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Physical Review

Further Deductions from the ℵ -Theorem

Harvey Einbinder

Published Aug 1, 1949• 5 citations
Publisher

Abstract

For an ideal statistical gas, where $PV=sE$, all thermodynamic functions can be derived from $G(V,T)$, i.e., $A(V,T)$ is given by a suitable integration over $G(V,T)$. Since $G={N}_{\ensuremath{\mu}}$, the normalization constant, $\ensuremath{\mu}(V,T)$, of the distribution function is sufficient to provide a complete thermodynamic description of such a system. With the aid of the $\ensuremath{\aleph}$-theorem it is shown that there is a sharp physical distinction between the degenerate and non-degenerate Bose-Einstein gases; to any given density there corresponds a transition temperature between them. The method developed is applied to the general (relativistic and non-relativistic) non-degenerate quantum and degenerate Fermi-Dirac gases.

DOI
10.1103/physrev.76.410

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For an ideal statistical gas, where PV=sEPV=sEPV=sE, all thermodynamic functions can be derived from G(V,T)G(V,T)G(V,T), i.e., A(V,T)A(V,T)A(V,T) is given by a suitable integration over G(V,T)G(V,T)G(V,T). Since G=N\ensuremathμG={N}_{\ensuremath{\mu}}G=N\ensuremathμ​, the normalization constant, \ensuremathμ(V,T)\ensuremath{\mu}(V,T)\ensuremathμ(V,T), of the distribution function is sufficient to provide a complete thermodynamic description of such a system. With the aid of the \ensuremathℵ\ensuremath{\aleph}\ensuremathℵ-theorem it is shown that there is a sharp physical distinction between the degenerate and non-degenerate Bose-Einstein gases; to any given density there corresponds a transition temperature between them. The method developed is applied to the general (relativistic and non-relativistic) non-degenerate quantum and degenerate Fermi-Dirac gases.
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
The theory of gases in nonstationary states, given by Lorentz and Enskog, is generalized for the quantum statistics to give the hydrodynamical equations and the distribution function in first and second approximation, and formal expressions for the viscosity and heat conductivity coefficients. These results are valid for all statistics and for all degrees of degeneration. Two essential points contribute to this generality: (a) Exact expressions independent of statistics and degeneration can be given for the coordinate and time derivatives of the coefficient AAA of the equilibrium distribution function in terms of the pressure and temperature gradients and time derivatives, though a closed expression for this coefficient as a function of vvv and TTT is known only in limiting cases; (b) The function WWW of the general equation of state for ideal gases in all statistics pv=(RTM)W(v23T)pv=(\frac{\mathrm{RT}}{M})W({v}^{\frac{2}{3}}T)pv=(MRT​)W(v32​T) is adiabatically invariant. Numerical values of the viscosity and heat conductivity coefficients, which should come out of the formal theory on the introduction of special assumptions about the molecular forces, have not yet been obtained. It is our hope that these results, when found, may furnish an experimental test of the existence of Einstein-Bose statistics in real gases, as is required by theory.
Calculations of chemical potentials for ideal monatomic gases with Bose-Einstein and Fermi-Dirac statistics as functions of temperature, across the temperature region that is typical for the collective quantum degeneracy effect, are presented. Numerical calculations are performed without any additional approximations, and explicit dependences of the chemical potentials on temperature are constructed at a fixed density of gas particles. Approximate polynomial dependences of chemical potentials on temperature are obtained that allow for the results to be used in further studies without re-applying the involved numerical methods. The ease of using the obtained representations is demonstrated on examples of deformation of distribution for a population of energy states at low temperatures, and on the impact of quantum statistics (exchange interaction) on the equations of state for ideal gases and some of the thermodynamic properties thereof. The results of this study essentially unify two opposite limiting cases in an intermediate region that are used to describe the equilibrium states of ideal gases, which are well known from university courses on statistical physics, thus adding value from an educational point of view.
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The topic of Bose–Einstein condensation in a uniform, non-interacting gas of bosons is treated in most textbooks on statistical mechanics. In the present chapter we discuss the properties of a non-interacting Bose gas in a trap. We shall calculate equilibrium properties of systems in a semi-classical approximation, in which the energy spectrum is treated as a continuum. For this approach to be valid the temperature must be large compared with Δ∊/ k , where Δ∊ denotes the separation between neighbouring energy levels. As is well known, at temperatures below the Bose–Einstein condensation temperature, the lowest-energy state is not properly accounted for if one simply replaces sums by integrals, and it must be included explicitly. The statistical distribution function is discussed in Sec. 2.1, as is the single-particle density of states, which is a key ingredient in the calculations of thermodynamic properties. Calculations of the transition temperature and the fraction of particles in the condensate are described in Sec. 2.2. In Sec. 2.3 the semi-classical distribution function is introduced. From this we obtain the density profile and the velocity distribution of particles, and use these to determine the shape of an anisotropic cloud after free expansion. Thermodynamic properties of Bose gases are calculated as functions of the temperature in Sec. 2.4. Finally, Sec. 2.5 treats corrections to the transition temperature due to a finite particle number.
We overwiev the properties of a quantum gas of particles with the intermediate statistics defined by Haldane. Although this statistics has no direct connection to the symmetry of the multiparticle wave function, the statistical distribution function interpolates continuously between the Fermi-Dirac and the Bose-Einstein limits. We present an explicit solution of the transcendental equation for the didtribution function in a general case, as well as determine the thermodynamic properties in both low- and high-temperature limits.