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Special Issue "Statistical Mechanics and Mathematical Physics"

A special issue of Entropy (ISSN 1099-4300). This special issue belongs to the section "Statistical Physics".

Deadline for manuscript submissions: 31 December 2019.

Special Issue Editor

Guest Editor
Dr. Eun-jin Kim

School of Mathematics and Statistics, University of Sheffield, Sheffield S3 7RH, UK
Website | E-Mail
Interests: fluid dynamics; magnetohydrodynamics (MHD); plasma physics; self-organisation; non-equilibrium statistical mechanics; turbulence; solar/stellar physics; magnetic fusion; information theory; homeostasis in biosystems; solar/stellar physics

Special Issue Information

Dear Colleagues,

In classical statistical mechanics, Gaussian (or normal) distribution and mean-field type theories based on such distributions have been widely used to describe equilibrium or near-equilibrium phenomena. The ubiquity of the Gaussian distribution stems from the central limit theorem that random variables governed by different distributions tend to follow Gaussian distribution in the limit of a large sample size. In such a limit, fluctuations are small and have a short correlation time, and mean values and variance completely describe all different moments, greatly facilitating analysis.

Many systems in nature and laboratories are, however, far from equilibrium, exhibiting significant fluctuations. Examples are found not only in astrophysical and laboratory plasmas, but also in forest fires, the stock market, and biological ecosystems. Such rare events of large amplitude (intermittency) can dominate the entire transport even if they occur infrequently. Gene expression and protein productions, which used to be thought of as smooth processes, have also been observed occurring in bursts. Far-from-equilibrium fluctuations produce dissipative patterns, shift or wipe-out phase transitions. The importance and consequences of strong fluctuations in far-from-equilibrium systems has attracted great attention in mathematical physics. This Special Issue aims to present different theories of statistical mechanics and mathematical physics. Submissions addressing non-equilibrium statistical physics and strong fluctuations are especially welcome.

Dr. Eun-jin Kim
Guest Editor

Manuscript Submission Information

Manuscripts should be submitted online at www.mdpi.com by registering and logging in to this website. Once you are registered, click here to go to the submission form. Manuscripts can be submitted until the deadline. All papers will be peer-reviewed. Accepted papers will be published continuously in the journal (as soon as accepted) and will be listed together on the special issue website. Research articles, review articles as well as short communications are invited. For planned papers, a title and short abstract (about 100 words) can be sent to the Editorial Office for announcement on this website.

Submitted manuscripts should not have been published previously, nor be under consideration for publication elsewhere (except conference proceedings papers). All manuscripts are thoroughly refereed through a single-blind peer-review process. A guide for authors and other relevant information for submission of manuscripts is available on the Instructions for Authors page. Entropy is an international peer-reviewed open access monthly journal published by MDPI.

Please visit the Instructions for Authors page before submitting a manuscript. The Article Processing Charge (APC) for publication in this open access journal is 1600 CHF (Swiss Francs). Submitted papers should be well formatted and use good English. Authors may use MDPI's English editing service prior to publication or during author revisions.

Keywords

  • Statistical mechanics
  • Mathematical physics
  • Non-equilibrium
  • Fluctuations
  • Intermittency
  • Phase transition
  • Pattern formation
  • Large deviation
  • Self-assembly
  • Hysteresis
  • Generalized statistical mechanics
  • q-Entropy
  • Fractional calculus

Published Papers (3 papers)

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Research

Open AccessArticle
On the Wave Turbulence Theory for the Nonlinear Schrödinger Equation with Random Potentials
Entropy 2019, 21(9), 823; https://doi.org/10.3390/e21090823 (registering DOI)
Received: 25 July 2019 / Revised: 13 August 2019 / Accepted: 19 August 2019 / Published: 23 August 2019
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Abstract
We derive new kinetic and a porous medium equations from the nonlinear Schrödinger equation with random potentials. The kinetic equation has a very similar form compared to the four-wave turbulence kinetic equation in the wave turbulence theory. Moreover, we construct a class of [...] Read more.
We derive new kinetic and a porous medium equations from the nonlinear Schrödinger equation with random potentials. The kinetic equation has a very similar form compared to the four-wave turbulence kinetic equation in the wave turbulence theory. Moreover, we construct a class of self-similar solutions for the porous medium equation. These solutions spread with time, and this fact answers the “weak turbulence” question for the nonlinear Schrödinger equation with random potentials. We also derive Ohm’s law for the porous medium equation. Full article
(This article belongs to the Special Issue Statistical Mechanics and Mathematical Physics)
Open AccessArticle
Comparing Information Metrics for a Coupled Ornstein–Uhlenbeck Process
Entropy 2019, 21(8), 775; https://doi.org/10.3390/e21080775
Received: 9 July 2019 / Revised: 30 July 2019 / Accepted: 6 August 2019 / Published: 8 August 2019
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Abstract
It is often the case when studying complex dynamical systems that a statistical formulation can provide the greatest insight into the underlying dynamics. When discussing the behavior of such a system which is evolving in time, it is useful to have the notion [...] Read more.
It is often the case when studying complex dynamical systems that a statistical formulation can provide the greatest insight into the underlying dynamics. When discussing the behavior of such a system which is evolving in time, it is useful to have the notion of a metric between two given states. A popular measure of information change in a system under perturbation has been the relative entropy of the states, as this notion allows us to quantify the difference between states of a system at different times. In this paper, we investigate the relaxation problem given by a single and coupled Ornstein–Uhlenbeck (O-U) process and compare the information length with entropy-based metrics (relative entropy, Jensen divergence) as well as others. By measuring the total information length in the long time limit, we show that it is only the information length that preserves the linear geometry of the O-U process. In the coupled O-U process, the information length is shown to be capable of detecting changes in both components of the system even when other metrics would detect almost nothing in one of the components. We show in detail that the information length is sensitive to the evolution of subsystems. Full article
(This article belongs to the Special Issue Statistical Mechanics and Mathematical Physics)
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Open AccessArticle
Information Geometry of Spatially Periodic Stochastic Systems
Entropy 2019, 21(7), 681; https://doi.org/10.3390/e21070681
Received: 8 June 2019 / Revised: 4 July 2019 / Accepted: 10 July 2019 / Published: 12 July 2019
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Abstract
We explore the effect of different spatially periodic, deterministic forces on the information geometry of stochastic processes. The three forces considered are f0=sin(πx)/π and f±=sin(πx)/π [...] Read more.
We explore the effect of different spatially periodic, deterministic forces on the information geometry of stochastic processes. The three forces considered are f 0 = sin ( π x ) / π and f ± = sin ( π x ) / π ± sin ( 2 π x ) / 2 π , with f - chosen to be particularly flat (locally cubic) at the equilibrium point x = 0 , and f + particularly flat at the unstable fixed point x = 1 . We numerically solve the Fokker–Planck equation with an initial condition consisting of a periodically repeated Gaussian peak centred at x = μ , with μ in the range [ 0 , 1 ] . The strength D of the stochastic noise is in the range 10 - 4 10 - 6 . We study the details of how these initial conditions evolve toward the final equilibrium solutions and elucidate the important consequences of the interplay between an initial PDF and a force. For initial positions close to the equilibrium point x = 0 , the peaks largely maintain their shape while moving. In contrast, for initial positions sufficiently close to the unstable point x = 1 , there is a tendency for the peak to slump in place and broaden considerably before reconstituting itself at the equilibrium point. A consequence of this is that the information length L , the total number of statistically distinguishable states that the system evolves through, is smaller for initial positions closer to the unstable point than for more intermediate values. We find that L as a function of initial position μ is qualitatively similar to the force, including the differences between f 0 = sin ( π x ) / π and f ± = sin ( π x ) / π ± sin ( 2 π x ) / 2 π , illustrating the value of information length as a useful diagnostic of the underlying force in the system. Full article
(This article belongs to the Special Issue Statistical Mechanics and Mathematical Physics)
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