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arXiv:2009.02178 · Ukrainian Journal of Physics

Bose–Einstein Condensation as a Deposition Phase Transition of Quantum Hard Spheres and New Relations between Bosonic and Fermionic Pressures

К.А. Bugaev, O. Ivanytskyi, B.E. Grinyuk, Ivan P Yakimenko

Published Nov 12, 2020• 1 citation

Abstract

We investigate the phase transition of Bose–Einstein particles with the hard-core repulsion in the grand canonical ensemble within the Van der Waals approximation. It is shown that the pressure of non-relativistic Bose–Einstein particles is mathematically equivalent to the pressure of simplified version of the statistical multifragmentation model of nuclei with the vanishing surface tension coefficient and the Fisher exponent тF = 5/2 , which for such parameters has the 1-st order phase transition. The found similarity of these equations of state allows us to show that within the present approach the high density phase of Bose-Einstein particles is a classical macro-cluster with vanishing entropy at any temperature which, similarly to the system of classical hard spheres, is a kind of solid state. To show this we establish new relations which allow us to identically represent the pressure of Fermi–Dirac particles in terms of pressures of Bose–Einstein particles of two sorts.

DOI: 10.15407/ujpe65.11.963

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We investigate the phase transition of Bose–Einstein particles with the hard-core repulsion in the grand canonical ensemble within the Van der Waals approximation. It is shown that the pressure of non-relativistic Bose–Einstein particles is mathematically equivalent to the pressure of simplified version of the statistical multifragmentation model of nuclei with the vanishing surface tension coefficient and the Fisher exponent тF = 5/2 , which for such parameters has the 1-st order phase transition. The found similarity of these equations of state allows us to show that within the present approach the high density phase of Bose-Einstein particles is a classical macro-cluster with vanishing entropy at any temperature which, similarly to the system of classical hard spheres, is a kind of solid state. To show this we establish new relations which allow us to identically represent the pressure of Fermi–Dirac particles in terms of pressures of Bose–Einstein particles of two sorts.

Phase transitions are investigated in the Bose-Fermi-Hubbard model in the mean field and hard-core boson approximations for the case of infinitely small fermion transfer and repulsive on-site boson-fermion interaction.The behavior of the Bose-Einstein condensate order parameter and grand canonical potential is analyzed as functions of the chemical potential of bosons at zero temperature.The possibility of change of order of the phase transition to the superfluid phase in the regime of fixed values of the chemical potentials of Bose-and Fermi-particles is established.The relevant phase diagrams are built.

The method developed in the previous paper for the treatment of the quantum-mechanical

N

-body hard-sphere problem is applied to a calculation of the grand partition function of an imperfect Bose gas with hard-sphere interactions. The grand partition function is calculated to second order in an expansion in powers of $\frac{a}{\ensuremath{\lambda}}$, where

a

is the hard-sphere diameter and $\ensuremath{\lambda}$ the thermal wavelength. The approximate equation of state for the gas phase is thereby obtained by calculating all the virial coefficients to order ${(\frac{a}{\ensuremath{\lambda}})}^{2}$.The first-order energy levels obtained in the previous paper embody some interesting physical properties. A fictitious system with exactly such energy levels is considered. The partition function for such a system can be calculated exactly and the exact equation of state obtained. It is shown that there is a phase transition, which more closely resembles an ordinary gas-liquid transition than the Bose-Einstein condensation.

The properties of Bose-Einstein-condensed systems of hard spheres at high density are studied by taking into account higher-order corrections beyond the Hartree-Fock-Bogoliubov approximation. Based on a diagrammatic technique of the perturbation expansion starting from the Bogoliubov approximation, we show that the effective interactions separate into two distinct parts: the screened and condensate-mediated interactions. The latter is described by the exchange of virtual quasiparticles between noncondensate bosons via the Bose-Einstein condensate and is attractive for small energy transfer. We calculate the condensation temperature, excitation spectrum, and condensate density in the whole region of the gas parameter. It is shown that the condensate temperature decreases compared to that of an ideal Bose gas due to hard-sphere interactions. We find that the condensate-mediated attraction between noncondensate bosons leads to an enhancement of the anomalous self-energy and, as a consequence, to the emergence of a roton minimum in the excitation spectrum and to the strong suppression of the condensate fraction in the region of high density.

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.