Editorial

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Entropic Aspects of Nonlinear Partial Differential Equations: Classical and Quantum Mechanical Perspectives

by Angelo Plastino

Entropy 2017, 19(4), 166; https://0-doi-org.brum.beds.ac.uk/10.3390/e19040166 - 12 Apr 2017

Cited by 3 | Viewed by 3282

There has been increasing research activity in recent years concerning the properties and the applications of nonlinear partial differential equations that are closely related to nonstandard entropic functionals, such as the Tsallis and Renyi entropies.[...] Full article

(This article belongs to the Special Issue Entropic Aspects of Nonlinear Partial Differential Equations: Classical and Quantum Mechanical Perspectives)

Research

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296 KiB

Open AccessArticle

Random Walks Associated with Nonlinear Fokker–Planck Equations

by Renio Dos Santos Mendes, Ervin Kaminski Lenzi, Luis Carlos Malacarne, Sergio Picoli and Max Jauregui

Entropy 2017, 19(4), 155; https://0-doi-org.brum.beds.ac.uk/10.3390/e19040155 - 01 Apr 2017

Cited by 17 | Viewed by 5039

Abstract

A nonlinear random walk related to the porous medium equation (nonlinear Fokker–Planck equation) is investigated. This random walk is such that when the number of steps is sufficiently large, the probability of finding the walker in a certain position after taking a determined [...] Read more.

A nonlinear random walk related to the porous medium equation (nonlinear Fokker–Planck equation) is investigated. This random walk is such that when the number of steps is sufficiently large, the probability of finding the walker in a certain position after taking a determined number of steps approximates to a q-Gaussian distribution (

G_{q, β} (x) \propto {[1 - (1 - q) β x^{2}]}^{1 / (1 - q)}

), which is a solution of the porous medium equation. This can be seen as a verification of a generalized central limit theorem where the attractor is a q-Gaussian distribution, reducing to the Gaussian one when the linearity is recovered (

q \to 1

). In addition, motivated by this random walk, a nonlinear Markov chain is suggested. Full article

(This article belongs to the Special Issue Entropic Aspects of Nonlinear Partial Differential Equations: Classical and Quantum Mechanical Perspectives)

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299 KiB

Open AccessArticle

Nonlinear Wave Equations Related to Nonextensive Thermostatistics

by Angel R. Plastino and Roseli S. Wedemann

Entropy 2017, 19(2), 60; https://0-doi-org.brum.beds.ac.uk/10.3390/e19020060 - 07 Feb 2017

Cited by 12 | Viewed by 5104

Abstract

We advance two nonlinear wave equations related to the nonextensive thermostatistical formalism based upon the power-law nonadditive

S_{q}

entropies. Our present contribution is in line with recent developments, where nonlinear extensions inspired on the q-thermostatistical formalism have been proposed for the [...] Read more.

We advance two nonlinear wave equations related to the nonextensive thermostatistical formalism based upon the power-law nonadditive

S_{q}

entropies. Our present contribution is in line with recent developments, where nonlinear extensions inspired on the q-thermostatistical formalism have been proposed for the Schroedinger, Klein–Gordon, and Dirac wave equations. These previously introduced equations share the interesting feature of admitting q-plane wave solutions. In contrast with these recent developments, one of the nonlinear wave equations that we propose exhibits real q-Gaussian solutions, and the other one admits exponential plane wave solutions modulated by a q-Gaussian. These q-Gaussians are q-exponentials whose arguments are quadratic functions of the space and time variables. The q-Gaussians are at the heart of nonextensive thermostatistics. The wave equations that we analyze in this work illustrate new possible dynamical scenarios leading to time-dependent q-Gaussians. One of the nonlinear wave equations considered here is a wave equation endowed with a nonlinear potential term, and can be regarded as a nonlinear Klein–Gordon equation. The other equation we study is a nonlinear Schroedinger-like equation. Full article

(This article belongs to the Special Issue Entropic Aspects of Nonlinear Partial Differential Equations: Classical and Quantum Mechanical Perspectives)

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940 KiB

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Intermittent Motion, Nonlinear Diffusion Equation and Tsallis Formalism

by Ervin K. Lenzi, Luciano R. Da Silva, Marcelo K. Lenzi, Maike A. F. Dos Santos, Haroldo V. Ribeiro and Luiz R. Evangelista

Entropy 2017, 19(1), 42; https://0-doi-org.brum.beds.ac.uk/10.3390/e19010042 - 21 Jan 2017

Cited by 11 | Viewed by 4665

Abstract

We investigate an intermittent process obtained from the combination of a nonlinear diffusion equation and pauses. We consider the porous media equation with reaction terms related to the rate of switching the particles from the diffusive mode to the resting mode or switching [...] Read more.

We investigate an intermittent process obtained from the combination of a nonlinear diffusion equation and pauses. We consider the porous media equation with reaction terms related to the rate of switching the particles from the diffusive mode to the resting mode or switching them from the resting to the movement. The results show that in the asymptotic limit of small and long times, the spreading of the system is essentially governed by the diffusive term. The behavior exhibited for intermediate times depends on the rates present in the reaction terms. In this scenario, we show that, in the asymptotic limits, the distributions for this process are given by in terms of power laws which may be related to the q-exponential present in the Tsallis statistics. Furthermore, we also analyze a situation characterized by different diffusive regimes, which emerges when the diffusive term is a mixing of linear and nonlinear terms. Full article

(This article belongs to the Special Issue Entropic Aspects of Nonlinear Partial Differential Equations: Classical and Quantum Mechanical Perspectives)

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549 KiB

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Perturbative Treatment of the Non-Linear q-Schrödinger and q-Klein–Gordon Equations

by Javier Zamora, Mario C. Rocca, Angelo Plastino and Gustavo L. Ferri

Entropy 2017, 19(1), 21; https://0-doi-org.brum.beds.ac.uk/10.3390/e19010021 - 31 Dec 2016

Cited by 4 | Viewed by 3920

Abstract

Interesting non-linear generalization of both Schrödinger’s and Klein–Gordon’s equations have been recently advanced by Tsallis, Rego-Monteiro and Tsallis (NRT) in Nobre et al. (Phys. Rev. Lett. 2011, 106, 140601). There is much current activity going on in this area. The non-linearity is [...] Read more.

Interesting non-linear generalization of both Schrödinger’s and Klein–Gordon’s equations have been recently advanced by Tsallis, Rego-Monteiro and Tsallis (NRT) in Nobre et al. (Phys. Rev. Lett. 2011, 106, 140601). There is much current activity going on in this area. The non-linearity is governed by a real parameter q. Empiric hints suggest that the ensuing non-linear q-Schrödinger and q-Klein–Gordon equations are a natural manifestations of very high energy phenomena, as verified by LHC-experiments. This happens for

q -

values close to unity (Plastino et al. (Nucl. Phys. A 2016, 955, 16–26, Nucl. Phys. A 2016, 948, 19–27)). It might thus be difficult for q-values close to unity to ascertain whether one is dealing with solutions to the ordinary Schrödinger equation (whose free particle solutions are exponentials and for which

q = 1

) or with its NRT non-linear q-generalizations, whose free particle solutions are q-exponentials. In this work, we provide a careful analysis of the

q \sim 1

instance via a perturbative analysis of the NRT equations. Full article

(This article belongs to the Special Issue Entropic Aspects of Nonlinear Partial Differential Equations: Classical and Quantum Mechanical Perspectives)

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338 KiB

Open AccessArticle

A Dissipation of Relative Entropy by Diffusion Flows

by Hiroaki Yoshida

Entropy 2017, 19(1), 9; https://0-doi-org.brum.beds.ac.uk/10.3390/e19010009 - 27 Dec 2016

Cited by 2 | Viewed by 3818

Abstract

Given a probability measure, we consider the diffusion flows of probability measures associated with the partial differential equation (PDE) of Fokker–Planck. Our flows of the probability measures are defined as the solution of the Fokker–Planck equation for the same strictly convex potential, which [...] Read more.

Given a probability measure, we consider the diffusion flows of probability measures associated with the partial differential equation (PDE) of Fokker–Planck. Our flows of the probability measures are defined as the solution of the Fokker–Planck equation for the same strictly convex potential, which means that the flows have the same equilibrium. Then, we shall investigate the time derivative for the relative entropy in the case where the object and the reference measures are moving according to the above diffusion flows, from which we can obtain a certain dissipation formula and also an integral representation of the relative entropy. Full article

(This article belongs to the Special Issue Entropic Aspects of Nonlinear Partial Differential Equations: Classical and Quantum Mechanical Perspectives)

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834 KiB

Open AccessArticle

Active and Purely Dissipative Nambu Systems in General Thermostatistical Settings Described by Nonlinear Partial Differential Equations Involving Generalized Entropy Measures

by T. D. Frank

Entropy 2017, 19(1), 8; https://0-doi-org.brum.beds.ac.uk/10.3390/e19010008 - 27 Dec 2016

Cited by 5 | Viewed by 4168

Abstract

In physics, several attempts have been made to apply the concepts and tools of physics to the life sciences. In this context, a thermostatistic framework for active Nambu systems is proposed. The so-called free energy Fokker–Planck equation approach is used to describe stochastic [...] Read more.

In physics, several attempts have been made to apply the concepts and tools of physics to the life sciences. In this context, a thermostatistic framework for active Nambu systems is proposed. The so-called free energy Fokker–Planck equation approach is used to describe stochastic aspects of active Nambu systems. Different thermostatistic settings are considered that are characterized by appropriately-defined entropy measures, such as the Boltzmann–Gibbs–Shannon entropy and the Tsallis entropy. In general, the free energy Fokker–Planck equations associated with these generalized entropy measures correspond to nonlinear partial differential equations. Irrespective of the entropy-related nonlinearities occurring in these nonlinear partial differential equations, it is shown that semi-analytical solutions for the stationary probability densities of the active Nambu systems can be obtained provided that the pumping mechanisms of the active systems assume the so-called canonical-dissipative form and depend explicitly only on Nambu invariants. Applications are presented both for purely-dissipative and for active systems illustrating that the proposed framework includes as a special case stochastic equilibrium systems. Full article

(This article belongs to the Special Issue Entropic Aspects of Nonlinear Partial Differential Equations: Classical and Quantum Mechanical Perspectives)

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258 KiB

Open AccessArticle

Existence of Solutions to a Nonlinear Parabolic Equation of Fourth-Order in Variable Exponent Spaces

by Bo Liang, Xiting Peng and Chengyuan Qu

Entropy 2016, 18(11), 413; https://0-doi-org.brum.beds.ac.uk/10.3390/e18110413 - 18 Nov 2016

Cited by 7 | Viewed by 3917

Abstract

This paper is devoted to studying the existence and uniqueness of weak solutions for an initial boundary problem of a nonlinear fourth-order parabolic equation with variable exponent [...] Read more.

This paper is devoted to studying the existence and uniqueness of weak solutions for an initial boundary problem of a nonlinear fourth-order parabolic equation with variable exponent

v_{t} + {div (| \nabla ▵ v |}^{p (x) - 2} {\nabla ▵ v) - | ▵ v |}^{q (x) - 2} ▵ v = g (x, v)

. By applying Leray-Schauder’s fixed point theorem, the existence of weak solutions of the elliptic problem is given. Furthermore, the semi-discrete method yields the existence of weak solutions of the corresponding parabolic problem by constructing two approximate solutions. Full article

(This article belongs to the Special Issue Entropic Aspects of Nonlinear Partial Differential Equations: Classical and Quantum Mechanical Perspectives)

1040 KiB

Open AccessArticle

The Analytical Solution of Parabolic Volterra Integro-Differential Equations in the Infinite Domain

by Yun Zhao and Fengqun Zhao

Entropy 2016, 18(10), 344; https://0-doi-org.brum.beds.ac.uk/10.3390/e18100344 - 23 Sep 2016

Cited by 6 | Viewed by 4864

Abstract

This article focuses on obtaining analytical solutions for d-dimensional, parabolic Volterra integro-differential equations with different types of frictional memory kernel. Based on Laplace transform and Fourier transform theories, the properties of the Fox-H function and convolution theorem, analytical solutions for the equations [...] Read more.

This article focuses on obtaining analytical solutions for d-dimensional, parabolic Volterra integro-differential equations with different types of frictional memory kernel. Based on Laplace transform and Fourier transform theories, the properties of the Fox-H function and convolution theorem, analytical solutions for the equations in the infinite domain are derived under three frictional memory kernel functions. The analytical solutions are expressed by infinite series, the generalized multi-parameter Mittag-Leffler function, the Fox-H function and the convolution form of the Fourier transform. In addition, graphical representations of the analytical solution under different parameters are given for one-dimensional parabolic Volterra integro-differential equations with a power-law memory kernel. It can be seen that the solution curves are subject to Gaussian decay at any given moment. Full article

(This article belongs to the Special Issue Entropic Aspects of Nonlinear Partial Differential Equations: Classical and Quantum Mechanical Perspectives)

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Entropy Generation on Nanofluid Flow through a Horizontal Riga Plate

by Tehseen Abbas, Muhammad Ayub, Muhammad Mubashir Bhatti, Mohammad Mehdi Rashidi and Mohamed El-Sayed Ali

Entropy 2016, 18(6), 223; https://0-doi-org.brum.beds.ac.uk/10.3390/e18060223 - 08 Jun 2016

Cited by 75 | Viewed by 6538

Abstract

In this article, entropy generation on viscous nanofluid through a horizontal Riga plate has been examined. The present flow problem consists of continuity, linear momentum, thermal energy, and nanoparticle concentration equation which are simplified with the help of Oberbeck-Boussinesq approximation. The resulting highly [...] Read more.

In this article, entropy generation on viscous nanofluid through a horizontal Riga plate has been examined. The present flow problem consists of continuity, linear momentum, thermal energy, and nanoparticle concentration equation which are simplified with the help of Oberbeck-Boussinesq approximation. The resulting highly nonlinear coupled partial differential equations are solved numerically by means of the shooting method (SM). The expression of local Nusselt number and local Sherwood number are also taken into account and discussed with the help of table. The physical influence of all the emerging parameters such as Brownian motion parameter, thermophoresis parameter, Brinkmann number, Richardson number, nanoparticle flux parameter, Lewis number and suction parameter are demonstrated graphically. In particular, we conferred their influence on velocity profile, temperature profile, nanoparticle concentration profile and Entropy profile. Full article

(This article belongs to the Special Issue Entropic Aspects of Nonlinear Partial Differential Equations: Classical and Quantum Mechanical Perspectives)

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Review

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Nonlinear q-Generalizations of Quantum Equations: Homogeneous and Nonhomogeneous Cases—An Overview

by Fernando D. Nobre, Marco Aurélio Rego-Monteiro and Constantino Tsallis

Entropy 2017, 19(1), 39; https://0-doi-org.brum.beds.ac.uk/10.3390/e19010039 - 21 Jan 2017

Cited by 10 | Viewed by 4052

Abstract

Recent developments on the generalizations of two important equations of quantum physics, namely the Schroedinger and Klein–Gordon equations, are reviewed. These generalizations present nonlinear terms, characterized by exponents depending on an index q, in such a way that the standard linear equations [...] Read more.

Recent developments on the generalizations of two important equations of quantum physics, namely the Schroedinger and Klein–Gordon equations, are reviewed. These generalizations present nonlinear terms, characterized by exponents depending on an index q, in such a way that the standard linear equations are recovered in the limit

q \to 1

. Interestingly, these equations present a common, soliton-like, traveling solution, which is written in terms of the q-exponential function that naturally emerges within nonextensive statistical mechanics. In both cases, the corresponding well-known Einstein energy-momentum relations, as well as the Planck and the de Broglie ones, are preserved for arbitrary values of q. In order to deal appropriately with the continuity equation, a classical field theory has been developed, where besides the usual

Ψ (\vec{x}, t)

, a new field

Φ (\vec{x}, t)

must be introduced; this latter field becomes

Ψ^{*} (\vec{x}, t)

only when

q \to 1

. A class of linear nonhomogeneous Schroedinger equations, characterized by position-dependent masses, for which the extra field

Φ (\vec{x}, t)

becomes necessary, is also investigated. In this case, an appropriate transformation connecting

Ψ (\vec{x}, t)

and

Φ (\vec{x}, t)

is proposed, opening the possibility for finding a connection between these fields in the nonlinear cases. The solutions presented herein are potential candidates for applications to nonlinear excitations in plasma physics, nonlinear optics, in structures, such as those of graphene, as well as in shallow and deep water waves. Full article

(This article belongs to the Special Issue Entropic Aspects of Nonlinear Partial Differential Equations: Classical and Quantum Mechanical Perspectives)

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