Distribution Functions and Quantum Statistics
R. U. Ayres · Physical Review · 1960
A formulation of quantum statistical mechanics is given, in terms of distribution functions. It is shown that all quantities of interest are obtained directly from nonlocal distribution functions ${\ensuremath{\gamma}}^{(1)}({{x}_{1}}^{\ensuremath{'}}, {x}_{1}, \ensuremath{\beta}), {\ensuremath{\gamma}}^{(2)}({{x}_{1}}^{\ensuremath{'}}, {{x}_{2}}^{\ensuremath{'}}, {x}_{1}{x}_{2}, \ensuremath{\beta}), \ensuremath{\cdots}.$ In a uniform system the Fourier transform of ${\ensuremath{\gamma}}^{(1)}({{x}_{1}}^{\ensuremath{'}}, {x}_{1}, \ensuremath{\beta})$ is the distribution of particles in $k$ space, $n(k, \ensuremath{\beta})$. Various schemes for calculating these distribution functions directly are discussed. An approximate integral equation for $n(k, \ensuremath{\beta})$ is found, which can be solved by a converging iteration process. Some remarks are included on the application of the virial theorem.