The mathematical theory of classical thermodynamics
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.
scientific article; zbMATH DE number 6158166
scientific article