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Consequences of the H Theorem from Nonlinear Fokker-Planck Equations

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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 introduc- ing 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, to- gether 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.
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