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Multivariate nonlinear Fokker–Planck equations and generalized thermostatistics
Physica A 292 (2001) 392–410
www.elsevier.com/locate/physa
Multivariate nonlinear Fokker–Planck equations and generalized thermostatistics T.D. Frank ∗ , A. Da)ertshofer Faculty of Human Movement Sciences, Vrije Universiteit, Van der Boechorststraat 9, 1081 BT Amsterdam, The Netherlands Received 2 August 2000
Abstract Multivariate nonlinear Fokker–Planck equations are derived which are solved by equilibrium distributions of generalized thermostatistics. The multivariate Fokker–Planck equations proposed by Kaniadakis and by Borland et al. are re-obtained as special cases. Furthermore, a Kramers equation is derived for particles obeying the nonextensive thermostatistics proposed by Tsallis. c 2001 Elsevier Science B.V. All rights reserved. PACS: 05.20−y; 05.40.+j Keywords: Nonlinear Fokker–Planck equation; Generalized entropy; Canonical ensembles; Bose and Fermi systems; Second law of thermodynamics
1. Introduction The study of irreversible processes is an integral part of the study of complex systems. For example, irreversible processes can describe chemical reactions [1–5], population dynamics [6], relaxation processes of assemblies of identical particles [7–9], or information processing in biological and arti?cial systems [10 –16]. Such stochastic processes are often formulated as solutions of master equations [3,4]. In statistical mechanics, properties of the transition rates that determine the explicit structure of master equations can be derived from assumed equilibrium distributions [8,9,17]. The equilibrium distributions, in turn, can be deduced, for example, by using the concepts of canonical ensembles [9,18]. Thus, irreversible stochastic processes that are described ∗
Corresponding author. Tel.: +31-20-444-8612; fax: +31-20-4448-509. E-mail address: [email protected] (T.D. Frank).
c 2001 Elsevier Science B.V. All rights reserved. 0378-4371/01/$ - see front matter PII: S 0 3 7 8 - 4 3 7 1 ( 0 0 ) 0 0 5 5 9 - 8
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by master equations can be linked to equilibrium statistics as well as to canonical ensembles. Furthermore, such a correspondence can be extended to irreversible processes described by Fokker–Planck equations since Fokker–Planck equations can be viewed as approximations of master equations [1,2,4,5,8]. When aiming at stochastic descriptions in terms of Fokker–Planck equations, there is a more direct route that may be followed: the relevant Kramers–Moyal coeFcients may be inferred from the assumed equilibrium statistics or, alternatively, from the assumed canonical ensembles. Such an approach has already been discussed by Kramers [19] and culminated in what is presently called as Kramers equation [1,2]. There are also several studies that propose to derive the relevant Kramers–Moyal coeFcients from steady-state or equilibrium-state autocorrelations, see, e.g., Refs. [20 –23]. Recently, the correspondence between stochastic descriptions via Fokker–Planck equations and canonical ensembles has been extended in order to encompass a variety of statistics that di)er from the Boltzmann–Gibbs–Shannon statistics: the statistics of fermions and of bosons [24,25], the nonextensive generalized thermostatistics proposed by Tsallis [26 –36] and the statistics based on the Renyi entropy [34]. A common feature of these stochastic evolution equations is that they are nonlinear with respect to their probability densities but recover conventional Fokker–Planck equations in well-de?ned limiting cases (see, however, [37,38]). We will therefore call the former kind of stochastic evolution equations nonlinear Fokker–Planck equations (NLFPEs) and the latter one linear Fokker–Planck equations. Many of the previously mentioned studies focus on one-dimensional Fokker–Planck equations. One may view these one-dimensional NLFPEs as possible generalizations of the Smoluchowski equation [1,2]. The present article extends the NLFPE/canonical ensemble correspondence principle established in Refs. [19,26 –36] to the multivariate case. To this end, we ?rst discuss drift forces that can be derived from potential functions and examine multivariate stochastic processes in the Smoluchowski limit. Subsequently, we consider more general random walks. We study stochastic processes described by generalized coordinates and generalized momenta of the system’s particles. As a special case, we discuss drift forces that are similar to those studied by Ebeling and Schimansky–Geier in the context of linear Fokker–Planck equations and that cannot be derived from a potential [39]. Finally, we propose generalized Kramers equations related to generalized thermostatistics. In the present article, we assume that the noise sources of the systems under considerations are homogeneous with respect to their state variables. In other words, we exclusively consider systems that are in thermal contact with spatially homogeneous heat baths and ignore phenomena related to local temperatures (blow torch e)ects). Consequently, the di)usion terms of the Fokker–Planck equations discussed here do not explicitly depend on the state variables. More elaborate discussions in this regard can be found in Refs. [32,40 – 42]. Before proceeding with our main objectives, we ?rst elucidate the relevance of generalized statistics such as nonextensive thermostatistics, see, e.g., Refs. [43– 48] and the extensive statistics induced by the Renyi entropy [49,50]. It is well known that classical thermodynamics applies to systems whose total energy is the sum of the
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energies of their macroscopic parts. Following Terletskii [51, Chapter 7] and Landsberg [52], systems characterized by long-range forces can fail to satisfy this property of additivity because interaction energies between macroscopic parts of these systems can make essential contributions to the total energy. Extensive thermostatistics can hardly cope with this problem. This is, for example, illustrated by the entropy of the Kerr– Newman black hole, which is a nonextensive variable [52]. In addition, Sharma and Garg pointed out that a number of biological and social systems should be viewed as nonextensive rather than as extensive systems and should therefore be modeled by means of nonextensive entropies [53]. In a nutshell, nonextensive systems seem to demand nonextensive statistical and stochastic descriptions. The establishment of such a theory, a nonextensive thermodynamics, lies at the heart of a current development in statistical physics which was inspired by the seminal work of Tsallis [43]. For reviews on this topic, the reader is referred to Refs. [54 –57]. Finally, entropies made up of terms like (pj ) , where pj represents the probability to ?nd a system in a state j, facilitate the investigation of fractal structures by means of the concepts of statistical mechanics. In this context, one can ?nd in the literature applications of the Renyi entropy [58] and the generalized nonextensive entropy proposed by Tsallis [55].
2. Generalized canonical ensembles and multivariate NLFPEs 2.1. Multivariate stochastic potential dynamics Consider a system whose state is described by the N -dimensional state vector q = (q1 ; : : : ; qN ), where the components qj represent dimensionless variables in the domain (−∞; ∞). Taking a statistical perspective, we assume that the equilibrium state of the system can be described in terms of a canonical ensemble involving a generalized entropy functional S and a potential V (q). We further assume that V (q) is a globally attractive potential, that is, V (|q| → ∞) → ∞, with V (q) ∈ C ∞ (RN ). Under these assumptions, we introduce the probability density P(q) of the system and the entropy functional N ˜ (1) S[P] := B · · · S(P(q)) d qi ; ˜ where the entropy kernel S(y) ∈ C ∞ [0; ∞) : y ∈ R+ → R assigns a particular amount of disorder to the function value y of the probability density P, while the outer function B(z) ∈ C 1 : z ∈ R → R ?xes the scale in which the overall degree of disorder is measured. Of course, the mere structure of the functional (1) does not imply that S represents an entropy functional. Throughout this article, however, it is assumed that the constituents B and S˜ are chosen in such a way that the functional (1) can be viewed as an entropy functional. 1 In line with the theory of canonical ensembles, we 1
For characteristic properties of entropies and entropy functionals, the reader is referred to Wehrl [59].
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then assume that the equilibrium statistics of the system Peq (q) makes the entropy functional (1) stationary under the constraints N 1 = · · · P(q) d qi and C = · · · V (q)P(q) d N qi : (2) To solve the corresponding variational problem, we use the Lagrange multipliers and and require that the variation of N N I := S + C − · · · V (q)P(q) d qi + 1 − · · · P(q) d qi (3) vanishes. From I = 0 it follows that ˜ d S(y) +V (q)=[Peq ] dy
y=Peq (q)
with
[P] :=
dB(z) d z z= ···
˜ S(P(q)) d N qi
; (4)
which implicitly determines the equilibrium statistics Peq (q). Having solved Eq. (4), the Lagrange multipliers and can be calculated from the constraints (2). In line with a previous study on the correspondence between canonical ensembles and one-dimensional NLFPEs [34], we now consider the multivariate Fokker–Planck equation N
@ @ P(q; t) = − @t @qj j=1
[P(q; t)] @ ˆ ˜ L{S(P(q; t))} × M [(P(q; t))] hj (q)P(q; t)− @qj
Y
(5) where the operator Lˆ acting on the function (y) is de?ned as d(y) ˆ L[(y)] := (y) − y : dy
(6)
In Eq. (5) the form M [(·)] denotes a function or functional of P(q; t) with M [(·)] ¿ 0. If B(z) = z ⇒ ≡ 1, we can con?ne our considerations to variables M that represent functions of the probability density P with M [(z)] = M (z) ∈ C ∞ [0; ∞), cf. Eq. (12) below. The drift terms hj (q) correspond to a deterministic force imposed on the system by the potential V (q ): hj (q) := − dV (q)= dqj . We now show that if Eq. (4) can be solved under the constraints (2), the obtained equilibrium statistics Peq (q) represents a stationary solution Pst (q) of the NLFPE (5). To this end, we di)erentiate the implicit de?nition of the equilibrium statistics given by Eq. (4) with respect to qj and obtain ˜ @ d S(y) [Peq ] = −hj (q) : (7) @qj dy y=Peq (q)
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Next, we evaluate the de?nition (6) of the operator Lˆ and ?nd ˜ @ d S(y) @ ˆ ˜ L{S(P(q; t))} = −P(q; t) : @qj dy y=P(q; t) @qj
(8)
Using Eqs. (7) and (8) one can further deduce that the term labeled Y in the NLFPE (5) vanishes for P = Peq . Consequently, the NLFPE (5) has a stationary solution Pst (q) that can be viewed as the equilibrium statistics Peq (q) induced by the functional S. In addition, if S is the Boltzmann–Gibbs–Shannon entropy, BGS S := − P(q) ln P(q) d N qi , we can ?nd a corresponding linear Fokker–Planck equation, namely, the Fokker–Planck equation (5) for S = BGS S and M [(·)] ≡ 1. Furthermore, we may consider the limit of vanishing noise ( → ∞) for M [(·)] ≡ 1. In this special case, the NLFPE (5) reduces to N
@ @ hj (q)P(q; t) : P(q; t) = − @t @qj
(9)
j=1
Eq. (9) can be solved in terms of a weak solution that reads P(C; t) = (C − q(t))
(10)
@ d V (q(t)) ; qj (t) = − dt @qj
(11)
with
where (z) denotes the N -dimensional delta distribution. 2 For this reason, the NLFPE (5) may be interpreted as stochastic description of a system whose deterministic evolution is described by the potential dynamics (11). Just as for the one-dimensional case [34], there are two special cases worth mentioning. First, for M [(·)] ≡ 1, the drift term in the NLFPE (5) is linear with respect to the probability density P, while, in general, the di)usion term is nonlinear. 2 Strictly speaking, we claim that the generalized function P(C; z)¿0 with t = z de?ned by Eqs. (10) and (11) satis?es the integral relation
∞
0=
dz 0
∞
∞
··· ∞
∞
dvjN P(C; z)
@ @ hj (C) (C; z) (C; z) + @z @vj N
j=1
for any test function (C; z) ∈ C ∞ (RN × [0; ∞)) that vanishes at the boundaries z = 0, z → ∞, and |C| → ∞. In fact, performing the integration over dvjN gives us
∞
dz 0
dqj @ @ (q; z) = (q; z) + @z d z @qj N
j=1
∞
dz 0
d (q(z); z) = (q; z)|z=∞ z=0 = 0 : dz
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Second, for
397
−1 ˜ d 2 S(y) M [(P)] = − [P(q; t)]P(q; t) dy2 y = P(q; t)
−1
˜ ˆ S(y)} d L{ = [P(q; t)] dy
(12)
y=P(q; t)
the NLFPE (5) reads
−1 N ˜ @ @ d 2 S(y) 1 @ hj (q) [P(q; t)] + P(q; t) P(q; t) = @qj dy2 P(q; t) @qj @t j=1
=−
N j=1
−1 ˜ ˆ S(y)} @ d L{ −1 @ hj (q) [P(q; t)] @qi dy @qj P(q; t)
×P(q; t) ;
(13)
the drift term may then be nonlinear with respect to the probability density P(q; t), whereas the di)usion term is linear. We illustrate the results obtained so far for the explicit entropy functional 1 T () {(P(q)) − P(q)} d N qi ; S := (14) 1− which has been studied in detail in the context of the generalized nonextensive thermostatistics proposed by Tsallis, cf., e.g., [43,54]. For M [(·)] ≡ 1 the corresponding multivariate NLFPE reads as N @ @ 1 @ P(q; t) = − hi (q)P(q; t) − : (15) (P(q; t)) @t @qi @qi i=1
By virtue of Eq. (12), we can also derive a NLFPE with a di)usion term that is linear with respect to the probability density. For S = T S() , however, we cannot ?nd a function or functional M [(·)] that yields both a linear drift and a linear di)usion term (except for the trivial case M [(·)] ≡ 0). Therefore, we may say that the NLFPE (5) related to T S() is essentially nonlinear. The stationary solution of Eq. (15) can be derived from Eq. (4) and is found as 1=(−1) N 1 1− Pst (q) = V (q) (16) ++ 1− i=1
for ∈ (0; 1) and potentials V (q) with a ?nite lower boundary, that is, ∀q: V (q)¿V0 . Other parameter values require more thorough analyses accounting for weak solutions (cut-o) solutions), see, e.g., Refs. [31,34,43]. Even for ∈ (0; 1) we have to check carefully whether a stationary solution of the form (16) exists at all. This cannot be
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taken for granted because even in the linear, one-dimensional case there are Fokker– Planck equations involving globally attractive potentials that do not admit for stationary solutions (consider, e.g., potentials that increase for large distances x according to ln|x|). Once the existence of a stationary solution is guaranteed, the constant can however, be computed from the normalization condition 1 = · · · Pst (q) d N qi . If analytical methods fail, might be estimated with the help of numerical techniques. Finally, note that in the limit → 1 the functional T S() recovers the Boltzmann–Gibbs–Shannon entropy functional BGS S. Similarly, in the limit → 1 the NLFPE (15) recovers a linear Fokker–Planck equation and the stationary probability density (16) reduces to a Boltzmann distribution. 2.2. Multivariate stochastic dynamics in phase space We now turn our attention to possible generalizations of the Kramers equation [1,2,19]. For that purpose, we take a somewhat broader point of view and examine a system whose state can be described by the generalized coordinate vector q and the generalized momentum vector p with q = (q1 ; : : : ; qN ) and p = (p1 ; : : : ; pN ), respectively. Consequently, we may assume that the components of q are invariant under time reversal, whereas the components of p change their signs when time is reversed, 3 that is, t → − t ⇒ (q; p) → (q; −p). Then, we introduce a function (q; p) in order to describe the equilibrium probability density Peq (q; p) of the system: Peq (q; p) is de?ned as the probability density that makes the entropy functional (1) stationary under the constraints 1 = · · · P(q; p) d 2N qi pi and E = · · · (q; p)P(q; p) d N qi d N pi : (17) Both constraints agree with the constraints commonly imposed on canonical ensembles. In particular, if (q; p) describes a Hamiltonian H (q; p), then the constant E is the mean energy of the system. Therefore, in general, we may consider (q; p) as a “generalized” Hamiltonian. The variational problem can be formulated by means of the Lagrange multipliers and and involves the functional N N I := S + E − (q; p)P(q; p) d qi d pi + 1 − P(q; p) d N qi d N pi :
(18)
Again, the solution of this variational problem implicitly de?nes the equilibrium probability density. Then, from I = 0 it follows that ˜ d S(y) : (19) + (q; p) = [Peq ] dy y=Peq (q; p)
3
Note that in what follows we will not refer to these properties. However, we will return to this subject in the Discussion (Section 3).
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If there exists a solution of Eq. (19), then it can be expressed in terms of a function F˜ as ˜ Peq = F( ) :
(20)
Note that the function F˜ depends solely on the function . Moreover, we may intro−1 ˜ where the normalization ˜ ln(z=N duce the function F(z) := F{− )} associatedN to F, −1 constant N is given by (N ) := · · · exp(− ) d qi d N pi . Eq. (20) can then be written as Peq = F(W) with the Boltzmann distribution W := N exp(− ). We realize that F(z) corresponds to the distortion function that has already been introduced in ˜ the context of one-variable NLFPEs [34]. Accordingly, F(z) as de?ned in Eq. (20) is interpreted as a function that maps the equilibrium density W, which is assumed by the system for S = BGS S, to the actual equilibrium probability density Peq induced by the generalized entropy functional under consideration. 4 In line with studies on stationary solutions of multivariate linear Fokker–Planck equations by van Kampen [60,61], Graham and Haken [62], Risken [63] and Graham [64], we introduce structural components of multivariate NLFPEs (cf. also Refs. [1,2,65 (qp) (pq) (pp) – 67]): the square matrices Kj;(qq) and the vector-like coeFcients r , Kj; r , Kj; r , Kj; r (q) (p) (q) (p) Ij , Ij , Dj , and Dj . Before interpreting these structural constituents, we formulate additional constraints. These constraints will guarantee that the multivariate NLFPEs (qp) (pq) (q) (p) (q) (p) based on Kj;(qq) are solved by Peq (q; p). First, r , Kj; r , Kj; r , Ij , Ij , Dj , and Dj (·) the coeFcients Ij are only implicitly de?ned. We require that they satisfy N @ (q) @ (p) ˜ ˜ [I (q; p)F( )] + [I (q; p)F( )] = 0 : (21) @qj j @pj j j=1
Note that the coeFcients Ij(·) are functions of q and p. Using Eq. (20), we obtain N @ (q) @ (p) [I (q; p)Peq ] + [I (q; p)Peq ] = 0 : (22) @qj j @pj j j=1
Second, we require that the vector-like coeFcients Dj(·) and the matrices Kj;(·)r ful?ll the relations N N @ (qp) @ Kj;(qq) (q; p) ; + K Dj(q) (q; p) :=: − r j; r @qr @pr j=1 r=1
Dj(p) (q; p) :=:
−
N N
Kj;(pq) r
j=1 r=1
@ @ + Kj;(pp) r @qr @pr
(q; p) :
(23)
Note that the elements Kj;(·)r are assumed to be constants, whereas the coeFcients Dj(·) are functions of q and p. In analogy to Section 2.1, from the de?nition of the operator 4 Similarly, using the Fokker–Planck equation perspective, the functions F˜ and F map the stationary solution W of the corresponding linear Fokker–Planck equation to the actual stationary probability Pst of the NLFPE under consideration.
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Lˆ (cf. Eq. (6)) we can derive Eq. (8), which now holds for derivatives with respect to qj and pj . Using this generalized version of Eq. (8) in combination with Eq. (19), we rewrite Eq. (23) as N N [Peq ] (qq) @ (qp) @ (q) ˜ eq )} ; ˆ S(P L{ Kj; r + Kj; r Dj Peq = − @qr @pr j=1 r=1
Dj(p) Peq = −
N N [Peq ] (pq) @ (pp) @ ˜ eq )} : ˆ S(P K L{ + K j; r j; r @qr @pr
(24)
j=1 r=1
We can now propose the multivariate NLFPE N @ @ (q) @ (p) P(q; p; t) = − (Ij + Dj(q) ) + (Ij + Dj(p) ) P(q; p; t) @pj @t @qj j=1
+
N N [P] (1) ˜ ˆ S(P(q; p; t))} ; Kj; r (@qj ; @pj ) L{
(25)
j=1 r=1
where the di)erential operator K(1) (@qj ; @pj ) is de?ned by (qq) K(1) j; r (@qj ; @pj ) := Kj; r
@2 @2 @2 (pq) + (Kj;(qp) + Kj;(pp) : r + Kr; j ) r @qj @qr @qj @pr @pj @pr
(26)
Since the coeFcients Ij(·) and Dj(·) occur in the NLFPE in combination with ?rst-order partial derivatives, they are referred to as drift coeFcients. Furthermore we can read o) from Eqs. (25) and (26) that the matrices Kj;(·)r weigh the components of the second-order di)erential operator K(1) (@qj ; @pj ). By analogy to linear Fokker–Planck equations, the matrices Kj;(·)r may be seen as measuring the strength of the Ructuations to which the multivariate stochastic process in question are subjected. In other words, the elements of the matrices Kj;(·)r are the strength of noise sources acting on the system. As stated in the introduction, we consider noise sources that are homogeneous with respect to the phase space in question – which is in agreement with our requirement of constant matrix elements Kj;(·)r . In this line, we will refer to the elements Kj;(·)r as di)usion coeFcients. Finally, from Eqs. (22) and (24) we can read o) that the equilibrium statistics Peq (q; p) is, indeed, a stationary solution of the NLFPE (25). The drift terms of the NLFPE (25) are linear with respect to the probability density P(q; p; t). Consequently, the NLFPE (25) fails to address NLFPEs as, for example, proposed by Kaniadakis in the context of classical stochastic descriptions of fermions and bosons [68]. To obtain NLFPEs with nonlinear drift terms we impose an additional constraint on the drift coeFcients Ij(·) , namely, that the divergence of the vector ?eld given by the components Ij(·) vanishes (see also Refs. [62,66]). That is, we require that N @ (q) @ (p) Ij (q; p) + Ij (q; p) = 0 (27) @qj @pj j=1
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holds in addition to the constraint (21). In this case, the drift functions Ij(·) obey the relation N @ @ (28) Ij(q) (q; p) Peq = 0 Peq + Ij(p) (q; p) @pj @qj j=1
instead of Eq. (22). From Eqs. (27) and (28) we can further conclude that N @ @ (q) (p) Ij (q; p) [M [(Peq )]Peq ] + Ij (q; p) [M [(Peq )]Peq ] = 0 @qj @pj j=1
⇒
N @ (q) @ (p) [Ij (q; p)M [(Peq )]Peq ] = 0 : [Ij (q; p)M [(Peq )]Peq ] + @pj @qj j=1
(29) In due recognition of relation (29), the NLFPE (25) can be replaced by N @ @ (q) @ (p) (q) (p) (I + Dj ) + (I + Dj ) M [(P)]P(q; p; t) P(q; p; t) = − @t @qj j @pj j j=1
+
N N [P] (M ) ˜ ˆ S(P(q; Kj; r (@qj ; @pj ) L{ p; t))} ;
(30)
j=1 r=1
where the di)erential operator K(M ) (@qj ; @pj ) is de?ned by @ ) (qq) @ (qp) @ K(M (@q ; @p ) := M [(P(q; p; t))] K + K j j j; r j; r j; r @qj @qr @pr @ (pq) @ (pp) @ : M [(P(q; p; t))] Kj; r + Kj; r + @pj @qr @pr
(31)
By virtue of Eqs. (24) and (29), one can verify that the equilibrium statistics Peq solves the NLFPE (30). The NLFPEs (25) and (30) may be regarded as stochastic descriptions of systems the evolution of which in the absence of noise (i.e., for → ∞) is determined by the drift functions Ij(·) and Dj(·) according to P(u; C; t) = (u − q(t)) · (C − p(t)) ; d qj (t) = Ij(q) (q(t); p(t)) + Dj(q) (q(t); p(t)) ; dt d (32) pj (t) = Ij(p) (q(t); p(t)) + Dj(p) (q(t); p(t)) : dt In general, the deterministic system (32) cannot be written as a potential dynamics such as given by Eq. (11). However, for (q; p) = V (q), Ij(q) = Ij(p) = Dj(p) ≡ 0, (pq) (pp) (qq) (q) Kj;(qp) r = Kj; r = Kj; r ≡ 0, Kj; r ≡ 1, Dj = −dV (q)=dqj , and = the NLFPE (30) recovers the NLFPE (5) – just as the system (32) recovers the potential dynamics (11).
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Three subtleties are worth mentioning. First, in this section we derived two distinct multivariate stochastic descriptions: the NLFPE (25) with drift functions Ij(·) satisfying the constraint (21) and the NLFPE (30) with drift functions Ij(·) satisfying both the constraints (21) and (27). Neither is the ?rst description a special case of the latter, nor is the latter description a special case of the ?rst. Second, for M [(·)] ≡ 1 the NLFPE (30) assumes the structure of the NLFPE (25) and exhibits a drift term that is linear with respect to the probability density P(q; p; t). In contrast, substituting −1 ˜ ˆ S(y)} d L{ (33) M [(P)] = [P(q; p; t)] dy y=P(q; p; t)
into Eq. (30) yields the NLFPE N 1 @ (q) d Lˆ @ (p) @ P(q; p; t) = − (Ij + Dj(p) ) P(q; p; t) (Ij + Dj(q) ) + @qj @pj dP @t [P] j=1
+
N N 1 (1) Kj; r (@qj ; @pj )P(q; p; t) :
(34)
j=1 r=1
Now, the di)usion term of the NLFPE (34) is linear with respect to P(q; p; t) but the drift term is, in general, nonlinear with respect to P(q; p; t). Third, substituting the Boltzmann–Gibbs–Shannon entropy functional BGS S into the NLFPE (25), we obtain a linear Fokker–Planck equation describing a multivariate stochastic process characterized by additive Ructuation forces. The same holds for the NLFPE (34) and the NLFPE (30) with M [(·)] ≡ 1. 2.3. Examples We consider a random walk in the six-dimensional Cartesian coordinate system described by the coordinate vector x = (x1 ; x2 ; x3 ) and the momentum vector p = (p1 ; p2 ; p3 ). 2.3.1. Ebeling/Schimansky–Geier systems Let the drift and di)usion coeFcients be de?ned by Ij(x) :=
@ H (x; p); @pj
Dj(x) := 0; Kj;(pp) r = &;
Ij(p) := −
Dj(p) := − &
@ H (x; p) ; @xj
@ H (x; p) ; @pj
(xp) (px) Kj;(xx) r = Kj; r = Kj; r ≡ 0 ;
(35)
where & denotes a semi-positive constant. The coeFcients Ij(·) constitute a vector ?eld with vanishing divergence, that is, Eq. (27) is satis?ed. Moreover, the coeFcients Ij(·)
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obey the constraint (21). Consequently, if there exists an equilibrium statistics Peq of the canonical ensemble de?ned by the entropy functional (1) and the constraints (17), then this equilibrium statistics corresponds to a stationary solution of the NLFPE (30) speci?ed by the coeFcients (35). In the absence of a friction force, that is, for & = 0, from Eq. (32) the deterministic evolution equations of the system in question can be obtained as @ d xj (t) = H (x(t); p(t)); dt @pj
d @ H (x(t); p(t)) : pj (t) = − @xj dt
(36)
Note that Eq. (36) describes a conservative system (dH=dt = 0). For M ≡ 1 and S = BGS S the NLFPE (30) becomes linear with respect to its probability density and the stochastic evolution equations of the corresponding random variables xi → 'i and pi → (i read @ d H (x; p) ; 'j (t) = dt @pj x=(t); p=(t) 2& @H @H d (j (t) = − −& + )(t) : (37) dt @xj x=(t); p=(t) @pj x=(t); p=(t)
Here, )(t) denotes a Langevin force with )(t))(t ) = (t − t ). The system (37) was addressed by Ebeling and Schimansky–Geier [39]. In sum, the NLFPE (30) in combination with the coeFcients (35) embodies the stochastic processes de?ned by Eq. (37) as a special case and features stochastic processes related to, for example, nonextensive thermostatistics and the thermostatistics based on the Renyi entropy. 2.3.2. Generalized Kramers equations We elaborate on the preceding example by specifying the Hamiltonian H (x; p) according to 3
H (x; p) :=
1 (pj )2 + U (x) ; 2m
(38)
j=1
where m ¿ 0 denotes the mass of the particle performing the random walk being studied and U (x) ∈ C ∞ (R3 ) stands for a globally attractive potential (i.e., U (|x| → ∞) → ∞). Inserting Eqs. (35) and (38) into the NLFPE (30), we obtain 3 pj @ dU (x) @ & @ P(x; p; t) = − M [(P)]P(x; p; t) − + pj @t m @xj @pj d xj m j=1
3
+
@ & @ ˆ ˜ L{S(P(x; p; t))} : [P] M [(P)] @pj @pj
(39)
j=1
The NLFPE (39) shall be viewed as a possible generalization of the Kramers equation in the context of generalized thermostatistics. Note that for M [(·)] ≡ 1 and S = BGS S
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the NLFPE (39) reduces to the ordinary Kramers equation [1,2] and can describe stochastic processes de?ned by the Langevin equation 1 d 'j (t) = (j (t) ; m dt d 2& & d (j (t) = − (j (t) − U (x) + )(t) ; (40) dt m d xj x=(t)
cf. also Eq. (37). 2.3.3. Kramers equations for classical bosons and fermions Next, we consider ensembles of classical bosons and fermions. Entropy functionals for bosons and fermions are usually formulated in the context of grand canonical ensembles [18]. Adapting these entropy functionals for canonical ensembles, we may consider the entropy functionals BE S and FD S de?ned by BE S = − {P ln P + (1 − P) ln(1 − P)} d 3 xi d 3 pi (41) and FD
S=−
{P ln P − (1 + P) ln(1 + P)} d 3 xi d 3 pi
(42)
for the Bose–Einstein statistics and the Fermi–Dirac statistics, respectively. Then, it ˜ = ln(1 + P) for bosons and L( ˜ = −ln(1 − P) for fermions. 5 ˆ S) ˆ S) follows that ≡ 1, L( For M [(·)] ≡ 1 we obtain 3 pj @ dU (x) @ & @ P(x; p; t) − + pj P(x; p; t) = − @t m @xj @pj d xj m j=1
±
3 & @2 ln {1 ± P(x; p; t)} : @pj2 j=1
(43)
In Eq. (43) the upper sign holds for classical bosons and the lower sign for classical fermions. Using Eq. (33), we can derive the Kramers equations with di)usion terms which are linear with respect to their probability densities. These Kramers equations read 3 pj @ @ dU @ & [1 ± P(x; p; t)]P(x; p; t) − + pj P(x; p; t) = − @t m @xj @pj d xj m j=1
+ 5
3 & @2 P(x; p; t) : @pj2 j=1
(44)
Note that in an earlier study of the NLFPE/canonical ensemble correspondence concerning the Bose– Einstein statistics and the Fermi–Dirac statistics some signs have been confused (cf. columns 3 and 4 of ˜ = ln(1 + P) and ˆ BE S) Table 1 [34]). In the present article Eqs. (41) and (42) in combination with L( FD ˜ ˆ L( S) = −ln(1 − P) give the correct formula.
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Again, the upper sign refers to Bose–Einstein statistics and the lower to Fermi–Dirac statistics. The NLFPE (44) was proposed by Kaniadakis as approximation of a master equation that takes speci?c exclusion/inclusion principles into account [68]. 2.3.4. Kramers equations for the generalized thermostatistics proposed by Tsallis Finally, we can derive Kramers equations for the nonextensive generalized thermostatistics proposed by Tsallis [43] by substituting the entropy functional (14) into Eq. (39). For M [(·)] ≡ 1 we obtain 3 3 pj @ & @2 dU & @ @P =− P+ + pj − (P) : (45) 2 m @xj @pj d xj m @t @p j j=1 j=1 For ∈ (0; 1) the stationary solution/equilibrium statistics can be derived from Eq. (19) and is found as 1=(−1) 1 1− Pst (x; p) = ; (46) + + H (x; p) 1− where H is given by Eq. (38). Note that for = 1 the NLFPE (45) recovers the classical Kramers equation. Similarly, in the limit → 1 the stationary solution (46) reduces to a Boltzmann distribution of the generalized Hamiltonian H . For ¿ 1 we deal with weak stationary solutions, which usually require more detailed analyses. The NLFPE (45) provides an explicit example of a multivariate NLFPE related to the thermostatistics based on the entropy functional T S() . Similar multivariate NLFPEs have been studied by Compte et al. [28,29] and Borland et al. [32]. Moreover, the multivariate NLFPE (45) can be considered as a possible extension of the one-dimensional NLFPE related to entropy functional T S() proposed by Plastino and Plastino [26]. We want to point out that Eq. (39) allows for the derivation of further Kramers equations related to the Renyi entropy [49], the entropy (U ) S proposed by Landsberg [46], the entropy related to stretched exponential distributions proposed by Anteneodo and Plastino [48], and the entropy of systems with statistical feedback [34]. 2.4. On the second law of thermodynamics In Sections 2.1 and 2.2 we derived multivariate NLFPEs which have generalized thermostatistics as stationary solutions. These NLFPEs were obtained under the explicit assumption that stationary solutions exist. Having derived these NLFPEs, however, one may explore their properties irrespective of the existence of stationary solutions. Hence, we now assume that there are particular stochastic processes that exist irrespective of whether or not they converge to stationary processes. If these processes converge to stationary processes, then they can be described by both canonical ensembles (for the stationary case) and multivariate NLFPEs (for the transient and stationary case). Furthermore, if the processes remain in transient regimes, then they can still be described by the NLFPEs despite the fact that in such cases the approach via canonical ensembles fails. In particular, we would like to answer the question whether the NLFPEs
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(5), (25) and (30) de?ne stochastic processes that are consistent with the second law of thermodynamics. To this end, we examine the evolution of the generalized entropy functional S for purely di)usive stochastic processes that are de?ned by the NLFPEs (5), (25) and (30) for vanishing drift forces. First of all, we observe that, in the absence of a drift, the NLFPE (25) can be regarded as a special case of the NLFPE (30). Consequently, we can con?ne our considerations to the NLFPEs (5) and (30). We then di)erentiate the entropy functional S given by Eqs. (1) with respect to the time variable t and obtain in analogy to the one-dimensional case [35] ˜ @ d d S(y) (47) P(q; t) d N qi S[P] = [P] · · · dy y=P(q; t) @t dt and d S[P] = [P] dt
˜ @ d S(y) P(q; p; t) d N qi d N pi ··· dy y=P(q; p; t) @t
(48)
for the entropy functionals related to the NLFPE (5) and the NLFPE (30), respectively. Inserting the NLFPE (5) into Eq. (47) with hj ≡ 0, and integrating by parts, gives us the ?nal result 2 N 2 1 d S˜ @ d 2 S[P]= {[P]} ··· P(q; t)M [(P)] d N qi : P(q; t) dt dP2 @qk k=1
(49) For ¿ 0 the entropy functional S is a nondecreasing function with respect to the time variable t. The positivity of the Lagrange multiplier agrees with its interpretation as a measure of the Ructuation strength (see above). Moreover, Plastino and Plastino [69] and Mendes [70] showed that, in the context of generalized thermostatistics de?ned by canonical ensembles, the Lagrange multiplier plays the role of a temperature measure. Turning to the NLFPE (30), a similar analysis can be carried out. For the sake of simplicity, we ?rst introduce the composed 2N -dimensional vector w = (w1 ; : : : ; w2N ) de?ned by w1 := q1 ; : : : ; wN := qN and wN +1 := p1 ; : : : ; w2N := pN and 6 the 2N × 2N di)usion matrix Kj;(w) r given by {Kj;(qq) {Kj;(qp) r } r } (w) ; (50) {Kj; r } := (pq) (pp) {Kj; r } {Kj; r } which agrees with the di)usion matrix de?ned in Ref. [62]. Then, for vanishing drift (i.e., Dj(·) ≡ Ij(·) ≡ 0) the NLFPE (30) can equivalently be expressed as 2N 2N [P] @ (w) @ @ ˆ ˜ P(w; t) = − L{S(P(w; t))} : (51) K M [(P)] @t @wj j; r @wr j=1 r=1
6
Here, {Aj; r } denotes the matrix A consisting of the elements Aj; r .
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Substituting (51) into the evolution equations of S given by Eq. (48) results in 2 2 (w) 2N 2N ˜ K d @P @P S d k; r S[P]={[P]}2 · · · P · M [(P)] d 2N wi : dP2 @wk @wr dt k=1 r=1
(52) 7 then dS¿0 holds. The Consequently, if the matrix Kj;(w) r = is semi-positive de?nite, requirement of a semi-positive di)usion matrix is well known in the theory of linear Fokker–Planck equations [2]. The immediate conclusions are, at least, twofold. First, the stochastic processes proposed in this article are consistent with the second law of thermodynamics provided that the di)usion coeFcients and Kj;(w) r = are appropriately chosen. Second, in turn, when starting o) with the requirement that the stochastic processes de?ned by the NLFPEs (5), (25) and (30) are consistent with the second law of thermodynamics, one can deduce necessary conditions for the di)usion coeFcients. As it turns out, these conditions are akin to the constraints that are usually imposed on the di)usion coeFcients of linear Fokker–Planck equations.
3. Discussion We derived stochastic evolution equations for time-dependent probability densities from equilibrium statistics of generalized canonical ensembles. We showed that the correspondence principle that assigns these stochastic evolution equations to canonical ensembles is not a peculiarity of the Boltzmann–Gibbs–Shannon statistics. It applies to a variety of generalized statistics irrespective of the dimensionality of the problem in hand. This basic feature reveals the generic nature of this approach in the study of irreversible stochastic processes. The derivation of Kramers equations related to generalized thermostatistics was carried out by adopting concepts from the theory of linear Fokker–Planck equations with stationary solutions satisfying the detailed balance condition. Although a deeper discussion of the implications of detailed balance for the nonlinear stochastic evolution equations discussed in Section 2.2 is beyond the scope of the present article, we want to conclude with a few general remarks. In Section 2.2, we obtained that if stochastic evolution equations can be cast under the guise of Eq. (25) and if their drift functions Dj(·) and Ij(·) are related to a function according to Eqs. (21) and (23), then the stationary probability densities can be expressed as functions of . Concerning stochastic evolution equations of the form (30), we have to require that the drift coeFcients Ij(·) also satisfy the constraint (27). In the context of linear Fokker–Planck equations, the constraints (21) and (27) imposed on the drift coeFcients Ij(·) and the 7
That is, ∀b: ( )−1
! j; r
(w)
Kj; r bj br ¿0 with b := (b1 ; : : : ; b2N ) and bj ∈ R.
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relations between the drift coeFcients Dj(·) and the di)usion matrices Kj;(·)r according to Eq. (23) can be derived from the assumption that the detailed balance condition holds (cf., e.g., Refs. [1,2,60 – 64,66]). Furthermore, the drift coeFcients Ij(·) correspond to processes that change sign if time is reversed (odd variables [60,61], negative parity [71]; cf. also the de?nition of pj in Section 2.2), whereas Dj(·) correspond to processes that are invariant under time reversal (even variables, positive parity; cf. also the definition of qj in Section 2.2). Then, Ij(·) are usually called the reversible parts of the drift coeFcient and Dj(·) the irreversible parts. Future works may clarify whether the concept of detailed balance and/or the concept of even and odd variables can also be applied to stochastic evolution equations related to generalized thermostatistics as formulated in Section 2.2. Irrespective of this issue, however, a decomposition of the drift terms of the multivariate stochastic evolution equations (25) and (30) into Ij(·) -terms and Dj(·) -terms is reminiscent of the decomposition of drift terms of multivariate linear Fokker–Planck equations. In the latter case, stationary solutions can be obtained by quadratures irrespective of whether or not the detail balance condition holds (cf., e.g., Ref. [72]). In the former case, stationary solutions can be derived by solving a corresponding variational problem. If analytical calculations fail, they can be determined at least implicitly, cf. Eqs. (7) and (19).
Acknowledgements We thank Peter Beek for his critical reading of the present article. We are indebted to Peter Beek for his constant support which made our studies of nonlinear Fokker–Planck equations possible at all. References [1] C.W. Gardiner, Handbook of Stochastic Methods, 2nd Edition, Springer, Berlin, 1997. [2] H. Risken, The Fokker–Planck Equation, Methods of Solution and Applications, Springer, Berlin, 1989. [3] N.G. van Kampen, Stochastic Processes in Physics and Chemistry, North-Holland Publ. Company, Amsterdam, 1981. [4] H. Haken, Synergetics, An Introduction, Springer, Berlin, 1977. [5] G. Nicolis, I. Prigogine, Self-Organization in Nonequilibrium System, Wiley, New York, 1977. [6] E.W. Montroll, B.J. West, in: H. Haken (Ed.), Synergetics – Cooperative Phenomena in MultiComponent Systems, Springer, Berlin, 1973, pp. 143–156. [7] A. Da)ertshofer, Phys. Rev. E 58 (1998) 327. [8] I. Oppenheimer, K.E. Shuler, G.H. Weis, Adv. Mol. Relaxation Process. 1 (1967) 13. [9] F. Reif, Fundamentals of Statistical and Thermal Physics, McGraw-Hill Book Company, New York, 1965. [10] W. Bialek, M. DeWeese, F. Rieke, D. Warland, Physica A 200 (1993) 581. [11] A. Da)ertshofer, C. van den Berg, P.J. Beek, Physica D 132 (1999) 243. [12] T.D. Frank, A. Da)ertshofer, P.J. Beek, H. Haken, Physica D 127 (1999) 233. [13] T.D. Frank, A. Da)ertshofer, C.E. Peper, P.J. Beek, H. Haken, Physica D 144 (2000) 62. [14] R. Landauer, J. Appl. Phys. 33 (1962) 2209. [15] R. Landauer, Physica A 263 (1999) 63.
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Multivariate nonlinear Fokker–Planck equations and generalized thermostatistics T.D. Frank ∗ , A. Da)ertshofer Faculty of Human Movement Sciences, Vrije Universiteit, Van der Boechorststraat 9, 1081 BT Amsterdam, The Netherlands Received 2 August 2000
Abstract Multivariate nonlinear Fokker–Planck equations are derived which are solved by equilibrium distributions of generalized thermostatistics. The multivariate Fokker–Planck equations proposed by Kaniadakis and by Borland et al. are re-obtained as special cases. Furthermore, a Kramers equation is derived for particles obeying the nonextensive thermostatistics proposed by Tsallis. c 2001 Elsevier Science B.V. All rights reserved. PACS: 05.20−y; 05.40.+j Keywords: Nonlinear Fokker–Planck equation; Generalized entropy; Canonical ensembles; Bose and Fermi systems; Second law of thermodynamics
1. Introduction The study of irreversible processes is an integral part of the study of complex systems. For example, irreversible processes can describe chemical reactions [1–5], population dynamics [6], relaxation processes of assemblies of identical particles [7–9], or information processing in biological and arti?cial systems [10 –16]. Such stochastic processes are often formulated as solutions of master equations [3,4]. In statistical mechanics, properties of the transition rates that determine the explicit structure of master equations can be derived from assumed equilibrium distributions [8,9,17]. The equilibrium distributions, in turn, can be deduced, for example, by using the concepts of canonical ensembles [9,18]. Thus, irreversible stochastic processes that are described ∗
Corresponding author. Tel.: +31-20-444-8612; fax: +31-20-4448-509. E-mail address: [email protected] (T.D. Frank).
c 2001 Elsevier Science B.V. All rights reserved. 0378-4371/01/$ - see front matter PII: S 0 3 7 8 - 4 3 7 1 ( 0 0 ) 0 0 5 5 9 - 8
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393
by master equations can be linked to equilibrium statistics as well as to canonical ensembles. Furthermore, such a correspondence can be extended to irreversible processes described by Fokker–Planck equations since Fokker–Planck equations can be viewed as approximations of master equations [1,2,4,5,8]. When aiming at stochastic descriptions in terms of Fokker–Planck equations, there is a more direct route that may be followed: the relevant Kramers–Moyal coeFcients may be inferred from the assumed equilibrium statistics or, alternatively, from the assumed canonical ensembles. Such an approach has already been discussed by Kramers [19] and culminated in what is presently called as Kramers equation [1,2]. There are also several studies that propose to derive the relevant Kramers–Moyal coeFcients from steady-state or equilibrium-state autocorrelations, see, e.g., Refs. [20 –23]. Recently, the correspondence between stochastic descriptions via Fokker–Planck equations and canonical ensembles has been extended in order to encompass a variety of statistics that di)er from the Boltzmann–Gibbs–Shannon statistics: the statistics of fermions and of bosons [24,25], the nonextensive generalized thermostatistics proposed by Tsallis [26 –36] and the statistics based on the Renyi entropy [34]. A common feature of these stochastic evolution equations is that they are nonlinear with respect to their probability densities but recover conventional Fokker–Planck equations in well-de?ned limiting cases (see, however, [37,38]). We will therefore call the former kind of stochastic evolution equations nonlinear Fokker–Planck equations (NLFPEs) and the latter one linear Fokker–Planck equations. Many of the previously mentioned studies focus on one-dimensional Fokker–Planck equations. One may view these one-dimensional NLFPEs as possible generalizations of the Smoluchowski equation [1,2]. The present article extends the NLFPE/canonical ensemble correspondence principle established in Refs. [19,26 –36] to the multivariate case. To this end, we ?rst discuss drift forces that can be derived from potential functions and examine multivariate stochastic processes in the Smoluchowski limit. Subsequently, we consider more general random walks. We study stochastic processes described by generalized coordinates and generalized momenta of the system’s particles. As a special case, we discuss drift forces that are similar to those studied by Ebeling and Schimansky–Geier in the context of linear Fokker–Planck equations and that cannot be derived from a potential [39]. Finally, we propose generalized Kramers equations related to generalized thermostatistics. In the present article, we assume that the noise sources of the systems under considerations are homogeneous with respect to their state variables. In other words, we exclusively consider systems that are in thermal contact with spatially homogeneous heat baths and ignore phenomena related to local temperatures (blow torch e)ects). Consequently, the di)usion terms of the Fokker–Planck equations discussed here do not explicitly depend on the state variables. More elaborate discussions in this regard can be found in Refs. [32,40 – 42]. Before proceeding with our main objectives, we ?rst elucidate the relevance of generalized statistics such as nonextensive thermostatistics, see, e.g., Refs. [43– 48] and the extensive statistics induced by the Renyi entropy [49,50]. It is well known that classical thermodynamics applies to systems whose total energy is the sum of the
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energies of their macroscopic parts. Following Terletskii [51, Chapter 7] and Landsberg [52], systems characterized by long-range forces can fail to satisfy this property of additivity because interaction energies between macroscopic parts of these systems can make essential contributions to the total energy. Extensive thermostatistics can hardly cope with this problem. This is, for example, illustrated by the entropy of the Kerr– Newman black hole, which is a nonextensive variable [52]. In addition, Sharma and Garg pointed out that a number of biological and social systems should be viewed as nonextensive rather than as extensive systems and should therefore be modeled by means of nonextensive entropies [53]. In a nutshell, nonextensive systems seem to demand nonextensive statistical and stochastic descriptions. The establishment of such a theory, a nonextensive thermodynamics, lies at the heart of a current development in statistical physics which was inspired by the seminal work of Tsallis [43]. For reviews on this topic, the reader is referred to Refs. [54 –57]. Finally, entropies made up of terms like (pj ) , where pj represents the probability to ?nd a system in a state j, facilitate the investigation of fractal structures by means of the concepts of statistical mechanics. In this context, one can ?nd in the literature applications of the Renyi entropy [58] and the generalized nonextensive entropy proposed by Tsallis [55].
2. Generalized canonical ensembles and multivariate NLFPEs 2.1. Multivariate stochastic potential dynamics Consider a system whose state is described by the N -dimensional state vector q = (q1 ; : : : ; qN ), where the components qj represent dimensionless variables in the domain (−∞; ∞). Taking a statistical perspective, we assume that the equilibrium state of the system can be described in terms of a canonical ensemble involving a generalized entropy functional S and a potential V (q). We further assume that V (q) is a globally attractive potential, that is, V (|q| → ∞) → ∞, with V (q) ∈ C ∞ (RN ). Under these assumptions, we introduce the probability density P(q) of the system and the entropy functional N ˜ (1) S[P] := B · · · S(P(q)) d qi ; ˜ where the entropy kernel S(y) ∈ C ∞ [0; ∞) : y ∈ R+ → R assigns a particular amount of disorder to the function value y of the probability density P, while the outer function B(z) ∈ C 1 : z ∈ R → R ?xes the scale in which the overall degree of disorder is measured. Of course, the mere structure of the functional (1) does not imply that S represents an entropy functional. Throughout this article, however, it is assumed that the constituents B and S˜ are chosen in such a way that the functional (1) can be viewed as an entropy functional. 1 In line with the theory of canonical ensembles, we 1
For characteristic properties of entropies and entropy functionals, the reader is referred to Wehrl [59].
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then assume that the equilibrium statistics of the system Peq (q) makes the entropy functional (1) stationary under the constraints N 1 = · · · P(q) d qi and C = · · · V (q)P(q) d N qi : (2) To solve the corresponding variational problem, we use the Lagrange multipliers and and require that the variation of N N I := S + C − · · · V (q)P(q) d qi + 1 − · · · P(q) d qi (3) vanishes. From I = 0 it follows that ˜ d S(y) +V (q)=[Peq ] dy
y=Peq (q)
with
[P] :=
dB(z) d z z= ···
˜ S(P(q)) d N qi
; (4)
which implicitly determines the equilibrium statistics Peq (q). Having solved Eq. (4), the Lagrange multipliers and can be calculated from the constraints (2). In line with a previous study on the correspondence between canonical ensembles and one-dimensional NLFPEs [34], we now consider the multivariate Fokker–Planck equation N
@ @ P(q; t) = − @t @qj j=1
[P(q; t)] @ ˆ ˜ L{S(P(q; t))} × M [(P(q; t))] hj (q)P(q; t)− @qj
Y
(5) where the operator Lˆ acting on the function (y) is de?ned as d(y) ˆ L[(y)] := (y) − y : dy
(6)
In Eq. (5) the form M [(·)] denotes a function or functional of P(q; t) with M [(·)] ¿ 0. If B(z) = z ⇒ ≡ 1, we can con?ne our considerations to variables M that represent functions of the probability density P with M [(z)] = M (z) ∈ C ∞ [0; ∞), cf. Eq. (12) below. The drift terms hj (q) correspond to a deterministic force imposed on the system by the potential V (q ): hj (q) := − dV (q)= dqj . We now show that if Eq. (4) can be solved under the constraints (2), the obtained equilibrium statistics Peq (q) represents a stationary solution Pst (q) of the NLFPE (5). To this end, we di)erentiate the implicit de?nition of the equilibrium statistics given by Eq. (4) with respect to qj and obtain ˜ @ d S(y) [Peq ] = −hj (q) : (7) @qj dy y=Peq (q)
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Next, we evaluate the de?nition (6) of the operator Lˆ and ?nd ˜ @ d S(y) @ ˆ ˜ L{S(P(q; t))} = −P(q; t) : @qj dy y=P(q; t) @qj
(8)
Using Eqs. (7) and (8) one can further deduce that the term labeled Y in the NLFPE (5) vanishes for P = Peq . Consequently, the NLFPE (5) has a stationary solution Pst (q) that can be viewed as the equilibrium statistics Peq (q) induced by the functional S. In addition, if S is the Boltzmann–Gibbs–Shannon entropy, BGS S := − P(q) ln P(q) d N qi , we can ?nd a corresponding linear Fokker–Planck equation, namely, the Fokker–Planck equation (5) for S = BGS S and M [(·)] ≡ 1. Furthermore, we may consider the limit of vanishing noise ( → ∞) for M [(·)] ≡ 1. In this special case, the NLFPE (5) reduces to N
@ @ hj (q)P(q; t) : P(q; t) = − @t @qj
(9)
j=1
Eq. (9) can be solved in terms of a weak solution that reads P(C; t) = (C − q(t))
(10)
@ d V (q(t)) ; qj (t) = − dt @qj
(11)
with
where (z) denotes the N -dimensional delta distribution. 2 For this reason, the NLFPE (5) may be interpreted as stochastic description of a system whose deterministic evolution is described by the potential dynamics (11). Just as for the one-dimensional case [34], there are two special cases worth mentioning. First, for M [(·)] ≡ 1, the drift term in the NLFPE (5) is linear with respect to the probability density P, while, in general, the di)usion term is nonlinear. 2 Strictly speaking, we claim that the generalized function P(C; z)¿0 with t = z de?ned by Eqs. (10) and (11) satis?es the integral relation
∞
0=
dz 0
∞
∞
··· ∞
∞
dvjN P(C; z)
@ @ hj (C) (C; z) (C; z) + @z @vj N
j=1
for any test function (C; z) ∈ C ∞ (RN × [0; ∞)) that vanishes at the boundaries z = 0, z → ∞, and |C| → ∞. In fact, performing the integration over dvjN gives us
∞
dz 0
dqj @ @ (q; z) = (q; z) + @z d z @qj N
j=1
∞
dz 0
d (q(z); z) = (q; z)|z=∞ z=0 = 0 : dz
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Second, for
397
−1 ˜ d 2 S(y) M [(P)] = − [P(q; t)]P(q; t) dy2 y = P(q; t)
−1
˜ ˆ S(y)} d L{ = [P(q; t)] dy
(12)
y=P(q; t)
the NLFPE (5) reads
−1 N ˜ @ @ d 2 S(y) 1 @ hj (q) [P(q; t)] + P(q; t) P(q; t) = @qj dy2 P(q; t) @qj @t j=1
=−
N j=1
−1 ˜ ˆ S(y)} @ d L{ −1 @ hj (q) [P(q; t)] @qi dy @qj P(q; t)
×P(q; t) ;
(13)
the drift term may then be nonlinear with respect to the probability density P(q; t), whereas the di)usion term is linear. We illustrate the results obtained so far for the explicit entropy functional 1 T () {(P(q)) − P(q)} d N qi ; S := (14) 1− which has been studied in detail in the context of the generalized nonextensive thermostatistics proposed by Tsallis, cf., e.g., [43,54]. For M [(·)] ≡ 1 the corresponding multivariate NLFPE reads as N @ @ 1 @ P(q; t) = − hi (q)P(q; t) − : (15) (P(q; t)) @t @qi @qi i=1
By virtue of Eq. (12), we can also derive a NLFPE with a di)usion term that is linear with respect to the probability density. For S = T S() , however, we cannot ?nd a function or functional M [(·)] that yields both a linear drift and a linear di)usion term (except for the trivial case M [(·)] ≡ 0). Therefore, we may say that the NLFPE (5) related to T S() is essentially nonlinear. The stationary solution of Eq. (15) can be derived from Eq. (4) and is found as 1=(−1) N 1 1− Pst (q) = V (q) (16) ++ 1− i=1
for ∈ (0; 1) and potentials V (q) with a ?nite lower boundary, that is, ∀q: V (q)¿V0 . Other parameter values require more thorough analyses accounting for weak solutions (cut-o) solutions), see, e.g., Refs. [31,34,43]. Even for ∈ (0; 1) we have to check carefully whether a stationary solution of the form (16) exists at all. This cannot be
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taken for granted because even in the linear, one-dimensional case there are Fokker– Planck equations involving globally attractive potentials that do not admit for stationary solutions (consider, e.g., potentials that increase for large distances x according to ln|x|). Once the existence of a stationary solution is guaranteed, the constant can however, be computed from the normalization condition 1 = · · · Pst (q) d N qi . If analytical methods fail, might be estimated with the help of numerical techniques. Finally, note that in the limit → 1 the functional T S() recovers the Boltzmann–Gibbs–Shannon entropy functional BGS S. Similarly, in the limit → 1 the NLFPE (15) recovers a linear Fokker–Planck equation and the stationary probability density (16) reduces to a Boltzmann distribution. 2.2. Multivariate stochastic dynamics in phase space We now turn our attention to possible generalizations of the Kramers equation [1,2,19]. For that purpose, we take a somewhat broader point of view and examine a system whose state can be described by the generalized coordinate vector q and the generalized momentum vector p with q = (q1 ; : : : ; qN ) and p = (p1 ; : : : ; pN ), respectively. Consequently, we may assume that the components of q are invariant under time reversal, whereas the components of p change their signs when time is reversed, 3 that is, t → − t ⇒ (q; p) → (q; −p). Then, we introduce a function (q; p) in order to describe the equilibrium probability density Peq (q; p) of the system: Peq (q; p) is de?ned as the probability density that makes the entropy functional (1) stationary under the constraints 1 = · · · P(q; p) d 2N qi pi and E = · · · (q; p)P(q; p) d N qi d N pi : (17) Both constraints agree with the constraints commonly imposed on canonical ensembles. In particular, if (q; p) describes a Hamiltonian H (q; p), then the constant E is the mean energy of the system. Therefore, in general, we may consider (q; p) as a “generalized” Hamiltonian. The variational problem can be formulated by means of the Lagrange multipliers and and involves the functional N N I := S + E − (q; p)P(q; p) d qi d pi + 1 − P(q; p) d N qi d N pi :
(18)
Again, the solution of this variational problem implicitly de?nes the equilibrium probability density. Then, from I = 0 it follows that ˜ d S(y) : (19) + (q; p) = [Peq ] dy y=Peq (q; p)
3
Note that in what follows we will not refer to these properties. However, we will return to this subject in the Discussion (Section 3).
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If there exists a solution of Eq. (19), then it can be expressed in terms of a function F˜ as ˜ Peq = F( ) :
(20)
Note that the function F˜ depends solely on the function . Moreover, we may intro−1 ˜ where the normalization ˜ ln(z=N duce the function F(z) := F{− )} associatedN to F, −1 constant N is given by (N ) := · · · exp(− ) d qi d N pi . Eq. (20) can then be written as Peq = F(W) with the Boltzmann distribution W := N exp(− ). We realize that F(z) corresponds to the distortion function that has already been introduced in ˜ the context of one-variable NLFPEs [34]. Accordingly, F(z) as de?ned in Eq. (20) is interpreted as a function that maps the equilibrium density W, which is assumed by the system for S = BGS S, to the actual equilibrium probability density Peq induced by the generalized entropy functional under consideration. 4 In line with studies on stationary solutions of multivariate linear Fokker–Planck equations by van Kampen [60,61], Graham and Haken [62], Risken [63] and Graham [64], we introduce structural components of multivariate NLFPEs (cf. also Refs. [1,2,65 (qp) (pq) (pp) – 67]): the square matrices Kj;(qq) and the vector-like coeFcients r , Kj; r , Kj; r , Kj; r (q) (p) (q) (p) Ij , Ij , Dj , and Dj . Before interpreting these structural constituents, we formulate additional constraints. These constraints will guarantee that the multivariate NLFPEs (qp) (pq) (q) (p) (q) (p) based on Kj;(qq) are solved by Peq (q; p). First, r , Kj; r , Kj; r , Ij , Ij , Dj , and Dj (·) the coeFcients Ij are only implicitly de?ned. We require that they satisfy N @ (q) @ (p) ˜ ˜ [I (q; p)F( )] + [I (q; p)F( )] = 0 : (21) @qj j @pj j j=1
Note that the coeFcients Ij(·) are functions of q and p. Using Eq. (20), we obtain N @ (q) @ (p) [I (q; p)Peq ] + [I (q; p)Peq ] = 0 : (22) @qj j @pj j j=1
Second, we require that the vector-like coeFcients Dj(·) and the matrices Kj;(·)r ful?ll the relations N N @ (qp) @ Kj;(qq) (q; p) ; + K Dj(q) (q; p) :=: − r j; r @qr @pr j=1 r=1
Dj(p) (q; p) :=:
−
N N
Kj;(pq) r
j=1 r=1
@ @ + Kj;(pp) r @qr @pr
(q; p) :
(23)
Note that the elements Kj;(·)r are assumed to be constants, whereas the coeFcients Dj(·) are functions of q and p. In analogy to Section 2.1, from the de?nition of the operator 4 Similarly, using the Fokker–Planck equation perspective, the functions F˜ and F map the stationary solution W of the corresponding linear Fokker–Planck equation to the actual stationary probability Pst of the NLFPE under consideration.
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Lˆ (cf. Eq. (6)) we can derive Eq. (8), which now holds for derivatives with respect to qj and pj . Using this generalized version of Eq. (8) in combination with Eq. (19), we rewrite Eq. (23) as N N [Peq ] (qq) @ (qp) @ (q) ˜ eq )} ; ˆ S(P L{ Kj; r + Kj; r Dj Peq = − @qr @pr j=1 r=1
Dj(p) Peq = −
N N [Peq ] (pq) @ (pp) @ ˜ eq )} : ˆ S(P K L{ + K j; r j; r @qr @pr
(24)
j=1 r=1
We can now propose the multivariate NLFPE N @ @ (q) @ (p) P(q; p; t) = − (Ij + Dj(q) ) + (Ij + Dj(p) ) P(q; p; t) @pj @t @qj j=1
+
N N [P] (1) ˜ ˆ S(P(q; p; t))} ; Kj; r (@qj ; @pj ) L{
(25)
j=1 r=1
where the di)erential operator K(1) (@qj ; @pj ) is de?ned by (qq) K(1) j; r (@qj ; @pj ) := Kj; r
@2 @2 @2 (pq) + (Kj;(qp) + Kj;(pp) : r + Kr; j ) r @qj @qr @qj @pr @pj @pr
(26)
Since the coeFcients Ij(·) and Dj(·) occur in the NLFPE in combination with ?rst-order partial derivatives, they are referred to as drift coeFcients. Furthermore we can read o) from Eqs. (25) and (26) that the matrices Kj;(·)r weigh the components of the second-order di)erential operator K(1) (@qj ; @pj ). By analogy to linear Fokker–Planck equations, the matrices Kj;(·)r may be seen as measuring the strength of the Ructuations to which the multivariate stochastic process in question are subjected. In other words, the elements of the matrices Kj;(·)r are the strength of noise sources acting on the system. As stated in the introduction, we consider noise sources that are homogeneous with respect to the phase space in question – which is in agreement with our requirement of constant matrix elements Kj;(·)r . In this line, we will refer to the elements Kj;(·)r as di)usion coeFcients. Finally, from Eqs. (22) and (24) we can read o) that the equilibrium statistics Peq (q; p) is, indeed, a stationary solution of the NLFPE (25). The drift terms of the NLFPE (25) are linear with respect to the probability density P(q; p; t). Consequently, the NLFPE (25) fails to address NLFPEs as, for example, proposed by Kaniadakis in the context of classical stochastic descriptions of fermions and bosons [68]. To obtain NLFPEs with nonlinear drift terms we impose an additional constraint on the drift coeFcients Ij(·) , namely, that the divergence of the vector ?eld given by the components Ij(·) vanishes (see also Refs. [62,66]). That is, we require that N @ (q) @ (p) Ij (q; p) + Ij (q; p) = 0 (27) @qj @pj j=1
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holds in addition to the constraint (21). In this case, the drift functions Ij(·) obey the relation N @ @ (28) Ij(q) (q; p) Peq = 0 Peq + Ij(p) (q; p) @pj @qj j=1
instead of Eq. (22). From Eqs. (27) and (28) we can further conclude that N @ @ (q) (p) Ij (q; p) [M [(Peq )]Peq ] + Ij (q; p) [M [(Peq )]Peq ] = 0 @qj @pj j=1
⇒
N @ (q) @ (p) [Ij (q; p)M [(Peq )]Peq ] = 0 : [Ij (q; p)M [(Peq )]Peq ] + @pj @qj j=1
(29) In due recognition of relation (29), the NLFPE (25) can be replaced by N @ @ (q) @ (p) (q) (p) (I + Dj ) + (I + Dj ) M [(P)]P(q; p; t) P(q; p; t) = − @t @qj j @pj j j=1
+
N N [P] (M ) ˜ ˆ S(P(q; Kj; r (@qj ; @pj ) L{ p; t))} ;
(30)
j=1 r=1
where the di)erential operator K(M ) (@qj ; @pj ) is de?ned by @ ) (qq) @ (qp) @ K(M (@q ; @p ) := M [(P(q; p; t))] K + K j j j; r j; r j; r @qj @qr @pr @ (pq) @ (pp) @ : M [(P(q; p; t))] Kj; r + Kj; r + @pj @qr @pr
(31)
By virtue of Eqs. (24) and (29), one can verify that the equilibrium statistics Peq solves the NLFPE (30). The NLFPEs (25) and (30) may be regarded as stochastic descriptions of systems the evolution of which in the absence of noise (i.e., for → ∞) is determined by the drift functions Ij(·) and Dj(·) according to P(u; C; t) = (u − q(t)) · (C − p(t)) ; d qj (t) = Ij(q) (q(t); p(t)) + Dj(q) (q(t); p(t)) ; dt d (32) pj (t) = Ij(p) (q(t); p(t)) + Dj(p) (q(t); p(t)) : dt In general, the deterministic system (32) cannot be written as a potential dynamics such as given by Eq. (11). However, for (q; p) = V (q), Ij(q) = Ij(p) = Dj(p) ≡ 0, (pq) (pp) (qq) (q) Kj;(qp) r = Kj; r = Kj; r ≡ 0, Kj; r ≡ 1, Dj = −dV (q)=dqj , and = the NLFPE (30) recovers the NLFPE (5) – just as the system (32) recovers the potential dynamics (11).
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Three subtleties are worth mentioning. First, in this section we derived two distinct multivariate stochastic descriptions: the NLFPE (25) with drift functions Ij(·) satisfying the constraint (21) and the NLFPE (30) with drift functions Ij(·) satisfying both the constraints (21) and (27). Neither is the ?rst description a special case of the latter, nor is the latter description a special case of the ?rst. Second, for M [(·)] ≡ 1 the NLFPE (30) assumes the structure of the NLFPE (25) and exhibits a drift term that is linear with respect to the probability density P(q; p; t). In contrast, substituting −1 ˜ ˆ S(y)} d L{ (33) M [(P)] = [P(q; p; t)] dy y=P(q; p; t)
into Eq. (30) yields the NLFPE N 1 @ (q) d Lˆ @ (p) @ P(q; p; t) = − (Ij + Dj(p) ) P(q; p; t) (Ij + Dj(q) ) + @qj @pj dP @t [P] j=1
+
N N 1 (1) Kj; r (@qj ; @pj )P(q; p; t) :
(34)
j=1 r=1
Now, the di)usion term of the NLFPE (34) is linear with respect to P(q; p; t) but the drift term is, in general, nonlinear with respect to P(q; p; t). Third, substituting the Boltzmann–Gibbs–Shannon entropy functional BGS S into the NLFPE (25), we obtain a linear Fokker–Planck equation describing a multivariate stochastic process characterized by additive Ructuation forces. The same holds for the NLFPE (34) and the NLFPE (30) with M [(·)] ≡ 1. 2.3. Examples We consider a random walk in the six-dimensional Cartesian coordinate system described by the coordinate vector x = (x1 ; x2 ; x3 ) and the momentum vector p = (p1 ; p2 ; p3 ). 2.3.1. Ebeling/Schimansky–Geier systems Let the drift and di)usion coeFcients be de?ned by Ij(x) :=
@ H (x; p); @pj
Dj(x) := 0; Kj;(pp) r = &;
Ij(p) := −
Dj(p) := − &
@ H (x; p) ; @xj
@ H (x; p) ; @pj
(xp) (px) Kj;(xx) r = Kj; r = Kj; r ≡ 0 ;
(35)
where & denotes a semi-positive constant. The coeFcients Ij(·) constitute a vector ?eld with vanishing divergence, that is, Eq. (27) is satis?ed. Moreover, the coeFcients Ij(·)
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obey the constraint (21). Consequently, if there exists an equilibrium statistics Peq of the canonical ensemble de?ned by the entropy functional (1) and the constraints (17), then this equilibrium statistics corresponds to a stationary solution of the NLFPE (30) speci?ed by the coeFcients (35). In the absence of a friction force, that is, for & = 0, from Eq. (32) the deterministic evolution equations of the system in question can be obtained as @ d xj (t) = H (x(t); p(t)); dt @pj
d @ H (x(t); p(t)) : pj (t) = − @xj dt
(36)
Note that Eq. (36) describes a conservative system (dH=dt = 0). For M ≡ 1 and S = BGS S the NLFPE (30) becomes linear with respect to its probability density and the stochastic evolution equations of the corresponding random variables xi → 'i and pi → (i read @ d H (x; p) ; 'j (t) = dt @pj x=(t); p=(t) 2& @H @H d (j (t) = − −& + )(t) : (37) dt @xj x=(t); p=(t) @pj x=(t); p=(t)
Here, )(t) denotes a Langevin force with )(t))(t ) = (t − t ). The system (37) was addressed by Ebeling and Schimansky–Geier [39]. In sum, the NLFPE (30) in combination with the coeFcients (35) embodies the stochastic processes de?ned by Eq. (37) as a special case and features stochastic processes related to, for example, nonextensive thermostatistics and the thermostatistics based on the Renyi entropy. 2.3.2. Generalized Kramers equations We elaborate on the preceding example by specifying the Hamiltonian H (x; p) according to 3
H (x; p) :=
1 (pj )2 + U (x) ; 2m
(38)
j=1
where m ¿ 0 denotes the mass of the particle performing the random walk being studied and U (x) ∈ C ∞ (R3 ) stands for a globally attractive potential (i.e., U (|x| → ∞) → ∞). Inserting Eqs. (35) and (38) into the NLFPE (30), we obtain 3 pj @ dU (x) @ & @ P(x; p; t) = − M [(P)]P(x; p; t) − + pj @t m @xj @pj d xj m j=1
3
+
@ & @ ˆ ˜ L{S(P(x; p; t))} : [P] M [(P)] @pj @pj
(39)
j=1
The NLFPE (39) shall be viewed as a possible generalization of the Kramers equation in the context of generalized thermostatistics. Note that for M [(·)] ≡ 1 and S = BGS S
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the NLFPE (39) reduces to the ordinary Kramers equation [1,2] and can describe stochastic processes de?ned by the Langevin equation 1 d 'j (t) = (j (t) ; m dt d 2& & d (j (t) = − (j (t) − U (x) + )(t) ; (40) dt m d xj x=(t)
cf. also Eq. (37). 2.3.3. Kramers equations for classical bosons and fermions Next, we consider ensembles of classical bosons and fermions. Entropy functionals for bosons and fermions are usually formulated in the context of grand canonical ensembles [18]. Adapting these entropy functionals for canonical ensembles, we may consider the entropy functionals BE S and FD S de?ned by BE S = − {P ln P + (1 − P) ln(1 − P)} d 3 xi d 3 pi (41) and FD
S=−
{P ln P − (1 + P) ln(1 + P)} d 3 xi d 3 pi
(42)
for the Bose–Einstein statistics and the Fermi–Dirac statistics, respectively. Then, it ˜ = ln(1 + P) for bosons and L( ˜ = −ln(1 − P) for fermions. 5 ˆ S) ˆ S) follows that ≡ 1, L( For M [(·)] ≡ 1 we obtain 3 pj @ dU (x) @ & @ P(x; p; t) − + pj P(x; p; t) = − @t m @xj @pj d xj m j=1
±
3 & @2 ln {1 ± P(x; p; t)} : @pj2 j=1
(43)
In Eq. (43) the upper sign holds for classical bosons and the lower sign for classical fermions. Using Eq. (33), we can derive the Kramers equations with di)usion terms which are linear with respect to their probability densities. These Kramers equations read 3 pj @ @ dU @ & [1 ± P(x; p; t)]P(x; p; t) − + pj P(x; p; t) = − @t m @xj @pj d xj m j=1
+ 5
3 & @2 P(x; p; t) : @pj2 j=1
(44)
Note that in an earlier study of the NLFPE/canonical ensemble correspondence concerning the Bose– Einstein statistics and the Fermi–Dirac statistics some signs have been confused (cf. columns 3 and 4 of ˜ = ln(1 + P) and ˆ BE S) Table 1 [34]). In the present article Eqs. (41) and (42) in combination with L( FD ˜ ˆ L( S) = −ln(1 − P) give the correct formula.
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Again, the upper sign refers to Bose–Einstein statistics and the lower to Fermi–Dirac statistics. The NLFPE (44) was proposed by Kaniadakis as approximation of a master equation that takes speci?c exclusion/inclusion principles into account [68]. 2.3.4. Kramers equations for the generalized thermostatistics proposed by Tsallis Finally, we can derive Kramers equations for the nonextensive generalized thermostatistics proposed by Tsallis [43] by substituting the entropy functional (14) into Eq. (39). For M [(·)] ≡ 1 we obtain 3 3 pj @ & @2 dU & @ @P =− P+ + pj − (P) : (45) 2 m @xj @pj d xj m @t @p j j=1 j=1 For ∈ (0; 1) the stationary solution/equilibrium statistics can be derived from Eq. (19) and is found as 1=(−1) 1 1− Pst (x; p) = ; (46) + + H (x; p) 1− where H is given by Eq. (38). Note that for = 1 the NLFPE (45) recovers the classical Kramers equation. Similarly, in the limit → 1 the stationary solution (46) reduces to a Boltzmann distribution of the generalized Hamiltonian H . For ¿ 1 we deal with weak stationary solutions, which usually require more detailed analyses. The NLFPE (45) provides an explicit example of a multivariate NLFPE related to the thermostatistics based on the entropy functional T S() . Similar multivariate NLFPEs have been studied by Compte et al. [28,29] and Borland et al. [32]. Moreover, the multivariate NLFPE (45) can be considered as a possible extension of the one-dimensional NLFPE related to entropy functional T S() proposed by Plastino and Plastino [26]. We want to point out that Eq. (39) allows for the derivation of further Kramers equations related to the Renyi entropy [49], the entropy (U ) S proposed by Landsberg [46], the entropy related to stretched exponential distributions proposed by Anteneodo and Plastino [48], and the entropy of systems with statistical feedback [34]. 2.4. On the second law of thermodynamics In Sections 2.1 and 2.2 we derived multivariate NLFPEs which have generalized thermostatistics as stationary solutions. These NLFPEs were obtained under the explicit assumption that stationary solutions exist. Having derived these NLFPEs, however, one may explore their properties irrespective of the existence of stationary solutions. Hence, we now assume that there are particular stochastic processes that exist irrespective of whether or not they converge to stationary processes. If these processes converge to stationary processes, then they can be described by both canonical ensembles (for the stationary case) and multivariate NLFPEs (for the transient and stationary case). Furthermore, if the processes remain in transient regimes, then they can still be described by the NLFPEs despite the fact that in such cases the approach via canonical ensembles fails. In particular, we would like to answer the question whether the NLFPEs
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(5), (25) and (30) de?ne stochastic processes that are consistent with the second law of thermodynamics. To this end, we examine the evolution of the generalized entropy functional S for purely di)usive stochastic processes that are de?ned by the NLFPEs (5), (25) and (30) for vanishing drift forces. First of all, we observe that, in the absence of a drift, the NLFPE (25) can be regarded as a special case of the NLFPE (30). Consequently, we can con?ne our considerations to the NLFPEs (5) and (30). We then di)erentiate the entropy functional S given by Eqs. (1) with respect to the time variable t and obtain in analogy to the one-dimensional case [35] ˜ @ d d S(y) (47) P(q; t) d N qi S[P] = [P] · · · dy y=P(q; t) @t dt and d S[P] = [P] dt
˜ @ d S(y) P(q; p; t) d N qi d N pi ··· dy y=P(q; p; t) @t
(48)
for the entropy functionals related to the NLFPE (5) and the NLFPE (30), respectively. Inserting the NLFPE (5) into Eq. (47) with hj ≡ 0, and integrating by parts, gives us the ?nal result 2 N 2 1 d S˜ @ d 2 S[P]= {[P]} ··· P(q; t)M [(P)] d N qi : P(q; t) dt dP2 @qk k=1
(49) For ¿ 0 the entropy functional S is a nondecreasing function with respect to the time variable t. The positivity of the Lagrange multiplier agrees with its interpretation as a measure of the Ructuation strength (see above). Moreover, Plastino and Plastino [69] and Mendes [70] showed that, in the context of generalized thermostatistics de?ned by canonical ensembles, the Lagrange multiplier plays the role of a temperature measure. Turning to the NLFPE (30), a similar analysis can be carried out. For the sake of simplicity, we ?rst introduce the composed 2N -dimensional vector w = (w1 ; : : : ; w2N ) de?ned by w1 := q1 ; : : : ; wN := qN and wN +1 := p1 ; : : : ; w2N := pN and 6 the 2N × 2N di)usion matrix Kj;(w) r given by {Kj;(qq) {Kj;(qp) r } r } (w) ; (50) {Kj; r } := (pq) (pp) {Kj; r } {Kj; r } which agrees with the di)usion matrix de?ned in Ref. [62]. Then, for vanishing drift (i.e., Dj(·) ≡ Ij(·) ≡ 0) the NLFPE (30) can equivalently be expressed as 2N 2N [P] @ (w) @ @ ˆ ˜ P(w; t) = − L{S(P(w; t))} : (51) K M [(P)] @t @wj j; r @wr j=1 r=1
6
Here, {Aj; r } denotes the matrix A consisting of the elements Aj; r .
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Substituting (51) into the evolution equations of S given by Eq. (48) results in 2 2 (w) 2N 2N ˜ K d @P @P S d k; r S[P]={[P]}2 · · · P · M [(P)] d 2N wi : dP2 @wk @wr dt k=1 r=1
(52) 7 then dS¿0 holds. The Consequently, if the matrix Kj;(w) r = is semi-positive de?nite, requirement of a semi-positive di)usion matrix is well known in the theory of linear Fokker–Planck equations [2]. The immediate conclusions are, at least, twofold. First, the stochastic processes proposed in this article are consistent with the second law of thermodynamics provided that the di)usion coeFcients and Kj;(w) r = are appropriately chosen. Second, in turn, when starting o) with the requirement that the stochastic processes de?ned by the NLFPEs (5), (25) and (30) are consistent with the second law of thermodynamics, one can deduce necessary conditions for the di)usion coeFcients. As it turns out, these conditions are akin to the constraints that are usually imposed on the di)usion coeFcients of linear Fokker–Planck equations.
3. Discussion We derived stochastic evolution equations for time-dependent probability densities from equilibrium statistics of generalized canonical ensembles. We showed that the correspondence principle that assigns these stochastic evolution equations to canonical ensembles is not a peculiarity of the Boltzmann–Gibbs–Shannon statistics. It applies to a variety of generalized statistics irrespective of the dimensionality of the problem in hand. This basic feature reveals the generic nature of this approach in the study of irreversible stochastic processes. The derivation of Kramers equations related to generalized thermostatistics was carried out by adopting concepts from the theory of linear Fokker–Planck equations with stationary solutions satisfying the detailed balance condition. Although a deeper discussion of the implications of detailed balance for the nonlinear stochastic evolution equations discussed in Section 2.2 is beyond the scope of the present article, we want to conclude with a few general remarks. In Section 2.2, we obtained that if stochastic evolution equations can be cast under the guise of Eq. (25) and if their drift functions Dj(·) and Ij(·) are related to a function according to Eqs. (21) and (23), then the stationary probability densities can be expressed as functions of . Concerning stochastic evolution equations of the form (30), we have to require that the drift coeFcients Ij(·) also satisfy the constraint (27). In the context of linear Fokker–Planck equations, the constraints (21) and (27) imposed on the drift coeFcients Ij(·) and the 7
That is, ∀b: ( )−1
! j; r
(w)
Kj; r bj br ¿0 with b := (b1 ; : : : ; b2N ) and bj ∈ R.
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relations between the drift coeFcients Dj(·) and the di)usion matrices Kj;(·)r according to Eq. (23) can be derived from the assumption that the detailed balance condition holds (cf., e.g., Refs. [1,2,60 – 64,66]). Furthermore, the drift coeFcients Ij(·) correspond to processes that change sign if time is reversed (odd variables [60,61], negative parity [71]; cf. also the de?nition of pj in Section 2.2), whereas Dj(·) correspond to processes that are invariant under time reversal (even variables, positive parity; cf. also the definition of qj in Section 2.2). Then, Ij(·) are usually called the reversible parts of the drift coeFcient and Dj(·) the irreversible parts. Future works may clarify whether the concept of detailed balance and/or the concept of even and odd variables can also be applied to stochastic evolution equations related to generalized thermostatistics as formulated in Section 2.2. Irrespective of this issue, however, a decomposition of the drift terms of the multivariate stochastic evolution equations (25) and (30) into Ij(·) -terms and Dj(·) -terms is reminiscent of the decomposition of drift terms of multivariate linear Fokker–Planck equations. In the latter case, stationary solutions can be obtained by quadratures irrespective of whether or not the detail balance condition holds (cf., e.g., Ref. [72]). In the former case, stationary solutions can be derived by solving a corresponding variational problem. If analytical calculations fail, they can be determined at least implicitly, cf. Eqs. (7) and (19).
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