Consequences of the H Theorem from Nonlinear Fokker-Planck Equations

PHYSICAL REVIEW E 76, 041123 ͑2007͒

Consequences of the H theorem from nonlinear Fokker-Planck equations

Veit Schwämmle,* Fernando D. Nobre,† and Evaldo M. F. Curado‡ Centro Brasileiro de Pesquisas Físicas, Rua Xavier Sigaud 150, 22290-180 Rio de Janeiro, Rio de Janeiro, Brazil ͑Received 18 June 2007; published 17 October 2007͒ A general type of nonlinear Fokker-Planck equation is derived directly from a master equation, by introducing generalized transition rates. The H theorem is demonstrated for systems that follow those classes of nonlinear Fokker-Planck equations, in the presence of an external potential. For that, a relation involving terms of Fokker-Planck equations and general entropic forms is proposed. It is shown that, at equilibrium, this relation is equivalent to the maximum-entropy principle. Families of Fokker-Planck equations may be related to a single type of entropy, and so, the correspondence between well-known entropic forms and their associated Fokker-Planck equations is explored. It is shown that the Boltzmann-Gibbs entropy, apart from its connection with the standard—linear Fokker-Planck equation—may be also related to a family of nonlinear Fokker-Planck equations.

DOI: 10.1103/PhysRevE.76.041123 PACS number͑s͒: 05.40.Fb, 05.20.Ϫy, 05.40.Jc, 66.10.Cb

I. INTRODUCTION nonlinear effects on its associated transition probabilities ͓21–23͔. It is well known that many real systems exhibit a dynami- The nonextensive statistical mechanics formalism has cal behavior that falls out of the scope of the standard linear emerged naturally as a strong candidate for dealing appropri- differential equations of physics. Although the linear Fokker- ately with many real systems that are not satisfactorily de- Planck equation ͑FPE͓͒1͔ is considered appropriate for the scribed within standard ͑extensive͒ statistical mechanics description of a wide variety of physical phenomena— ͓24–26͔. The powerlike probability distribution that maxi- typically those associated with normal diffusion—it is well mizes the entropy proposed by Tsallis ͓27–29͔ is very often accepted that this equation is not adequate for describing found as a solution of nonlinear FPEs ͓12–16,18͔, suggesting anomalous diffusion. An example consists of particle trans- that the nonextensive statistical mechanics formalism should port in disordered media ͓2͔, such as amorphous materials, or be intimately related to nonlinear FPEs. In what concerns some other kind of media containing impurities and/or de- thermodynamics, one requires the extensivity of the entropy; fects. In such systems, particles are driven by highly irregu- however, there are several examples in the literature for ͓ ͔ lar forces, which lead to transport coefficients that may vary which the Boltzmann-Gibbs entropy is not extensive 26,30 , ͓ ͔ locally in a nontrivial manner. In addition to those, various including quantum spin systems 31,32 . This suggests the other phenomena fall out of the scope of the linear FPE, such utility of alternative entropic forms that may be, in certain as surface growth ͓3͔, diffusion of micelles in salted water cases, extensive. Many important equations and properties of standard sta- ͓4͔, polytropic distributions of self-gravitating stellar systems tistical mechanics have been extended within the formalism ͓5͔, the heartbeat histograms in a healthy individual ͓6͔, re- of nonextensive statistical mechanics. An example is the H laxation of two-dimensional turbulence in a pure-electron theorem, which was shown to be valid, taking into account plasma ͓7͔, anomalous diffusion in an optical lattice ͓8͔, and ͓ ͔ certain restrictions on the parameters of the corresponding black-hole radiation 9 . These kinds of phenomena became entropic form ͓11,18,33–35͔. Usually, one proves the H theo- one of the most investigated topics in physics nowadays, rem by defining previously an entropic form, and then con- leading to many interesting new aspects and to a wide range sidering either the master equation or a FPE, when dealing of open problems. with the time derivative of the probability distribution. In order to cope with such anomalous systems, modifica- The main motivation of this paper is to prove the H theo- tions in the linear FPE have been carried out, and this subject rem for a system in the presence of an external potential and has attracted the attention of many researchers recently. Es- following a general type of nonlinear FPE. In order to sentially, there are two alternatives for introducing modifica- achieve this, we introduce a relation involving quantities of tions in the linear FPE: ͑i͒ a procedure that leads to the ͑ ͓ ͔ ͒ the FPE with an entropic form; in principle, one may have fractional FPE see Ref. 10 for a review , where one con- classes of Fokker-Planck equations associated with a single siders a linear theory with nonlocal operators carrying the ͑ ͒ ͓ ͔ entropic form. We show that, when considered at equilib- anomalous nature of the process; ii the nonlinear FPEs 11 rium, this relation is equivalent to the maximum-entropy that in most of the cases come out as simple phenomenologi- ͓ ͔ principle. In the next section we derive a general FPE di- cal generalizations of the standard linear FPE 12–20 . Re- rectly from a master equation by introducing nonlinear terms cently, it has been shown that nonlinear FPEs may be derived in its transition probabilities; such a FPE will be used directly from a standard master equation, by introducing throughout most of this paper. In Sec. III we prove the H theorem by using this FPE, and show that the validity of this theorem requires a relation involving a general entropic form *[email protected] and the parameters of this nonlinear FPE. In Sec. IV we †Corresponding author; [email protected] discuss particular cases of this FPE and their associated en- ‡[email protected] tropic forms. In Sec. V we introduce a modified FPE that is

1539-3755/2007/76͑4͒/041123͑8͒ 041123-1 ©2007 The American Physical Society SCHWÄMMLE, NOBRE, AND CURADO PHYSICAL REVIEW E 76, 041123 ͑2007͒ compatible with the definition of a “generalized internal en- in such a way that considering the limit ⌬→0, one gets the ergy,” as used within the context of nonextensive statistical nonlinear FPE, mechanics. The same relation introduced previously is also P͑x,t͒ץ ץ A͑x͒⌿͓P͑x,t͔͖͕͒ץ P͑x,t͒ץ necessary in this case, in order to prove the H theorem. Fi- =− + ͭ⍀͓P͑x,t͔͒ ͮ, xץ xץ xץ tץ .nally, in Sec. VI we present our conclusions ͑ ͒ II. DERIVATION OF THE NONLINEAR FOKKER- 2.5 PLANCK EQUATION FROM A MASTER EQUATION with In this section we will derive, directly from the master ⌿͓P͑x,t͔͒ = P͑x,t͒a͓P͑x,t͔͒, ͑2.6a͒ equation, the nonlinear FPE that will be investigated throughout most of the present paper; we will follow closely ⌼͓P,R͔ץ ⌼͓P,R͔ץ ͔ ͓ the approach used in Refs. 21,22 . Let us then consider the ⍀͓P͑x,t͔͒ = ͫ⌼͓P,R͔ + P͑x,t͒ͩ − ͪͬ , ץ ץ standard master equation, associated with a discrete spec- P R R=P trum, ͑2.6b͒ ϱ .xץ/R͑x,t͒ץxϵץ/P͑x,t͒ץ P͑n,t͒ where we have used the fact thatץ = ͚ ͓P͑m,t͒w ͑t͒ − P͑n,t͒w ͑t͔͒, ͑2.1͒ ͒ m,n n,m ͑ ͒ ␾͑ ץ t m=−ϱ The external force A x is associated with a potential x ͓A͑x͒=−d␾͑x͒/dx, ␾͑x͒=−͐x A͑xЈ͒dxЈ͔, and we are as- ͑ ͒ −ϱ with P n,t representing the probability for finding a given suming analyticity of the potential ␾͑x͒, as well as integra- system in a state characterized by a variable n, at time t.We bility of the force A͑x͒ in all space. Furthermore, the func- introduce nonlinearities in the system through the following tionals ⌿͓P͑x,t͔͒ and ⍀͓P͑x,t͔͒ are supposed to be both transition rates: positive finite quantities, integrable as well as differentiable 1 ͑at least once͒ with respect to the probability distribution w ͑⌬͒ =− ␦ A͑k⌬͒ a͓P͑k⌬,t͔͒ ͑ ͒ ⍀͓ ͔ ⌿͓ ͔ʦ 1 k,l ⌬ k,l+1 P x,t ; i.e., they should be at least, P , P C .Inad- dition to that, ⌿͓P͑x,t͔͒ should be also a monotonically in- 1 creasing functional of P͑x,t͒. + ͑␦ + ␦ ͒ ⌼͓P͑k⌬,t͒,R͑l⌬,t͔͒. ͑2.2͒ ⌬2 k,l+1 k,l−1 As usual, we assume that the probability distribution, together with its first derivative, as well as the product ͑ ⌬͒ In the equation above, A k represents an external dimen- A͑x͒⌿͓P͑x,t͔͒, should all be zero at infinity, sionless force, a͓P͔ is a functional of the probability P͑n,t͒, P͑x,t͒ץ ,whereas the functional ⌼͓P,R͔ depends on two probabilities ͉ ͑ ͉͒ ͯ ͯ P x,t → ϱ =0, =0, ץ ± P and R, that are associated with two different states, al- x x x→±ϱ though R͑k⌬,t͒ϵP͑k⌬,t͒. Substituting this transition rate in ͉ ͑ ͒⌿͓ ͑ ͔͉͒ ͑∀ ͒ ͑ ͒ Eq. ͑2.1͒, performing the sums, and defining x=k⌬, one gets A x P x,t x→±ϱ =0 t . 2.7 -P͑x,t͒ 1 The conditions above guarantee the preservation of the norץ =− ͕P͑x + ⌬,t͒A͑x + ⌬͒a͓P͑x + ⌬,t͔͒ malization for the probability distribution, i.e., if for a given ⌬ ץ t ͐ϱ ͑ ͒ time t0 one has that −ϱdx P x,t0 =1, then a simple integra- − P͑x,t͒A͑x͒a͓P͑x,t͔͖͒ tion of Eq. ͑2.5͒ with respect to the variable x yields 1 ϱ ϱ ץ ͔͒ ͑ ͒ ⌬ ͑ ⌼͓͒ ⌬ ͑ ͕ + 2 P x + ,t P x + ,t ,R x,t ͵ dx P͑x,t͒ =−͕A͑x͒⌿͓P͑x,t͔͖͒ ϱ− ץ ⌬ t −ϱ 1 ϱ P͑x,t͒ץ P͑x − ⌬,t͒⌼͓P͑x − ⌬,t͒,R͑x,t͔͖͒ − P͑x,t͒ + ⌬2 + ͩ⍀͓P͑x,t͔͒ ͪ =0, ͑2.8͒ ץ x −ϱ ϫ͕⌼͓P͑x,t͒,R͑x + ⌬,t͔͒ + ⌼͓P͑x,t͒,R͑x − ⌬,t͔͖͒. and so, ͑2.3͒ ϱ ϱ The quantities depending on ⌬ may be expanded for small ⌬, ͵ ͑ ͒ ͵ ͑ ͒ ͑∀ ͒ ͑ ͒ dx P x,t = dx P x,t0 =1 t . 2.9 e.g., −ϱ −ϱ R͑x,t͒ It is important to stress that the nonlinear FPE of Eq. ͑2.5͒ isץ ⌼͓P͑x,t͒,R͑x ± ⌬,t͔͒ = ͫ⌼͓P͑x,t͒,R͑x,t͔͒ + ͩ±⌬ -x very general and reproduces well-known FPEs of the literaץ ture. As examples, one should mention the particular cases: ͒ ͑ ͔͒ ͑ ⌼͓P,R͔ ͑ ͒ ⌿͓ץ 2R͑x,t͒ץ 2⌬ + ͪ i the linear FPE is recovered for P x,t = P x,t and R ⍀=D ͑constant͒; ͑ii͒ the nonlinear FPE that presents Tsallisץ x2ץ 2 distribution as a solution ͓12,13͔, is obtained by setting 2⌼͓P,R͔ץ R͑x,t͒ 2ץ 2⌬ + ͩ ͪ + ͬ , ⌿͓P͑x,t͔͒= P͑x,t͒ and ⍀͓P͑x,t͔͒=qD͓P͑x,t͔͒q−1, where q is ¯ 2 ץ ץ 2 x R R=P the well-known entropic index ͓27͔, characteristic of the ͑2.4͒ nonextensive statistical mechanics formalism; ͑iii͒ the non-

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ϱ Pץ linear FPE derived previously from the master equation dF d␾͑x͒ ͓21–23͔ is recovered for ⌿͓P͑x,t͔͒= P͑x,t͒. =−͵ dxͩ ⌿͓P͔ + ⍀͓P͔ ͪ xץ dt ϱ dx From the experimental point of view, particular cases of − Pץ the nonlinear FPE of Eq. ͑2.5͒ were shown to be important d␾͑x͒ 1 d⌳͓Q͔ d2g͓P͔ ͑ ͒ ϫͩ − ͪ. ͑3.4͒ xץ for the description of several physical phenomena: i par- dx ␤ dQ dP2 ticle transport in porous media ͓A͑x͒=0, ⍀͓P͑x,t͔͒ =D͓P͑x,t͔͒␮, ␮ʦR͔͓2͔ and a specific application of this In most of the cases, one is interested in verifying the H equation corresponds to a diffusion of micelles in salted wa- theorem by using a well-defined FPE, together with particu- ter, in which case ␮=−3/2 ͓4͔; ͑ii͒ surface growth, which lar entropic forms, in such a way that some of the quantities, may be governed by nonlinear equations characterized by ⌳͓ ͔ ⍀͓ ͔ ⌿͓ ͔ 2 ͓ ͔ 2 ͓ ͔ Q , P , P , and d g P /dP , are previously defined complicated force and diffusive terms 3 . ͑ ͓ ͔͒ In the next section we prove the H theorem for a system see, e.g., Refs. 18,35 . Herein, we follow a more general approach, i.e., we assume that Eqs. ͑2.5͒, ͑2.7͒, ͑3.1͒, and in the presence of an external potential and following the ͑ ͒ general type of nonlinear FPE of Eq. ͑2.5͒. 3.2 are satisfied, and then, we impose the condition

1 d⌳͓Q͔ d2g͓P͔ ⍀͓P͔ III. H THEOREM − = . ͑3.5͒ ␤ dQ dP2 ⌿͓P͔ Herein, we will consider a general type of entropic form, satisfying the following conditions: Using this condition, Eq. ͑3.4͒ may be written as

ϱ ϱ P 2ץ dF d␾͑x͒ ⍀͓P͔ ͔͒ ͑ ͓ ͔ ͓ ͔͒ ͓ ͑⌳ ͔ ͓ S P = Q P , Q P = ͵ dx g P x,t , =−͵ dx ⌿͓P͔ͩ + ͪ Յ 0, ͑3.6͒ ץ ͔ ϱ ⌿͓− dt −ϱ dx P x

d2g and we remind one that ⌿͓P͔ is a positive, monotonically g͑0͒ = g͑1͒ =0, Յ 0, ͑3.1͒ ͑ ͒ dP2 increasing functional of P x,t . It should be stressed that Eq. ͑3.5͒ expresses an important where ⌳͓Q͔ represents a monotonically increasing outer relation involving quantities of the FPE and possible entropic functional with dimensions of entropy that is supposed to forms, for the case of a system in the presence of an external satisfy, at least, ⌳͓Q͔ʦC1, whereas the inner functional potential. It leads to a correspondence between whole families of FPEs, defined in terms of the functionals ⍀͓P͔ and g͓P͑x,t͔͒ should be also, at least, g͓P͑x,t͔͒ʦC2 in the inter- ⌿͓P͔, with a single entropic form. Therefore, it allows the val 0Ͻ P͑x,t͒Ͻ1 ͑end points excluded͒. Since we are deal- calculation of the entropic form associated with a given class ing with a system that exchanges energy with its surround- of FPEs; on the other hand, one may also start by considering, herein represented by the potential ␾͑x͒, it is important ing a given entropic form and then find the class of FPEs to define also the free-energy functional, associated with it. In fact, since the FPE is a phenomenologi- ϱ cal equation that specifies the dynamical evolution associated 1 ͑ ͒ F = U − S, U = ͵ dx ␾͑x͒P͑x,t͒, ͑3.2͒ with a given physical system, Eq. 3.5 may be useful in the ␤ −ϱ identification of the entropic form associated with such a system. In particular, one may identify entropic forms asso- where ␤ represents a positive Lagrange multiplier. ciated with some anomalous systems, exhibiting unusual be- The H theorem, for a system subject to an external poten- havior that are appropriately described by nonlinear FPEs, tial, corresponds to a well-defined sign for the time deriva- like the one of Eq. ͑2.5͒. Within the present approach, the tive of the above free-energy functional, which we will con- relation of Eq. ͑3.5͒ should hold for the H theorem to be sider as ͑dF/dt͒Յ0. Using the definitions above, valid; even though the relation of Eq. ͑3.5͒ may not be unique, we shall argue its relevance in what follows. ϱ First of all, let us show that at equilibrium, Eq. ͑3.5͒ is 1 ץ dF = ͩ͵ dx ␾͑x͒P͑x,t͒ − ⌳͑Q͓P͔͒ͪ equivalent to the maximum-entropy principle. For that, we ␤ ץ dt t −ϱ introduce the functional ϱ Pץ d⌳͓Q͔ dg͓P͔ 1 = ͵ dxͩ␾͑x͒ − ͪ , ͑3.3͒ ϱ ץ ␤ −ϱ dQ dP t I͓P͑x,t͔͒ = ⌳͑Q͓P͔͒ + ␣ͩ1−͵ dx P͑x,t͒ͪ −ϱ where we remind one that ⌳͓Q͔ and d⌳͓Q͔/dQ do not de- ϱ pend on the variable x. Now, one may use the FPE of Eq. + ␤ͩU − ͵ dx ␾͑x͒P͑x,t͒ͪ, ͑3.7͒ ͑2.5͒ for the time derivative of the probability distribution; −ϱ carrying an integration by parts, and assuming the conditions of Eq. ͑2.7͒, one gets where ␣ and ␤ are Lagrange multipliers. Then, one has that

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dI͓P͔ d⌳͓Q͔ dg͓P͔ An important—and complementary—property required ͯ ͯ =0⇒ ͯ ͯ − ␣ − ␤␾͑x͒ for a functional satisfying the H theorem is that it should be dP ͑ ͒ dQ dP ͑ ͒ P=Peq x P=Peq x bounded from below, =0, ͑3.8͒ ͑ ͒ where Peq x represents the equilibrium probability distribu- ͑ ͒ Ն ͑ ͒ ͑∀ ͒ ͑ ͒ F„P x,t … F„Peq x … t . 3.12 tion. From the general FPE of Eq. ͑2.5͒, one gets that, at equilibrium, Herein, we assume the presence of a unique equilibrium state ⍀͓ ͔ ͑ ͒ in the functional F(P͑x,t͒). In this case, Eq. ͑3.12͒ together Peq dPeq x A͑x͒ = , ͑3.9͒ with the imposition from the H theorem, for a time- ⌿͓P ͔ dx eq decreasing functional F, ensure that, after a long time, the which, after integration, yields system will always reach equilibrium. Therefore, it is sufﬁ- cient to prove that the requirement of Eq. ͑3.12͒ holds only x ⍀͓P ͔ dP ͑x͒ ␾ − ␾͑x͒ = ͵ dx eq eq in the nearness of the global equilibrium. Let us then con- 0 ⌿͓P ͔ dx sider, x0 eq ͑ ͒ Peq x ⍀͓P ͑xЈ͔͒ = ͵ eq dP ͑xЈ͒, ͑3.10͒ ⌿͓ ͑ Ј͔͒ eq ϱ P ͑x ͒ Peq x 1 eq 0 ͑ ͒ ͑ ͒ ͵ ␾͑ ͒͑ ͒ ͕⌳͑ ͓ ͔͒ F P − F Peq = dx x P − Peq − Q P ␾ ϵ␾͑ ͒ ͑ ͒ ϱ ␤ where 0 x0 is a constant. Integrating Eq. 3.5 , at equi- − librium, ⌳͑ ͓ ͔͖͒ ͑ ͒ − Q Peq , 3.13 1 d⌳͓Q͔ dg͓P͔ ͯ ͯ ␾͑ ͒ ͑ ͒ = x + C1, 3.11 ␤ dQ dP ͑ ͒ Peq x which may be expanded, near the equilibrium, up to O͓͑P ͑ ͒ ͒2͔ where we have used Eq. 3.10 , and C1 is a constant resulting − Peq . It should be noticed that an expansion on the prob- ͑ ͒ ͑ ͒ from the above integration. One notices that the equation ability P x,t , near Peq x , implies an expansion of the func- ⌳͑ ͓ ͔͒ ͓ ͔ ͓ ͔ above is equivalent to the one obtained from the maximum- tional Q P in powers of Q P −Q Peq ; carrying out entropy principle ͓cf. Eq. ͑3.8͔͒. such an expansion, one gets that

ϱ 1 d⌳͓Q͔ dg͓P͔ 1 1 d⌳͓Q͔ d2g͓P͔ F͑P͒ − F͑P ͒ = ͵ dxͭ͑P − P ͒ͩ␾͑x͒ − ͯ ͯ ͪ + ͑P − P ͒2ͩ− ͯ ͯ ͪͮ eq eq ␤ eq ␤ 2 −ϱ dQ dP ͑ ͒ 2 dQ dP ͑ ͒ Peq x Peq x ϱ 1 d2⌳͓Q͔ dg͓P͔ 2 − ͯ ͯ ͩ͵ dxͯ ͯ ͑P − P ͒ͪ + . ͑3.14͒ ␤ 2 eq ¯ 2 dQ ͑ ͒ −ϱ dP ͑ ͒ Peq x Peq x

͑ ͒ ͑ ͒ For the term inside the first integral that appears multiplying P− Peq , one may use Eq. 3.11 in order to get an arbitrary constant; after integration, using the normalization condition of Eq. ͑2.9͒, this first-order term yields zero. For the term inside ͑ ͒2 ͑ ͒ the first integral that multiplies P− Peq , one may use Eq. 3.5 at equilibrium, in such a way that

ϱ ϱ 1 ⍀͓P͔ 1 d2⌳͓Q͔ dg͓P͔ 2 F͑P͒ − F͑P ͒ = ͵ dxͯ ͑P − P ͒2ͭ ͮͯ − ͯ ͯ ͩ͵ dxͯ ͯ ͑P − P ͒ͪ + . eq eq ⌿͓ ͔ ␤ 2 eq ¯ −ϱ 2 P ͑ ͒ 2 dQ ͑ ͒ −ϱ dP ͑ ͒ P=Peq x Peq x Peq x ͑3.15͒

͓ ͑ ͒ ͑ ͔͒Ն The equation above yields F P −F Peq 0 provided that Let us now analyze the situation of an isolated system, one uses the previously deﬁned properties for the quantities i.e., ␾͑x͒=constant; in this case, the H theorem should be ⍀͓P͔ and ⌿͓P͔, and additionally, one supposes that expressed in terms of the time derivative of the entropy, in 2 2 ͉͑d ⌳͓Q͔/dQ ͉͒ ͑ ͒ Ͻ0. such a way that Eq. ͑3.4͒ should be replaced by Peq x

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ϱ -P tributions, but still associated with the Boltzmann-Gibbs enץ P d⌳͓Q͔ d2g͓P͔ץ dS͓P͔ =−͵ dxͩ⍀͓P͔ ͪͩ ͪ -x tropy. This whole family of FPEs presents the Boltzmannץ x dQ dP2ץ dt −ϱ Gibbs distribution as the stationary-state solution. As a ϱ P 2 simple example of this class, one may have the nonlinearץ d⌳͓Q͔ d2g͓P͔ ͵ ⍀͓ ͔ ͩ ͪ Ն ␯ =− dx P 2 0. FPE characterized by a͓P͔= P ͑␯ʦR͒ and b͓P͔=D ͑con- xץ ϱ dQ dP − stant͒. ͑3.16͒ ͑b͒ The class of FPEs associated with Tsallis’ entropy:It is important to notice that the simplest FPE of this class was As expected, the proof of the H theorem for an isolated sys- originally proposed with ⌿͓P͑x,t͔͒= P͑x,t͒ and ⍀͓P͑x,t͔͒ tem becomes much simpler than that for the system in the =͑2−q͒D͓P͑x,t͔͒1−q, where D is a constant ͓12͔, however, it presence of an external potential. In particular, there is no ͓ ͔ requirement for a relation involving the parameters of the is very common in the literature 24–26 to find this FPE ͑ ͒ with the replacement 2−q→q. Herein we shall consider this FPE and the entropy, like the one of Eq. 3.5 ; all that one ͑ ͒ needs is a standard condition associated with the FPE, i.e., class of FPEs in such a way to satisfy Eq. 4.1 , with ͓ ͑ ͔͒ ͓ ͑ ͔͒q−1 ͑ ͒ ⍀͓P͔Ն0, the restriction d⌳͓Q͔/dQՆ0 for the outer func- b P x,t =qD P x,t ; integrating Eq. 4.2 , tional of the entropy, as well as the general restrictions of Eq. ␤D P − Pq ͑3.1͒ for the entropy. g͓P͔ =− Pq + CP ⇒ g͓P͔ = k , ͑4.4͒ q −1 q −1 where we have set ␤=k/D ͑k is a constant with dimensions IV. SOME FAMILIES OF FPEs AND THEIR of entropy͒ and have also used the conditions g͑0͒=g͑1͒=0 ASSOCIATED ENTROPIES to eliminate the constant C.InEq.͑4.4͒ one readily recog- In this section we will explore further the correspondence nizes the entropy proposed by Tsallis ͓27͔ that depends on between the nonlinear FPE of Eq. ͑2.5͒ and general entropic the well-known entropic index q. Similarly to example ͑a͒, forms, established through Eq. ͑3.5͒. This equation shows one has a whole class of FPEs, corresponding to different clearly that there may be families of FPEs, corresponding to choices for the functional a͓P͔ of Eq. ͑4.1͒, some of which the same ratio ͑⍀͓P͔/⌿͓P͔͒, associated with a single en- exhibit time-dependent solutions different from the ones pre- tropic form; i.e., the same entropy may be associated with sented in Refs. ͓12,13͔, but all of them associated with the different dynamical processes. In the following examples, we entropic form of Eq. ͑4.4͒. This whole family of FPEs pre- consider classes of FPEs satisfying sents the Tsallis distribution ͑also known as q exponential͒ ͓24–26͔ as the stationary-state solution. ⍀͓ ͔ ͓ ͔ ͓ ͔ ⌿͓ ͔ ͓ ͔ ͑ ͒ P = a P b P , P = a P P, 4.1 ͑c͒ The class of FPEs associated with the entropy of Refs. where the functionals a͓P͔ and b͓P͔ are restricted by the ͓36,37͔. : In this example we proceed in an inverse way with conditions imposed previously for the functionals ⍀͓P͔ and respect to the previous two cases; i.e., we start from a given ⌿͓P͔. In addition to that, in the first three examples we will entropic form, in order to find the class of FPEs associated ͓ ͔ consider entropic forms characterized by ⌳͑Q͓P͔͒=Q͓P͔; with it. Let us then consider 36,37 for these cases Eq. ͑3.5͒ becomes ͓ ͔ ͓ ͑ ͒ ͔ ͓ ͑ ͒ ͔ g P = k 1 − exp − cP + Pg0 , g0 = exp − c −1 , d2g͓P͔ b͓P͔ ͑4.5͒ ␤ ͑ ͒ 2 =− . 4.2 dP P where c is an arbitrary dimensionless constant, and k is a Therefore one has a freedom for choosing different forms for constant with dimensions of entropy. Substituting into Eq. ͑ ͒ the functional a͓P͔, leading to the same entropic form. Next, 4.2 , one gets we work out some examples. b͓P͔ =−DP͓1−c2 exp͑− cP͔͒, ͑4.6͒ ͑a͒ The class of FPEs associated with the Boltzmann- Gibbs entropy: This class corresponds to the functionals where we have set D=k/␤. The functional form above de- ⍀͓P͔ and ⌿͓P͔ satisfying Eq. ͑4.1͒, with b͓P͔=D ͑con- fines the family of FPEs associated with different definitions ͓ ͔ stant͒. Integrating Eq. ͑4.2͒ one gets for the functional a P , all of them related to the entropic form of Eq. ͑4.5͒; the simplest of these equations corre- dg sponds to a͓P͔=1. =−␤D ln P + C ⇒ g͓P͔ =−k P ln P, ͑4.3͒ dP B ͑d͒ The class of FPEs associated with the Renyi entropy ͓38͔: Similarly to the previous example, we start from the ͑ ͒ ͑ ͒ where we have used the conditions g 0 =g 1 =0 to elimi- entropic form, in order to find the class of FPEs associated ␤ nate the constant C, and set the Lagrange multiplier with it. In this case we have that =kB /D, where kB represents the Boltzmann constant. It should be stressed that usually one associates the Boltzmann- ln Q͓P͔ d⌳͓Q͔ k ⌳͑Q͓P͔͒ = k , = , g͓P͔ = Pq, Gibbs entropy with the linear FPE, which represents the sim- 1−q dQ ͑1−q͒Q͓P͔ plest equation within the present class. Herein we show that, ͑4.7͒ by properly defining the functionals ⍀͓P͔ and ⌿͓P͔, one may get nonlinear FPEs, with time-dependent solutions that where k is a constant with dimensions of entropy. It is im- may be different from standard exponential probability dis- portant to stress that in order to satisfy the H theorem, en-

041123-5 SCHWÄMMLE, NOBRE, AND CURADO PHYSICAL REVIEW E 76, 041123 ͑2007͒ tropic forms characterized by an outer functional ⌳͓Q͔ are fusion in an optical lattice ͑in this case, the value of q differs restricted to the condition that ͑d⌳͓Q͔/dQ͒ should present a from 1 by an amount that depends on the ratio of typical sign different from the one of ͑d2g͓P͔/dP2͒͑as assumed in energy parameters of the system, like the recoil energy and the beginning of Sec. III͒, as can be seen from simple analy- the potential depth͓͒8͔; ͑iv͒ seismic time series ͑qϷ2.98͒ ses of Eq. ͑3.5͒, for the case of a system in the presence of an ͓40͔; and ͑v͒ distribution of daily returns of financial stock external potential, or of Eq. ͑3.16͒, for the case of an isolated markets ͑e.g., SP100 ͑q=1.4͓͒41͔ and DJ30 ͑qϷ1.45͓͒42͔͒. system. Substituting the functionals of Eq. ͑4.1͒ into Eq. Another type of one-parameter entropic form ͑whose corre- ͑3.5͒ one gets sponding nonlinear FPE appears as a particular case of Eq. ͑2.5͒ and has already been discussed in detail elsewhere ͓23͔͒ 1 d⌳͓Q͔ d2g͓P͔ b͓P͔ − = , ͑4.8͒ seems to be appropriate for a description of some relativistic ␤ dQ dP2 P phenomena ͓9͔; in this case, the dimensionless entropic parameter is associated to the ratio of velocities of special rela- ͑ ͒ and using Eq. 4.7 , tivity ͑v/c͒. Further discussions on applications of general- q−1 ized entropies for stellar dynamics, two-dimensional Dq q−1 DqP b͓P͔ = P = ϱ , ͑4.9͒ turbulence, and bacterial populations may also be found in Q͓P͔ ͓ ͔ ͵ dx Pq Refs. 43,44 . −ϱ where we have set D=k/␤. It is important to remember that V. FPE FOR A MORE GENERAL the functionals ⍀͓P͔ and ⌿͓P͔ are supposed to be both posi- FREE-ENERGY FUNCTIONAL tive, for a well-defined FPE, which implies a͓P͔,b͓P͔Ͼ0 ͓cf. Eq. ͑4.1͔͒. From Eq. ͑4.9͒ this condition is not satisfied if In this section we will consider a slightly different FPE, qՅ0. Notice that this entropic form satisfies the condition with respect to the one of Eq. ͑2.5͒, namely, 2 2 ͑d ⌳͓Q͔/dQ ͒ ͑ ͒ Ͻ0, required by the H theorem ͓cf. Eq. Peq x Pץ ץ ץ ץ P͑x,t͒ץ for qϽ1; therefore, one can assure the validity of ,3.15͔͒͑ such an entropic form, from the physical point of view, for = ͩ⌿͓P͔ ␾͑x͒␹͓P͔ ͪ + ͩ⍀͓P͔ ͪ, xץ xץ … „xץ xץ tץ the interval 0ϽqϽ1. From the entropic forms discussed above, surely the ͑5.1͒ Boltzmann-Gibbs form represents the most common in nature, being applicable for systems characterized by short- where a new functional ␹͓P͔ was introduced ͓notice that Eq. range interactions and/or weak correlations. For these sys- ͑2.5͒ is recovered for ␹͓P͔=1͔ that should be finite and posi- tems, such an entropy is extensive, as required by tive definite. The interesting point about such a FPE is that it thermodynamics. However, for strongly correlated systems, is consistent with the definition of a “generalized internal one may have a Boltzmann-Gibbs entropy that turns out to energy” ͓24–26͔, be nonextensive, and so, in order to match satisfactorily thermodynamics, one should look for other types of entropic forms, like the ones defined above, in such a way to recover ϱ ␾͑ ͒⌫͓ ͑ ͔͒ ͑ ͒ extensivity. From these, the one proposed by Tsallis appears U = ͵ dx x P x,t , 5.2 ϱ as the most found in natural systems; as a typical example, − one could mention quantum spin chains, for which Tsallis entropy is extensive for a value of q1 ͓31,32͔. Besides where we are assuming that ⌫͓P͔ represents a positive, that, several examples in the literature suggest that other en- monotonically increasing functional of P͑x,t͒, that should be tropic forms ͑distinct from the Boltzmann-Gibbs one͒ should at least ⌫͓P͔ʦC1. be applicable; this is usually indicated by an analysis of the Now, we take this internal energy in the free-energy func- corresponding distribution functions, which are associated to tional of Eq. ͑3.2͒ and consider the same entropic form of given entropic forms, through standard entropy- Eq. ͑3.1͒. Let us then prove the H theorem for such a system, maximization procedures. Many systems exhibit some kind following the same steps of Sec. III; one gets that of distribution that has been associated with Tsallis distribution; it is important to mention that any entropic form given ϱ dF d 1 by a monotonically increasing functional of Tsallis entropy = ͩ͵ dx ␾͑x͒⌫͓P͑x,t͔͒ − ⌳͑Q͓P͔͒ͪ ␤ leads to the same distribution, under maximization. As an dt dt −ϱ example, Tsallis’ and Renyi’s entropies share the same dis- ϱ Pץ d⌫͓P͔ 1 d⌳͓Q͔ dg͓P͔ tribution; however, as we have seen above, this later entropy = ͵ dxͩ␾͑x͒ − ͪ . ץ ␤ Ͻ Ͻ is well defined only for the interval 0 q 1, leading to a −ϱ dP dQ dP t significant restriction for its applicability. As concrete ex- ͑5.3͒ amples of possible applications for Tsallis entropy, we mention ͑i͒ self-gravitating systems, characterized by qϽ7/9 ͓5,39͔; ͑ii͒ relaxation of two-dimensional turbulence in a Using the FPE of Eq. ͑5.1͒ and integrating by parts, one pure-electron plasma, for which q=2 ͓7͔; ͑iii͒ anomalous dif- obtains

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ϱ ϵ␾͑ ͒␹͓ ͑ ͔͒ .P where C x0 Peq x0 is a constant. Integrating Eqץ ץ dF =−͵ dxͭ⌿͓P͔ ␾͑x͒␹͓P͔ + ⍀͓P͔ ͮ at equilibrium, and using the equation above, one gets ,3.5͒͑ ץ … „ ץ dt −ϱ x x 1 d⌳͓Q͔ dg͓P͔ P ͯ ͯ ␾͑ ͒␹͓ ͑ ͔͒ Јץ d⌫͓P͔ 1 d⌳͓Q͔ d2g͓P͔ ץ ϫͭ ͩ␾͑ ͒ ͪ ͮ = x Peq x + C x − 2 . ␤ dQ dP ͑ ͒ x Peq xץ x dP ␤ dQ dPץ ͑5.4͒ d⌫͓P͔ = ͯ␾͑x͒ ͯ + CЈ, dP ͑ ͒ The H theorem applies; i.e., P=Peq x ͑ ͒ ϱ 5.11 P 2ץ ⍀͓P͔ ץ dF =−͵ dx ⌿͓P͔ͭ ␾͑x͒␹͓P͔ + ͮ Յ 0, where we have used Eq. ͑5.6͒ and CЈ represents another xץ x„ … ⌿͓P͔ץ dt −ϱ integration constant. The equation above is equivalent to Eq. ͑5.5͒ ͑5.8͒, obtained from the maximum-entropy principle. Therefore, in what concerns the H theorem, the necessary provided that Eq. ͑3.5͒ holds, with an additional restriction relation involving quantities of the FPE with a general en- for the functional ␹͓P͔, tropic form and its equivalence with the maximum-entropy principle, the FPE of Eq. ͑5.1͒ is consistent with the defini- d⌫͓P͔ ␹͓P͔ = . ͑5.6͒ tion of a generalized internal energy that is sometimes used dP in the context of nonextensive statistical mechanics ͓24–26͔. It should be mentioned that the constraint above, relating the functional ␹͓P͔ of the FPE with the quantity ⌫͓P͔ that ap- VI. CONCLUSIONS pears in the definition of the generalized internal energy We have proved the H theorem by using general nonlinear ͑with ⌫͓P͔ P͒ has to be introduced, in such a way to satisfy Fokker-Planck equations. In order to prove the H theorem the H theorem. for a system in the presence of an external potential, a rela- Let us now show that, at equilibrium, the condition of Eq. tion involving terms of the Fokker-Planck equation and the ͑3.5͒ is equivalent to the maximum-entropy principle, when ͑ ͒ entropy of the system was proposed. In principle, one may one uses the FPE of Eq. 5.1 . Defining the functional have classes of Fokker-Planck equations related to a single ϱ entropic form. Since the Fokker-Planck equation is a phe- I͓P͑x,t͔͒ = ⌳͑Q͓P͔͒ + ␣ͩ1−͵ dx P͑x,t͒ͪ nomenological equation that specifies the dynamical evolu- −ϱ tion associated with a given physical system, this relation ϱ may be useful in the identification of the entropic form as- + ␤ͩU − ͵ dx ␾͑x͒⌫͓P͑x,t͔͒ͪ ͑5.7͒ sociated with such a system. In particular, the present ap- −ϱ proach makes it possible to identify entropic forms associated with some anomalous systems, exhibiting unusual ͑␣ ␤ ͒ and are Lagrange multipliers one has that behavior, that are known to be appropriately described by nonlinear Fokker-Planck equations, like the ones considered dI͓P͔ d⌳͓Q͔ dg͓P͔ ͯ ͯ =0⇒ ͯ ͯ herein. By considering a modified Fokker-Planck equation, dP ͑ ͒ dQ dP ͑ ͒ we have also proved the H theorem for a type of generalized P=Peq x P=Peq x internal energy, like the one used within the nonextensive d⌫͓P͔ − ␣ − ␤␾͑x͒ͯ ͯ =0, statistical-mechanics formalism. For that, the same relation dP ͑ ͒ connecting the parameters of the Fokker-Planck equation and P=Peq x ͑ ͒ the corresponding entropic form had to be introduced. To our 5.8 knowledge, it is the first time that the H theorem has been ͑ ͒ verified, for a system in the presence of an external potential, where Peq x represents the probability distribution at equilibrium. Considering Eq. ͑5.1͒ at equilibrium one gets by considering a nonlinear weight in the definition of the internal energy. Making use of the relation mentioned, we ⍀͓ ͔ ͑ ͒ have calculated well-known entropic forms, associated with ץ Peq dPeq x − ͑␾͑x͒␹͓P ͔͒ = , ͑5.9͒ given Fokker-Planck equations. In the case of the standard x eq ⌿͓P ͔ dxץ eq Boltzmann-Gibbs entropy, apart from the simplest linear and after integration, Fokker-Planck equation, one may have a whole class of nonlinear Fokker-Planck equations, whose time-dependent prob- x ⍀͓P ͔ dP ͑x͒ ability distributions may be distinct from simple exponential ␾͑ ͒␹͓ ͑ ͔͒ ͵ eq eq − x Peq x + C = dx distributions, but all of them are related to this particular ⌿͓P ͔ dx x0 eq entropic form; the stationary-state solution is the same as the ͑ ͒ Peq x ⍀͓P ͑xЈ͔͒ one of the linear Fokker-Planck equation, i.e., a Boltzmann- ͵ eq ͑ Ј͒ = dPeq x , Gibbs distribution. A similar behavior is verified for more ͑ ͒ ⌿͓P ͑xЈ͔͒ Peq x0 eq general, nonadditive, entropic forms, e.g., the Tsallis’ ͑5.10͒ entropy. Although this relation involving families of

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Fokker-Planck equations and entropic forms may not be ACKNOWLEDGMENTS unique, we have shown that, when considered at equilibrium, it is equivalent to the principle of maximum entropy. The We thank Professor Angel Plastino and Professor Con- present results suggest that behind such a relation there may stantino Tsallis for fruitful conversations. The partial ﬁnan- be a deep physical insight that deserves further investiga- cial support from CNPq and Pronex/MCT/FAPERJ ͑Brazil- tions. ian agencies͒ are acknowledged.

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