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Laser Physics

Ideal bose gas in an oscillator potential (exact solution)

Е. А. Титов

Published Jun 28, 2006• 0 citations
Publisher

Abstract

Exact expressions for the statistical sum of the grand canonical ensemble and the one-particle density matrix are derived based on the definition of the density matrix for a system of N identical noninteracting Bose particles in an oscillator potential as a sum with respect to the symmetric exchange of the density matrix coordinates of distinguishable particles. A quasi-classical scenario is analyzed in detail.

DOI
10.1134/s1054660x06070103

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Exact expressions for the statistical sum of the grand canonical ensemble and the one-particle density matrix are derived based on the definition of the density matrix for a system of N identical noninteracting Bose particles in an oscillator potential as a sum with respect to the symmetric exchange of the density matrix coordinates of distinguishable particles. A quasi-classical scenario is analyzed in detail.
The ideal Bose-gas with finite number NNN of particles is investigated. The exact expressions for the partition functions and occupation numbers in the grand canonical, canonical and microcanonical ensembles are found. The asymp\-totic expressions (in the case N≫1N\gg1N≫1) for the partition functions and occupation numbers in the canonical and microcanonical ensembles are evaluated. It is shown that the chemical potential μμμ of the ideal Bose-gas can lie in the range $-\infty
Abstract This chapter describes the mechanism of Bose–Einstein condensation in the simplest ideal Bose gas case. An explicit expression for the critical temperature is derived in the analytically soluble case of a gas confined in a box. Various thermodynamic quantities are calculated above and below the critical temperature. Emphasis is given to the isothermal compressibility which diverges below the critical temperature. The behaviour of the off-diagonal one-body density matrix is discussed. Results for the fluctuations of the single-particle occupation numbers are derived in both the canonical and grand canonical ensembles.
We discuss the exact particle number counting statistics of degenerate ideal Bose gases in the microcanonical, canonical, and grand-canonical ensemble, respectively, for various trapping potentials. We then invoke the Maxwell's Demon ensemble [Navez et el., Phys. Rev. Lett. (1997)] and show that for large total number of particles the root-mean-square fluctuation of the condensate occupation scales n0 / [T=Tc] r N s with scaling exponents r = 3=2, s = 1=2 for the3D harmonic oscillator trapping potential, and r = 1,s= 2=3 for the 3D box. We derive an explicit expression for r and s in terms of spatial dimension D and spectral index sigma of the single-particle energy spectrum. Our predictions also apply to systems where Bose-Einstein condensation does not occur. We point out that the condensate fluctuations in the microcanonical and canonical ensemble respect the principle of thermodynamic equivalence.
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
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The paper is devoted to the theory of an ideal Bose-gas with a finite number N of particles. The exact expressions for the partition functions and occupation numbers of the model in the grand canonical, canonical, and microcanonical ensembles are found. From the calculations, it is followed that, oppositely to the accepted opinion that the chemical potential μ of an ideal Bose-gas is only negative, it can take values in the range −∞ < μ < ∞. The asymptotic expressions (in the case N ≫ 1) for the partition functions and occupation numbers for all above-mentioned thermodynamic ensembles are also evaluated.