involving the parameters of the FPE n =1: = aP2 − aP1 + bP2P1 − bP1P2 , (37) and an entropic form was introduced. Since the FPE is ∂t ∂PW µ µ ν−1 ν−1 a phenomenological equation that specifies the dynamical n = W : = aPW−1 −aPW +bPW−1PW −bPW PW−1, evolution associated with a given physical system, such a ∂t relation may be useful for identifying entropic forms as- (38) sociated with real systems and, in particular, anomalous ∂Pn µ µ µ n =2, ..., (W − 1): = a Pn+1 + Pn−1 − 2aPn systems that exhibit unusual behavior and are appropri- ∂t ν−1 ν−1 ν−1 ately described by nonlinear FPEs. It is shown that, in the + bPn (Pn+1 + Pn−1) − bPn Pn+1 + Pn−1 , simple case of a linear FPE, the Boltzmann-Gibbs entropy (39) comes out straightforwardly from this relation. Consid- ering the above-mentioned nonlinear diffusion term, this wherewehavetreatedtheborders of the spectrum (n =1 relation yields a very general entropic form, which simi- and n = W ) separately from the rest. Let us now consider larly to its corresponding FPE, depends on two distinct W the entropy, S = g[Pn], satisfying the same conditions powers of the probability distribution. Apart from Tsal- n=1 lis’ entropy, other entropic forms introduced in the lit- specified in the text [see Eq. (11)]; the H-theorem, to be erature are recovered as particular cases of the present proven below, is expressed in terms of its time derivative, one, essentially those characterized by two entropic in- W dices. Nonlinear FPEs (as well as their associated entropic ∂S dg[Pn] dPn = ≥ 0. (40) forms) whose nonlinear terms depend essentially on two ∂t dPn dt different powers of the probability distribution, like the n=1 ones discussed in the present paper, are good candidates Then, using equations (37–39) one can write this time for describing anomalous-diffusion phenomena where a derivative as, crossover between two different diffusion regimes may take W−1 place. As a typical example, one could have a particle ∂S dg[Pn] µ µ ν−1 ν−1 = aPn+1 − aPn − bPnPn+1 + bPn+1 Pn transport in a system composed by two types of disor- ∂t n=1 dPn dered media, characterized respectively, by significantly W different grains (e.g., different average sizes), arranged in dg[Pn] µ µ ν−1 ν−1 + aP − aP + bPn−1P −bPnP . such a way that the diffusion process is dominated by one n−1 n n n−1 n=2 dPn medium, at high densities, and by the other one, at low (41) densities. The sum indices can be rearranged to yield all summations W−1 in the range , and thus the time derivative of the n=1 Appendix A entropy becomes, W−1 ∂S dg[Pn] dg[Pn+1] µ µ In this appendix we will prove the H-theorem directly from = − a Pn+1 − Pn the master equation, for an isolated system (i.e., no exter- ∂t n=1 dPn dPn+1 nal forces). Let us consider a system described in terms of ν−2 ν−2 −bPnPn+1 Pn+1 − Pn . (42) W discrete stochastic variables; then, Pn(t)representsthe probability of finding this system in a state characterized The negative curvature of the entropic function [cf. by the variable n at time t. This probability evolves in Eq. (11)] implies that its first derivative decays mono- time according to a master equation, tonically with Pn. Hence, for Pn+1 >Pn, the condition of equation (40) is satisfied for W µ µ ν−2 ν−2 ∂Pn Pn+1 − Pn Pn+1 − Pn = (Pmwm,n − Pnwn,m) , (35) a − bPnPn+1 ≥ 0, (43) ∂t m=1 Pn+1 − Pn Pn+1 − Pn where we divided the term inside the brackets in equa- where wk,l(t) represents the transition probability rate tion (42) by the difference ∆P = Pn+1 − Pn.Nowwe from state k to state l. The nonlinear effects were intro- consider the limit ∆P → 0andobtain, duced through the transition rates [38], aµP µ−1 + b(2 − ν)P ν−1 ≥ 0, (44)

1 µ−1 ν−1 which corresponds to the condition Ω[P ] ≥ 0 found in wk,l(∆)= (δk,l+1 +δk,l−1)[aP (t)+bP (t)], (36) ∆2 k l the text, when proving the H-theorem by making use of the FPE of equation (1). It should be mentioned that the where ∆ represents the size of the step of the random walk. procedure above works also for Pn >Pn+1; therefore, in Herein, we shall consider a random walk characterized by this appendix we have proven that the H-theorem holds for ∆ = 1; substituting such a transition rate in the master an arbitrary state n and all times t of an isolated system. V. Schw¨ammle et al.: A general nonlinear Fokker-Planck equation and its associated entropy 165

References 20. T.D. Frank, Nonlinear Fokker-Planck equations: Fundamentals and Applications (Springer, Berlin, 2005) 1. L.E. Reichl, A Modern Course in Statistical Physics,2nd 21. A.R. Plastino, A. Plastino, Physica A 222, 347 (1995) edn. (Wiley, New York, 1998) 22. C. Tsallis, D.J. Bukman, Phys. Rev. E R2197, 54 (1996) 2. N.G. Van Kampen, Stochastic Processes in Physics and 23. L. Borland, Phys. Rev. E 57, 6634 (1998) Chemistry (North-Holland, Amsterdam, 1981) 24. L. Borland, F. Pennini, A.R. Plastino, A. Plastino, Eur. 3. H. Haken, Synergetics (Springer-Verlag, Berlin, 1977) Phys.J.B12, 285 (1999) 4. H. Risken, The Fokker-Planck Equation: Methods of 25. E.K. Lenzi, R.S. Mendes, C. Tsallis, Phys. Rev. E 67, Solution and Applications (Springer-Verlag, Berlin, 1989) 031104 (2003) 5. T.M. Cover, J.A. Thomas, Elements of Information 26. T.D. Frank, A. Daﬀertshofer, Physica A 272, 497 (1999) Theory (Wiley, New York, 1991) 27. T.D. Frank, A. Daﬀertshofer, Physica A 295, 455 (2001) 6. Nonextensive Statistical Mechanics and Thermodynamics, 28. T.D. Frank, Physica A 301, 52 (2001) edited by S.R.A. Salinas, C. Tsallis, Vol. 29,Braz.J.Phys. 29. L.C. Malacarne, R.S. Mendes, I.T. Pedron, E.K. Lenzi, (1999) Phys. Rev. E 63, 030101 (2001) 7. Nonextensive Entropy - Interdisciplinary Applications 30. L.C. Malacarne, R.S. Mendes, I.T. Pedron, E.K. Lenzi, edited by M. Gell-Mann, C. Tsallis (Oxford University Phys. Rev. E 65, 052101 (2002) Press., New York, 2004) 31. P.H. Chavanis, Phys. Rev. E 68, 036108 (2003) 8. Nonextensive Statistical Mechanics: New Trends, New 32. P.H. Chavanis, Physica A 340, 57 (2004) Perspectives,Vol.36, Europhys. News (2005) 33. A.R. Plastino, L. Borland, C. Tsallis, J. Math. Phys. 39, 9. C. Tsallis, J. Stat. Phys. 52, 479 (1988) 6490 (1998) 10. S. Abe, Phys. Lett. A 224, 326 (1997) 34.A.Compte,D.Jou,J.Phys.A29, 4321 (1996) 11. C. Anteneodo, J. Phys. A 32, 1089 (1999) 35. T.D. Frank, Phys. Lett. A 267, 298 (2000) 12. E.P. Borges, I. Roditi, Phys. Lett. A 246, 399 (1998) 36. T.D. Frank, Physica A 292, 392 (2001) 13. P.T. Landsberg, V. Vedral, Phys. Lett. A 247, 211 (1998) 37. S. Martinez, A.R. Plastino, A. Plastino, Physica A 259, 14. E.M.F. Curado, Braz. J. Phys. 29, 36 (1999) 183 (1998) 15. E.M.F. Curado, F.D. Nobre, Physica A 335, 94 2004 38. E.M.F. Curado, F.D. Nobre, Phys. Rev. E 67, 021107 16. A. Renyi, Probability Theory (North-Holland, Amsterdam, (2003) 1970) 39. F.D. Nobre, E.M.F. Curado, G. Rowlands, Physica A 334, 17. G. Kaniadakis, Physica A 296, 405 (2001) 109 (2004) 18. G. Kaniadakis, Phys. Rev. E 66, 056125 (2002) 40. M. Shiino, J. Math. Phys. 42, 2540 (2001) 19. G. Kaniadakis, Phys. Rev. E 72, 036108 (2005)