The mathematical theory of classical thermodynamics
type: Item
instance of: scholarly article
title: The mathematical theory of classical thermodynamics
zbMATH Open document ID: 1268.80002
DOI: 10.1134/S0001434613010112
author: V. P. Maslov
published in: Mathematical Notes
publication date: 2013-04-26T00:00:00Z
review text: The author gathers some partly new ideas concerning the use of number-theoretical tools in the modelling of classical thermodynamics. The author especially focuses on the computations of number of degrees of freedom for a thermodynamic system, linked to the number of particles. Throughout the note, he presents examples of thermodynamic systems and their properties. He starts with a system consisting of \(N\) interacting particles and gives the structure of the Hamiltonian associated to this system whose energy is bounded by some \(M\). The first main result describes the behaviour of the entropy \(S\) of the system in terms of \(N\) and \(M\), under some hypotheses. The author then considers a Bose gas of fractional dimension \(D=2(1+\gamma )\). He writes the thermodynamic potential in terms of the polylogarithm or the Fermi-Dirac integral \(Li_{2+\gamma }(a)\), where \(a\) is the activity. He computes the dimensionless pressure and the number of particles associated to this potential and compares these computations to experimental results. The author then considers a phase transition to an ideal liquid and holes in a Bose condensate, which may be associated to the particles. Then, he moves on to the study of the attraction process between originally scattered particles. He here gives expressions of the energy and of the Lennard-Jones potential. He then introduces the compressibility factor \(Z=\frac{PV}{NT}\) and the notion of Zeno line as the curve defined by \(Z=1\), also describing some properties of this Zeno line. In the last part of the note, the author describes some properties of these notions using further examples of thermodynamic systems. Throughout the note, many figures illustrate the theoretical computations which are drawn by the author.
reviewed by: Alain Brillard
Mathematics Subject Classification ID: 80A10
Mathematics Subject Classification ID: 80A05
Mathematics Subject Classification ID: 82C22
zbMATH DE Number: 6158166
zbMATH Keywords: measuring device
zbMATH Keywords: Bose condensate
zbMATH Keywords: phase transition
zbMATH Keywords: fluid
zbMATH Keywords: negative pressure
zbMATH Keywords: influence of vessel walls
zbMATH Keywords: Knudsen criterion
zbMATH Keywords: Clapeyron-Clausius gas law
zbMATH Keywords: Zeno line
zbMATH Keywords: number theory
zbMATH Keywords: Hamiltonian
zbMATH Keywords: Lennard-Jones potential
MaRDI profile type: Publication
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full work available at URL: s0001434613010112
OpenAlex ID: W2081455632
description: scientific article; zbMATH DE number 6158166