The formulation of the previous paper (paper IV) is extended so that it becomes applicable in an interacting system in the presence of a Bose-Einstein degeneracy. This extension is carried out by the introduction of an $x$ -ensemble, which enables one to utilize an Ursell-type expansion even in the presence of a Bose-Einstein degeneracy. The variational principle of the previous paper is also extended. It is proved that in the presence of a Bose-Einstein degeneracy, the average occupation number of a single particle state with momentum p approaches infinity as p\ensuremath{\rightarrow}0. The method is applied to a dilute system of Bose hard spheres.The formulation of the previous paper (paper IV) is extended so that it becomes applicable in an interacting system in the presence of a Bose-Einstein degeneracy. This extension is carried out by the introduction of an $x$-ensemble, which enables one to utilize an Ursell-type expansion even in the presence of a Bose-Einstein degeneracy. The variational principle of the previous paper is also extended. It is proved that in the presence of a Bose-Einstein degeneracy, the average occupation number of a single particle state with momentum p approaches infinity as p\ensuremath{\rightarrow}0. The method is applied to a dilute system of Bose hard spheres.

Physical ReviewFeb 15, 196059 citations

Transport Phenomena in Einstein-Bose and Fermi-Dirac Gases. I

Edwin A. Uehling, G. E. Uhlenbeck

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 $A$ 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 $v$ and $T$ is known only in limiting cases; (b) The function $W$ of the general equation of state for ideal gases in all statistics $pv=(\frac{\mathrm{RT}}{M})W({v}^{\frac{2}{3}}T)$ 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.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 $A$ 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 $v$ and $T$ is known only in limiting cases; (b) The function $W$ of the general equation of state for ideal gases in all statistics $pv=(\frac{\mathrm{RT}}{M})W({v}^{\frac{2}{3}}T)$ 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.

Physical ReviewApr 1, 1933673 citations

x

-ensemble, which enables one to utilize an Ursell-type expansion even in the presence of a Bose-Einstein degeneracy. The variational principle of the previous paper is also extended. It is proved that in the presence of a Bose-Einstein degeneracy, the average occupation number of a single particle state with momentum p approaches infinity as p\ensuremath{\rightarrow}0. The method is applied to a dilute system of Bose hard spheres.

A

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

v

and

T

is known only in limiting cases; (b) The function

W

of the general equation of state for ideal gases in all statistics

pv=(\frac{\mathrm{RT}}{M})W({v}^{\frac{2}{3}}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.