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arXiv:1207.0019

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

Виктор Павлович Маслов

Published Jun 25, 2012• 1 citation
arXivHTMLPDF

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.

DOI
10.48550/arxiv.1207.0019

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The number-theoretical problem of partition of an integer corresponds to D=2D=2D=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 NNN of the order of 10210^2102 is considered by heuristic considerations like those Fock used to derive the Hartree--Fock equation. The case of N≪1N\ll1N≪1 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.
Abstract Chapter 11 presents a statistical mechanical treatment of the quantum ideal gases, i.e., the ideal Boltzmann, fermion, and boson gases. The discussion begins with the microscopic description and the solution of the eigenvalue problem for a quantum ideal gas. It is argued that calculating the partition function is most readily accomplished in the grand canonical ensemble using a second-quantized formulation. When the particles are distinguishable, the equation of state is identical to that of a classical ideal gas. For fermions and bosons, however, the problem of computing thermodynamic properties is significantly more complex and can only be solved exactly in certain limits. Away from these limits approximations are needed and are discussed in detail. The relevant distributions - the Fermi-Dirac and Bose-Einstein distributions are derived. The local density approximation of density functional theory is derived for the ideal electron gas. For the ideal boson gas, the phenomenon of Bose-Einstein condensation is discussed
Bose–Einstein and Fermi–Dirac statistics versus Boltzmann statistics It was remarked in Chapter 2, §2.1, that there are circumstances in which the “Boltzmann statistics” for an ideal gas – the approximation in Eq. (2.6), in which the total partition function is decomposed into a product of single-molecule partition functions, with division by N 1 ! N 2 ! … to correct for particle identity – is inaccurate even if the gas is still ideal (non-interacting particles). That happens when the number of single-particle states within the energy kT of the ground state is no longer very much greater than the number of particles. (Recall Fig. 2.1 and the accompanying discussion.) We noted that when the Boltzmann statistics is no longer an adequate approximation one must distinguish the particles as fermions, no two of which can be in the same single-particle state (counting states of different spin as different), or as bosons, for which there is no restriction on multiple occupancy of the singleparticle states. Fermions are particles of half-odd-integer spin quantum number; they could be composite particles consisting of an odd number of spin-½ particles. Protons, neutrons, and electrons, all of which are of spin ½, are fermions. Bosons are particles of integer spin; they could be spinless, or of spin 1, etc., or could be composites consisting of an even number of fermions. The nuclei of helium atoms of mass number 4 consist of two protons and two neutrons, and the neutral atom has two electrons outside the nucleus; 4 He atoms are bosons.
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In this work we revisit the concept of chemical potential μμμ in both classical and quantum gases from a perspective of Equilibrium Statistical Mechanics (ESM). Two new results regarding the equation of state μ=μ(n,T)μ=μ(n,T)μ=μ(n,T), where nnn is the particle density and TTT the absolute temperature, are given for the classical interacting gas and for the weakly-interacting quantum Bose gas. In order to make this review self-contained and adequate for a general reader we provide all the basic elements in an advanced-undergraduate or graduate statistical mechanics course required to follow all the calculations. We start by presenting a calculation of μ(n,T)μ(n,T)μ(n,T) for the classical ideal gas in the canonical ensemble. After this, we consider the interactions between particles and compute the effects of them on μ(n,T)μ(n,T)μ(n,T) for the van der Waals gas. For quantum gases we present an alternative approach to calculate the Bose-Einstein (BE) and Fermi-Dirac (FD) statistics. We show that this scheme can be straightforwardly generalized to determine what we have called Intermediate Quantum Statistics (IQS) which deal with ideal quantum systems where a single-particle energy can be occupied by at most jjj particles with 0⩽j⩽N0 \leqslant j \leqslant N0⩽j⩽N with NNN the total number of particles. In the final part we address general considerations that underlie the theory of weakly interacting quantum gases. In the case of the weakly interacting Bose gas, we focus our attention to the equation of state μ=μ(n,T)μ=μ(n,T)μ=μ(n,T) in the Hartree-Fock mean-field approximation (HF) and the implications of such results in the elucidation of the order of the phase transitions involved in the BEC phase for non-ideal Bose gases.