Generalized entropy production phenomena: a master-equation approach

Non-Additive Entropic Forms and Evolution Equations for Continuous and Discrete Probabilities

Increasing interest has been shown in the subject of non-additive entropic forms during recent years, which has essentially been due to their potential applications in the area of complex systems. Based on the fact that a given entropic form should depend only on a set of probabilities, its time evolution is directly related to the evolution of these probabilities. In the present work, we discuss some basic aspects related to non-additive entropies considering their time evolution in the cases of continuous and discrete probabilities, for which nonlinear forms of Fokker-Planck and master equations are considered, respectively. For continuous probabilities, we discuss an H-theorem, which is proven by connecting functionals that appear in a nonlinear Fokker-Planck equation with a general entropic form. This theorem ensures that the stationary-state solution of the Fokker-Planck equation coincides with the equilibrium solution that emerges from the extremization of the entropic form. At equilibrium, we show that a Carnot cycle holds for a general entropic form under standard thermodynamic requirements. In the case of discrete probabilities, we also prove an H-theorem considering the time evolution of probabilities described by a master equation. The stationary-state solution that comes from the master equation is shown to coincide with the equilibrium solution that emerges from the extremization of the entropic form. For this case, we also discuss how the third law of thermodynamics applies to equilibrium non-additive entropic forms in general. The physical consequences related to the fact that the equilibrium-state distributions, which are obtained from the corresponding evolution equations (for both continuous and discrete probabilities), coincide with those obtained from the extremization of the entropic form, the restrictions for the validity of a Carnot cycle, and an appropriate formulation of the third law of thermodynamics for general entropic forms are discussed.

Thermodynamic Definition of Time: Considerations on the EPR Paradox

Causality is the relationship between causes and effects. Following Relativity, any cause of an event must always be in the past light cone of the event itself, but causes and effects must always be related to some interactions. In this paper, causality is developed as a consequence of the analysis of the Einstein, Podolsky, and Rosen paradox. Causality is interpreted as the result of time generation, due to irreversible interactions of real systems among them. Time results as a consequence of irreversibility; so, any state function of a system in its space cone, when affected by an interaction with an observer, moves into a light cone or within it, with the consequence that any cause must precede its effect in a common light cone.

Time, Irreversibility and Entropy Production in Nonequilibrium Systems

The aim of this review is to shed light on time and irreversibility, in order to link macroscopic to microscopic approaches to these complicated problems. After a brief summary of the standard notions of thermodynamics, we introduce some considerations about certain fundamental aspects of temporal evolution of out-of-equilibrium systems. Our focus is on the notion of entropy generation as the marked characteristic of irreversible behaviour. The concept of time and the basic aspects of the thermalization of thermal radiation, due to the interaction of thermal radiation with matter, are explored concisely from complementary perspectives. The implications and relevance of time for the phenomenon of thermal radiation and irreversible thermophysics are carefully discussed. The concept of time is treated from a different viewpoint, in order to make it as clear as possible in relation to its different fundamental problems.

H theorem for generalized entropic forms within a master-equation framework

The H theorem is proven for generalized entropic forms, in the case of a discrete set of states. The associated probability distributions evolve in time according to a master equation, for which the corresponding transition rates depend on these entropic forms. An important equation describing the time evolution of the transition rates and probabilities in such a way as to drive the system towards an equilibrium state is found. In the particular case of Boltzmann-Gibbs entropy, it is shown that this equation is satisfied in the microcanonical ensemble only for symmetric probability transition rates, characterizing a single path to the equilibrium state. This equation fulfils the proof of the H theorem for generalized entropic forms, associated with systems characterized by complex dynamics, e.g., presenting nonsymmetric probability transition rates and more than one path towards the same equilibrium state. Some examples considering generalized entropies of the literature are discussed, showing that they should be applicable to a wide range of natural phenomena, mainly those within the realm of complex systems.

Second law and entropy production in a nonextensive system

A model of superconducting vortices under overdamped motion is currently used for describing type-II superconductors. Recently, this model has been identified to a nonlinear Fokker-Planck equation and associated to an entropic form characteristic of nonextensive statistical mechanics, S(2)(t)≡S((q)=2)(t). In the present work, we consider a system of superconducting vortices under overdamped motion, following an irreversible process, so that by using the corresponding nonlinear Fokker-Planck equation, the entropy time rate [dS(2)(t)/dt] is investigated. Both entropy production and entropy flux from the system to its surroundings are analyzed. Molecular dynamics simulations are carried for this process, showing a good agreement between the numerical and analytical results. It is shown that the second law holds within the present framework, and we exhibit the increase of S(2)(t) with time, up to its stationary-state value.