Bose Condensate in the D-Dimensional Case, in Particular, for D=2. Semiclassical transition to the classical thermodynamics

TL;DR

This paper explores Bose-Einstein and Fermi-Dirac statistics in various dimensions, analyzing thermodynamic potentials, particle decay, and quantum effects, with applications to nuclear physics and mathematical frameworks.

Contribution

It introduces a novel analysis of Bose condensates in different dimensions, connecting thermodynamics, number theory, and quantum statistics with new insights into particle decay and nuclear energy.

Findings

Partition problem corresponds to D=2 and obeys Bose-Einstein statistics.

Derived a relation for neutron separation energy using de Broglie wavelength.

Studied particle decay and quantum effects using nonstandard analysis and tropical geometry.

Abstract

The number-theoretical problem of partition of an integer corresponds to $D=2$. This problem obeys the Bose--Eeinstein statistics, where repeated terms are admissible in the partition, and to the Fermi--Dirac statistics, where they are inadmissible. The Hougen--Watson P,Z-diagram shows that this problem splits into two cases: the positive pressure domain corresponds to the Fermi system, and the negative, to the Bose system. This analogy can be applied to the van der Waals thermodynamics. The thermodynamic approach is related to four potentials corresponding to the energy, free energy, thermodynamic Gibbs potential, enthalpy. The important notion of de Broglie's wavelength permits passing from particle to wave packet, in particular, to Bose and Fermi distributions. Particles of ideal Bose and Fermi gases and the decay of a boson consisting of two fermions into separate fermions are studied. The case of finitely many particles $N$ of the order of $10^2$ is considered by heuristic considerations like those Fock used to derive the Hartree--Fock equation. The case of $N\ll1$ is studied by Gentile statistics, tropical geometry and nonstandard analysis (Leibnitz differential or monad). A relation for the energy of neutron separation from the atomic nucleus is obtained when the atomic nucleus volume and de Broglie's wavelength are known. The Appendix is author's paper written in 1995.