Nonlinear Fokker–Planck equations, H – theorem, and entropies

Santos, Maike A. F. dos; Lenzi, Marcelo Kaminski; Lenzi, E. K.

doi:10.1016/j.cjph.2017.07.003

Cited by 8 publications

(8 citation statements)

References 24 publications

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“…We show that these equations gain a foundation when they are involved with the proposal of non-extensive statistical mechanics [31]. Nowadays, the Tsallis statistic [31] and nonlinear diffusion have assumed important roles in the application of more subtle problems in thermodynamics, such as black holes [21,143], generalised forms of H-theorem [19,20,144], financial market [145,146], and many other systems [94,147,148]. Here, it is worth mentioning to the reader that from the point of view of the H theorem [11] the porous media equations may imply the Tsallis entropy.…”

Section: Brief Discussion and Some Considerationsmentioning

confidence: 91%

“…The connection between the Tsallis statistic and nonlinear Fokker-Planck equation (FPE) is deeper than the Boltzmann factor. It was proved through the Htheorem in works [11,20] to nonlinear FPE, and in Refs. [19,10] to nonlinear Klein-Kramers equation.…”

Section: Non-linear Equations and The Generalised Walkermentioning

confidence: 91%

“…The nonlinear diffusion equations imply in a series of generalisations, in particular, the nonlinear dynamics can imply in anomalous diffusive processes and in generalised entropies [19,20]. These entropic forms recover the Boltzmann-Gibbs entropy, so the type of treatment to describe the anomalous phenomena is associated with generalised statistics.…”

Section: Introductionmentioning

confidence: 99%

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Analytic approaches of the anomalous diffusion: A review

Santos

2019

Chaos, Solitons & Fractals

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This review article aims to stress and reunite some of the analytic formalism of the anomalous diffusive processes that have succeeded in their description. Also, it has the objective to discuss which of the new directions they have taken nowadays. The discussion is started by a brief historical report that starts with the studies of thermal machines and combines in theories such as the statistical mechanics of Boltzmann-Gibbs and the Brownian Movement. In this scenario, in the twentieth century, a series of experiments were reported that were not described by the usual model of diffusion. Such experiments paved the way for deeper investigation into anomalous diffusion, i.e. (∆x) 2 ∼ t α to α = 1. These processes are very abundant in physics, and the mechanisms for them to occur are diverse. For this reason, there are many possible ways of modelling the diffusive processes. This article discusses three analytic approaches to investigate anomalous diffusion: fractional diffusion equation, nonlinear diffusion equation and Langevin equation in the presence of fractional, coloured or multiplicative noises. All these formalisms presented different degrees of complexity and for this reason, they have succeeded in describing anomalous diffusion phenomena.

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Section: Brief Discussion and Some Considerationsmentioning

confidence: 91%

Section: Non-linear Equations and The Generalised Walkermentioning

confidence: 91%

Section: Introductionmentioning

confidence: 99%

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Analytic approaches of the anomalous diffusion: A review

Santos

2019

Chaos, Solitons & Fractals

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“…By using the previous equations, we have

We verify that

which implies

Consequently, by solving Equation ( 13 ) with

under the conditions defined in Refs. [ 28 , 29 , 30 , 31 ], we obtain

The entropy for the composite system is given by

which can also be rewritten as

and, consequently, as

Equation ( 18 ) has several particular cases, such as the Tsallis and Kaniadakis entropies, depending on the values of the parameters

, and

. It is noteworthy that this result preserves the additivity in the Penrose sense [ 3 ], i.e.,

required for a system composed of independent subsystems when the standard entropy is employed.…”

Section: The Problemmentioning

confidence: 99%

Nonlinear Fokker–Planck Equations, H-Theorem and Generalized Entropy of a Composed System

Evangelista,

Lenzi

2023

Entropy

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We investigate the dynamics of a system composed of two different subsystems when subjected to different nonlinear Fokker–Planck equations by considering the H–theorem. We use the H–theorem to obtain the conditions required to establish a suitable dependence for the system’s interaction that agrees with the thermodynamics law when the nonlinearity in these equations is the same. In this framework, we also consider different dynamical aspects of each subsystem and investigate a possible expression for the entropy of the composite system.

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“…defines a uniform reaction process in space and was extensively investigated in literature [56][57][58][59][60][61]. Here, we propose that a reaction process between two species occurs in a specific point which defines a controlled-reaction for two species.…”

Section: Controlled-diffusion and Localised Reactionmentioning

confidence: 99%

From continuous-time random walks to controlled-diffusion reaction

Santos¹

2019

J. Stat. Mech.

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Daily, are reported systems in nature that present anomalous diffusion phenomena due to irregularities of medium, traps or reactions process. In this scenario, the diffusion with traps or localised-reactions emerge through various investigations that include numerical, analytical and experimental techniques. In this work, we construct a model which involves a coupling of two diffusion equations to approach the random walkers in a medium with localised reaction point (or controlled diffusion). We present the exact analytical solutions to the model. In the following, we obtain the survival probability and mean square displacement. Moreover, we extend the model to include memory effects in reaction points. Thereby, we found a simple relation that connects the power-law memory kernels with anomalous diffusion phenomena, i.e. (x − x ) 2 ∝ t µ . The investigations presented in this work uses recent mathematical techniques to introduces a form to represent the coupled random walks in context of reaction-diffusion problem to localised reaction.

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Nonlinear Fokker–Planck equations, H – theorem, and entropies

Cited by 8 publications

References 24 publications

Analytic approaches of the anomalous diffusion: A review

Analytic approaches of the anomalous diffusion: A review

Nonlinear Fokker–Planck Equations, H-Theorem and Generalized Entropy of a Composed System

From continuous-time random walks to controlled-diffusion reaction

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