Topical Collection "Foundations of Statistical Mechanics"

A topical collection in Entropy (ISSN 1099-4300). This collection belongs to the section "Statistical Physics".

Editor

Dr. Antonio M. Scarfone
E-Mail Website
Collection Editor
Istituto dei Sistemi Complessi, Consiglio Nazionale delle Ricerche (ISC-CNR), c/o DISAT, Politecnico di Torino, Corso Duca degli Abruzzi 24, I-10129 Torino, Italy
Interests: nonextensive statistical mechanics; nonlinear Fokker–Planck equations; geometry information; nonlinear Schroedinger equation; quantum groups and quantum algebras; complex systems
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Topical Collection Information

Dear Colleagues,

Statistical mechanics, covered in the section of Statistical Physics, aims to relate the microscopic to the macroscopic properties of matter by using the concepts developed in the field of probability theory and thermodynamics. It is a successful combination of the statistics and mechanics arising from the union of the basic laws of classical or quantum mechanics, with the laws of large numbers.

The foundations of statistical mechanics lie in the thermodynamics theory developed at the end of the nineteenth century. The first person to analyse transport phenomena with statistical methods was Clausius, who introduced the concept of a mean free path. He also introduced the famous “Stosszahlansatz” hypothesis, which played a prominent role in the succeeding works of Boltzmann. In the pioneering paper, “Zusammeuhang zwischen den Satzen iiber das Verhalten mehratomiger Gasmolekiile mit Jacobi's Princip des letzten Multiplicators”, Boltzmann considers explicitly a great number of systems, their distribution in phase space, and the permanence of this distribution in time. Another impressive contribution to the theory is represented by Maxwell’s work on the kinetic theory of gases derived from what is now called the Maxwell velocity distribution. Finally, Gibbs, in his book “Elementary principles in statistical mechanics”, published in 1902, definitively established the equivalence between the statistical physics and thermodynamics.

From then, statistical mechanics has been developed in several aspects, becoming so general that its methods still hold in a much wider context than that on which the original theory was developed. In fact, thanks to its impressive success, considerable efforts have been made in recent years to extend the formalism of statistical mechanics beyond its application limits. Traditional statistical mechanics focuses on systems with many degrees of freedom, and has become exact in the thermodynamic limit, although, nowadays, an increasing amount of physical systems seem to not comply with this limit imposed by the large numbers. Definitively, such systems reach a meta-equilibrium configuration, which appears to be better described by generalized entropic forms different from the traditional Boltzmann–Gibbs one.

This collection intends to present mainly theoretical oriented material (even purely mathematical) on the foundation of statistical mechanics. It focuses on the challenges of modern theory incorporating a high degree of mathematical rigor, in order to provide relevance not only to statistical physicists, but also to mathematicians and theoretical physicists. The papers submitted should have real and concrete applications in statistical mechanics, or provide clear evidence of possible applications.

Dr. Antonio M. Scarfone
Collection Editor

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Keywords

  • foundations of classical and quantum statistical mechanics
  • Maxwell–Boltzmann, Bose–Einstein, and Fermi–Dirac statistics
  • exotic statistics—Haldane, Gentile, and Quons
  • generalizations of statistical mechanics
  • non-Gibbsian distributions and power–law distributions
  • statistical mechanics of non-equilibrium and meta-equilibrium—critical phenomena and phase transitions
  • geometric foundations of statistical mechanics

Published Papers (5 papers)

2022

Jump to: 2021, 2020, 2019

Article
On the Thermodynamics of the q-Particles
Entropy 2022, 24(2), 159; https://doi.org/10.3390/e24020159 - 20 Jan 2022
Viewed by 1096
Abstract
Since the grand partition function Zq for the so-called q-particles (i.e., quons), q(1,1), cannot be computed by using the standard 2nd quantisation technique involving the full Fock space construction for q=0 [...] Read more.
Since the grand partition function Zq for the so-called q-particles (i.e., quons), q(1,1), cannot be computed by using the standard 2nd quantisation technique involving the full Fock space construction for q=0, and its q-deformations for the remaining cases, we determine such grand partition functions in order to obtain the natural generalisation of the Plank distribution to q[1,1]. We also note the (non) surprising fact that the right grand partition function concerning the Boltzmann case (i.e., q=0) can be easily obtained by using the full Fock space 2nd quantisation, by considering the appropriate correction by the Gibbs factor 1/n! in the n term of the power series expansion with respect to the fugacity z. As an application, we briefly discuss the equations of the state for a gas of free quons or the condensation phenomenon into the ground state, also occurring for the Bose-like quons q(0,1). Full article
Article
Boltzmann Configurational Entropy Revisited in the Framework of Generalized Statistical Mechanics
Entropy 2022, 24(2), 140; https://doi.org/10.3390/e24020140 - 18 Jan 2022
Cited by 1 | Viewed by 400
Abstract
As known, a method to introduce non-conventional statistics may be realized by modifying the number of possible combinations to put particles in a collection of single-particle states. In this paper, we assume that the weight factor of the possible configurations of a system [...] Read more.
As known, a method to introduce non-conventional statistics may be realized by modifying the number of possible combinations to put particles in a collection of single-particle states. In this paper, we assume that the weight factor of the possible configurations of a system of interacting particles can be obtained by generalizing opportunely the combinatorics, according to a certain analytical function f{π}(n) of the actual number of particles present in every energy level. Following this approach, the configurational Boltzmann entropy is revisited in a very general manner starting from a continuous deformation of the multinomial coefficients depending on a set of deformation parameters {π}. It is shown that, when f{π}(n) is related to the solutions of a simple linear difference–differential equation, the emerging entropy is a scaled version, in the occupational number representation, of the entropy of degree (κ,r) known, in the framework of the information theory, as Sharma–Taneja–Mittal entropic form. Full article
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2021

Jump to: 2022, 2020, 2019

Article
Thermodynamic Definitions of Temperature and Kappa and Introduction of the Entropy Defect
Entropy 2021, 23(12), 1683; https://doi.org/10.3390/e23121683 - 15 Dec 2021
Cited by 1 | Viewed by 782
Abstract
This paper develops explicit and consistent definitions of the independent thermodynamic properties of temperature and the kappa index within the framework of nonextensive statistical mechanics and shows their connection with the formalism of kappa distributions. By defining the “entropy defect” in the composition [...] Read more.
This paper develops explicit and consistent definitions of the independent thermodynamic properties of temperature and the kappa index within the framework of nonextensive statistical mechanics and shows their connection with the formalism of kappa distributions. By defining the “entropy defect” in the composition of a system, we show how the nonextensive entropy of systems with correlations differs from the sum of the entropies of their constituents of these systems. A system is composed extensively when its elementary subsystems are independent, interacting with no correlations; this leads to an extensive system entropy, which is simply the sum of the subsystem entropies. In contrast, a system is composed nonextensively when its elementary subsystems are connected through long-range interactions that produce correlations. This leads to an entropy defect that quantifies the missing entropy, analogous to the mass defect that quantifies the mass (energy) associated with assembling subatomic particles. We develop thermodynamic definitions of kappa and temperature that connect with the corresponding kinetic definitions originated from kappa distributions. Finally, we show that the entropy of a system, composed by a number of subsystems with correlations, is determined using both discrete and continuous descriptions, and find: (i) the resulted entropic form expressed in terms of thermodynamic parameters; (ii) an optimal relationship between kappa and temperature; and (iii) the correlation coefficient to be inversely proportional to the temperature logarithm. Full article
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2020

Jump to: 2022, 2021, 2019

Article
Solving Equations of Motion by Using Monte Carlo Metropolis: Novel Method Via Random Paths Sampling and the Maximum Caliber Principle
Entropy 2020, 22(9), 916; https://doi.org/10.3390/e22090916 - 21 Aug 2020
Viewed by 1106
Abstract
A permanent challenge in physics and other disciplines is to solve Euler–Lagrange equations. Thereby, a beneficial investigation is to continue searching for new procedures to perform this task. A novel Monte Carlo Metropolis framework is presented for solving the equations of motion in [...] Read more.
A permanent challenge in physics and other disciplines is to solve Euler–Lagrange equations. Thereby, a beneficial investigation is to continue searching for new procedures to perform this task. A novel Monte Carlo Metropolis framework is presented for solving the equations of motion in Lagrangian systems. The implementation lies in sampling the path space with a probability functional obtained by using the maximum caliber principle. Free particle and harmonic oscillator problems are numerically implemented by sampling the path space for a given action by using Monte Carlo simulations. The average path converges to the solution of the equation of motion from classical mechanics, analogously as a canonical system is sampled for a given energy by computing the average state, finding the least energy state. Thus, this procedure can be general enough to solve other differential equations in physics and a useful tool to calculate the time-dependent properties of dynamical systems in order to understand the non-equilibrium behavior of statistical mechanical systems. Full article
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2019

Jump to: 2022, 2021, 2020

Article
Scaling of the Berry Phase in the Yang-Lee Edge Singularity
Entropy 2019, 21(9), 836; https://doi.org/10.3390/e21090836 - 26 Aug 2019
Cited by 2 | Viewed by 1135
Abstract
We study the scaling behavior of the Berry phase in the Yang-Lee edge singularity (YLES) of the non-Hermitian quantum system. A representative model, the one-dimensional quantum Ising model in an imaginary longitudinal field, is selected. For this model, the dissipative phase transition (DPT), [...] Read more.
We study the scaling behavior of the Berry phase in the Yang-Lee edge singularity (YLES) of the non-Hermitian quantum system. A representative model, the one-dimensional quantum Ising model in an imaginary longitudinal field, is selected. For this model, the dissipative phase transition (DPT), accompanying a parity-time (PT) symmetry-breaking phase transition, occurs when the imaginary field changes through the YLES. We find that the real and imaginary parts of the complex Berry phase show anomalies around the critical points of YLES. In the overlapping critical regions constituted by the (0 + 1)D YLES and (1 + 1)D ferromagnetic-paramagnetic phase transition (FPPT), we find that the real and imaginary parts of the Berry phase can be described by both the (0 + 1)D YLES and (1 + 1)D FPPT scaling theory. Our results demonstrate that the complex Berry phase can be used as a universal order parameter for the description of the critical behavior and the phase transition in the non-Hermitian systems. Full article
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