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Foundations of Statistical Mechanics

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

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Dr. Antonio M. Scarfone

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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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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 (12 papers)

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2024

Jump to: 2023, 2022, 2021, 2020, 2019

28 pages, 3508 KiB

Open AccessReview

On Casimir and Helmholtz Fluctuation-Induced Forces in Micro- and Nano-Systems: Survey of Some Basic Results

by Daniel Dantchev

Entropy 2024, 26(6), 499; https://doi.org/10.3390/e26060499 - 7 Jun 2024

Cited by 2 | Viewed by 1206

Abstract

Fluctuations are omnipresent; they exist in any matter, due either to its quantum nature or to its nonzero temperature. In the current review, we briefly cover the quantum electrodynamic Casimir (QED) force as well as the critical Casimir (CC) and Helmholtz (HF) forces. In the QED case, the medium is usually a vacuum and the massless excitations are photons, while in the CC and HF cases the medium is usually a critical or correlated fluid and the fluctuations of the order parameter are the cause of the force between the macroscopic or mesoscopic bodies immersed in it. We discuss the importance of the presented results for nanotechnology, especially for devising and assembling micro- or nano-scale systems. Several important problems for nanotechnology following from the currently available experimental findings are spelled out, and possible strategies for overcoming them are sketched. Regarding the example of HF, we explicitly demonstrate that when a given integral quantity characterizing the fluid is conserved, it has an essential influence on the behavior of the corresponding fluctuation-induced force. Full article

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10 pages, 290 KiB

Open AccessFeature PaperArticle

Memory Corrections to Markovian Langevin Dynamics

by Mateusz Wiśniewski, Jerzy Łuczka and Jakub Spiechowicz

Entropy 2024, 26(5), 425; https://doi.org/10.3390/e26050425 - 16 May 2024

Cited by 1 | Viewed by 926

Abstract

Analysis of non-Markovian systems and memory-induced phenomena poses an everlasting challenge in the realm of physics. As a paradigmatic example, we consider a classical Brownian particle of mass M subjected to an external force and exposed to correlated thermal fluctuations. We show that the recently developed approach to this system, in which its non-Markovian dynamics given by the Generalized Langevin Equation is approximated by its memoryless counterpart but with the effective particle mass

M^{*} < M

, can be derived within the Markovian embedding technique. Using this method, we calculate the first- and the second-order memory correction to Markovian dynamics of the Brownian particle for the memory kernel represented as the Prony series. The second one lowers the effective mass of the system further and improves the precision of the approximation. Our work opens the door for the derivation of higher-order memory corrections to Markovian Langevin dynamics. Full article

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2023

Jump to: 2024, 2022, 2021, 2020, 2019

9 pages, 285 KiB

Open AccessArticle

Correspondence between the Energy Equipartition Theorem in Classical Mechanics and Its Phase-Space Formulation in Quantum Mechanics

by Esteban Marulanda, Alejandro Restrepo and Johans Restrepo

Entropy 2023, 25(6), 939; https://doi.org/10.3390/e25060939 - 15 Jun 2023

Viewed by 1391

Abstract

In classical physics, there is a well-known theorem in which it is established that the energy per degree of freedom is the same. However, in quantum mechanics, due to the non-commutativity of some pairs of observables and the possibility of having non-Markovian dynamics, the energy is not equally distributed. We propose a correspondence between what is known as the classical energy equipartition theorem and its counterpart in the phase-space formulation in quantum mechanics based on the Wigner representation. Further, we show that in the high-temperature regime, the classical result is recovered. Full article

10 pages, 369 KiB

Open AccessArticle

Radial Basis Function Finite Difference Method Based on Oseen Iteration for Solving Two-Dimensional Navier–Stokes Equations

by Liru Mu and Xinlong Feng

Entropy 2023, 25(5), 804; https://doi.org/10.3390/e25050804 - 16 May 2023

Cited by 2 | Viewed by 1345

Abstract

In this paper, the radial basis function finite difference method is used to solve two-dimensional steady incompressible Navier–Stokes equations. First, the radial basis function finite difference method with polynomial is used to discretize the spatial operator. Then, the Oseen iterative scheme is used to deal with the nonlinear term, constructing the discrete scheme for Navier–Stokes equation based on the finite difference method of the radial basis function. This method does not require complete matrix reorganization in each nonlinear iteration, which simplifies the calculation process and obtains high-precision numerical solutions. Finally, several numerical examples are obtained to verify the convergence and effectiveness of the radial basis function finite difference method based on Oseen Iteration. Full article

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20 pages, 426 KiB

Open AccessArticle

Uniform Error Estimates of the Finite Element Method for the Navier–Stokes Equations in

R^{2}

with L² Initial Data

by Shuyan Ren, Kun Wang and Xinlong Feng

Entropy 2023, 25(5), 726; https://doi.org/10.3390/e25050726 - 27 Apr 2023

Viewed by 1451

Abstract

In this paper, we study the finite element method of the Navier–Stokes equations with the initial data belonging to the

L^{2}

space for all time

t > 0

. Due to the poor smoothness of the initial data, the solution of the [...] Read more.

In this paper, we study the finite element method of the Navier–Stokes equations with the initial data belonging to the

L^{2}

space for all time

t > 0

. Due to the poor smoothness of the initial data, the solution of the problem is singular, although in the

H^{1}

-norm, when

t \in [0, 1)

. Under the uniqueness condition, by applying the integral technique and the estimates in the negative norm, we deduce the uniform-in-time optimal error bounds for the velocity in

H^{1}

-norm and the pressure in

L^{2}

-norm. Full article

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2022

Jump to: 2024, 2023, 2021, 2020, 2019

25 pages, 544 KiB

Open AccessFeature PaperEditor’s ChoiceReview

Diffusion Coefficient of a Brownian Particle in Equilibrium and Nonequilibrium: Einstein Model and Beyond

by Jakub Spiechowicz, Ivan G. Marchenko, Peter Hänggi and Jerzy Łuczka

Entropy 2023, 25(1), 42; https://doi.org/10.3390/e25010042 - 26 Dec 2022

Cited by 23 | Viewed by 6108

Abstract

The diffusion of small particles is omnipresent in many processes occurring in nature. As such, it is widely studied and exerted in almost all branches of sciences. It constitutes such a broad and often rather complex subject of exploration that we opt here to narrow our survey to the case of the diffusion coefficient for a Brownian particle that can be modeled in the framework of Langevin dynamics. Our main focus centers on the temperature dependence of the diffusion coefficient for several fundamental models of diverse physical systems. Starting out with diffusion in equilibrium for which the Einstein theory holds, we consider a number of physical situations outside of free Brownian motion and end by surveying nonequilibrium diffusion for a time-periodically driven Brownian particle dwelling randomly in a periodic potential. For this latter situation the diffusion coefficient exhibits an intriguingly non-monotonic dependence on temperature. Full article

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16 pages, 964 KiB

Open AccessArticle

How, Why and When Tsallis Statistical Mechanics Provides Precise Descriptions of Natural Phenomena

by Alberto Robledo and Carlos Velarde

Entropy 2022, 24(12), 1761; https://doi.org/10.3390/e24121761 - 1 Dec 2022

Cited by 4 | Viewed by 4632

Abstract

The limit of validity of ordinary statistical mechanics and the pertinence of Tsallis statistics beyond it is explained considering the most probable evolution of complex systems processes. To this purpose we employ a dissipative Landau–Ginzburg kinetic equation that becomes a generic one-dimensional nonlinear iteration map for discrete time. We focus on the Renormalization Group (RG) fixed-point maps for the three routes to chaos. We show that all fixed-point maps and their trajectories have analytic closed-form expressions, not only (as known) for the intermittency route to chaos but also for the period-doubling and the quasiperiodic routes. These expressions have the form of q-exponentials, while the kinetic equation’s Lyapunov function becomes the Tsallis entropy. That is, all processes described by the evolution of the fixed-point trajectories are accompanied by the monotonic progress of the Tsallis entropy. In all cases the action of the fixed-point map attractor imposes a severe impediment to access the system’s built-in configurations, leaving only a subset of vanishing measure available. Only those attractors that remain chaotic have ineffective configuration set reduction and display ordinary statistical mechanics. Finally, we provide a brief description of complex system research subjects that illustrates the applicability of our approach. Full article

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13 pages, 313 KiB

Open AccessArticle

On the Thermodynamics of the q-Particles

by Fabio Ciolli and Francesco Fidaleo

Entropy 2022, 24(2), 159; https://doi.org/10.3390/e24020159 - 20 Jan 2022

Cited by 3 | Viewed by 2724

Abstract

Since the grand partition function

Z_{q}

for the so-called q-particles (i.e., quons),

q \in (- 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

Z_{q}

for the so-called q-particles (i.e., quons),

q \in (- 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 \in [- 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 \in (0, 1)

. Full article

16 pages, 379 KiB

Open AccessArticle

Boltzmann Configurational Entropy Revisited in the Framework of Generalized Statistical Mechanics

by Antonio Maria Scarfone

Entropy 2022, 24(2), 140; https://doi.org/10.3390/e24020140 - 18 Jan 2022

Cited by 2 | Viewed by 2100

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.

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: 2024, 2023, 2022, 2020, 2019

19 pages, 2013 KiB

Open AccessArticle

Thermodynamic Definitions of Temperature and Kappa and Introduction of the Entropy Defect

by George Livadiotis and David J. McComas

Entropy 2021, 23(12), 1683; https://doi.org/10.3390/e23121683 - 15 Dec 2021

Cited by 21 | Viewed by 3148

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 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: 2024, 2023, 2022, 2021, 2019

8 pages, 1576 KiB

Open AccessArticle

Solving Equations of Motion by Using Monte Carlo Metropolis: Novel Method Via Random Paths Sampling and the Maximum Caliber Principle

by Diego González Diaz, Sergio Davis and Sergio Curilef

Entropy 2020, 22(9), 916; https://doi.org/10.3390/e22090916 - 21 Aug 2020

Cited by 3 | Viewed by 3163

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 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: 2024, 2023, 2022, 2021, 2020

9 pages, 3101 KiB

Open AccessArticle

Scaling of the Berry Phase in the Yang-Lee Edge Singularity

by Liang-Jun Zhai, Huai-Yu Wang and Guang-Yao Huang

Entropy 2019, 21(9), 836; https://doi.org/10.3390/e21090836 - 26 Aug 2019

Cited by 2 | Viewed by 3103

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), 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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Published Papers (12 papers)

2024

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2023

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2022

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2021

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2020

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2019

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