On the relevance of the maximum entropy principle in non-equilibrium statistical mechanics

2017 ◽  
Vol 226 (10) ◽  
pp. 2327-2343 ◽  
Author(s):  
Gennaro Auletta ◽  
Lamberto Rondoni ◽  
Angelo Vulpiani
Keyword(s):  
Statistical Mechanics ◽  
Maximum Entropy ◽  
Maximum Entropy Principle ◽  
Equilibrium Statistical Mechanics ◽  
Entropy Principle ◽  
Non Equilibrium

scholarly journals An Introduction to the Non-Equilibrium Steady States of Maximum Entropy Spike Trains

Entropy ◽  
2019 ◽  
Vol 21 (9) ◽  
pp. 884 ◽  
Author(s):  
Rodrigo Cofré ◽  
Leonardo Videla ◽  
Fernando Rosas
Keyword(s):  
Statistical Mechanics ◽  
Maximum Entropy ◽  
Spike Trains ◽  
Biological Processes ◽  
Comprehensive Overview ◽  
Equilibrium Statistical Mechanics ◽  
Temporal Asymmetry ◽  
Key Concepts ◽  
Mathematical Foundations ◽  
Non Equilibrium

Although most biological processes are characterized by a strong temporal asymmetry, several popular mathematical models neglect this issue. Maximum entropy methods provide a principled way of addressing time irreversibility, which leverages powerful results and ideas from the literature of non-equilibrium statistical mechanics. This tutorial provides a comprehensive overview of these issues, with a focus in the case of spike train statistics. We provide a detailed account of the mathematical foundations and work out examples to illustrate the key concepts and results from non-equilibrium statistical mechanics.

FROM THERMODYNAMICS TO MAXENT

2007 ◽  
Vol 21 (13n14) ◽  
pp. 2557-2563
Author(s):  
A. PLASTINO ◽  
E. M. F. CURADO
Keyword(s):  
Statistical Mechanics ◽  
Maximum Entropy ◽  
Body Problem ◽  
Maximum Entropy Principle ◽  
Many Body ◽  
Laws Of Thermodynamics ◽  
Entropy Principle

The maximum entropy principle (MaxEnt) is one of the most powerful approaches in the theorist arsenal. It has been used in many areas of science, and, in particular, in both statistical mechanics and the quantum many body problem. Here we show how to derive MaxEnt from the laws of thermodynamics.

A PROPER NONLOCAL FORMULATION OF QUANTUM MAXIMUM ENTROPY PRINCIPLE IN STATISTICAL MECHANICS

2012 ◽  
Vol 26 (12) ◽  
pp. 1241007 ◽  
Author(s):  
M. TROVATO ◽  
L. REGGIANI
Keyword(s):  
Statistical Mechanics ◽  
Maximum Entropy ◽  
Maximum Entropy Principle ◽  
Quantum Statistical Mechanics ◽  
Fundamental Principle ◽  
Quantum Entropy ◽  
Closure Condition ◽  
Entropy Principle ◽  
Quantum Hydrodynamic ◽  
Reduced Density

By considering Wigner formalism, the quantum maximum entropy principle (QMEP) is here asserted as the fundamental principle of quantum statistical mechanics when it becomes necessary to treat systems in partially specified quantum states. From one hand, the main difficulty in QMEP is to define an appropriate quantum entropy that explicitly incorporates quantum statistics. From another hand, the availability of rigorous quantum hydrodynamic (QHD) models is a demanding issue for a variety of quantum systems. Relevant results of the present approach are: (i) The development of a generalized three-dimensional Wigner equation. (ii) The construction of extended quantum hydrodynamic models evaluated exactly to all orders of the reduced Planck constant ℏ. (iii) The definition of a generalized quantum entropy as global functional of the reduced density matrix. (iv) The formulation of a proper nonlocal QMEP obtained by determining an explicit functional form of the reduced density operator, which requires the consistent introduction of nonlocal quantum Lagrange multipliers. (v) The development of a quantum-closure procedure that includes nonlocal statistical effects in the corresponding quantum hydrodynamic system. (vi) The development of a closure condition for a set of relevant quantum regimes of Fermi and Bose gases both in thermodynamic equilibrium and nonequilibrium conditions.

On Singular Closures for the 5-Moment System in Kinetic Gas Theory

2015 ◽  
Vol 17 (2) ◽  
pp. 371-400 ◽  
Author(s):  
Roman Pascal Schaerer ◽  
Manuel Torrilhon
Keyword(s):  
Maximum Entropy ◽  
Maximum Entropy Principle ◽  
Second Order ◽  
Gas Flows ◽  
Moment Equations ◽  
Moment System ◽  
Entropy Principle ◽  
Kinetic Gas Theory ◽  
Closure Theory ◽  
Non Equilibrium

AbstractMoment equations provide a flexible framework for the approximation of the Boltzmann equation in kinetic gas theory. While moments up to second order are sufficient for the description of equilibrium processes, the inclusion of higher order moments, such as the heat flux vector, extends the validity of the Euler equations to non-equilibrium gas flows in a natural way.Unfortunately, the classical closure theory proposed by Grad leads to moment equations, which suffer not only from a restricted hyperbolicity region but are also affected by non-physical sub-shocks in the continuous shock-structure problem if the shock velocity exceeds a critical value. Amore recently suggested closure theory based on the maximum entropy principle yields symmetric hyperbolic moment equations. However, if moments higher than second order are included, the computational demand of this closure can be overwhelming. Additionally, it was shown for the 5-moment system that the closing flux becomes singular on a subset of moments including the equilibrium state.Motivated by recent promising results of closed-form, singular closures based on the maximum entropy approach, we study regularized singular closures that become singular on a subset of moments when the regularizing terms are removed. In order to study some implications of singular closures, we use a recently proposed explicit closure for the 5-moment equations. We show that this closure theory results in a hyperbolic system that can mitigate the problem of sub-shocks independent of the shock wave velocity and handle strongly non-equilibrium gas flows.

scholarly journals Derivation of some new distributions in statistical mechanics using maximum entropy approach

2014 ◽  
Vol 24 (1) ◽  
pp. 145-155 ◽  
Author(s):  
Amritansu Ray ◽  
S.K. Majumder
Keyword(s):  
Statistical Mechanics ◽  
Maximum Entropy ◽  
Maximum Entropy Principle ◽  
The Other ◽  
Central Idea ◽  
Entropy Principle ◽  
Bose Einstein ◽  
The Way

The maximum entropy principle has been earlier used to derive the Bose Einstein(B.E.), Fermi Dirac(F.D.) & Intermediate Statistics(I.S.) distribution of statistical mechanics. The central idea of these distributions is to predict the distribution of the microstates, which are the particle of the system, on the basis of the knowledge of some macroscopic data. The latter information is specified in the form of some simple moment constraints. One distribution differs from the other in the way in which the constraints are specified. In the present paper, we have derived some new distributions similar to B.E., F.D. distributions of statistical mechanics by using maximum entropy principle. Some proofs of B.E. & F.D. distributions are shown, and at the end some new results are discussed.

scholarly journals An Introduction to the Non-Equilibrium Steady States of Maximum Entropy Spike Trains

2019 ◽  
Author(s):  
Rodrigo Cofre ◽  
Leonardo Videla ◽  
Fernando Rosas
Keyword(s):  
Statistical Mechanics ◽  
Maximum Entropy ◽  
Spike Trains ◽  
Biological Processes ◽  
Comprehensive Overview ◽  
Equilibrium Statistical Mechanics ◽  
Temporal Asymmetry ◽  
Key Concepts ◽  
Time Irreversibility ◽  
Non Equilibrium

Although most biological processes are characterized by a strong temporal asymmetry, several popular mathematical models neglect this issue. Maximum entropy methods provide a principled way of addressing time irreversibility, which leverages powerful results and ideas from the3literature of non-equilibrium statistical mechanics. This article provides a comprehensive overview of these issues, with a focus in the case of spike train statistics. We provide a detailed account of the5mathematical foundations and work out examples to illustrate the key concepts and results from non-equilibrium statistical mechanics

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