## Abstract

An ideal gas of non-degenerate electrons of mass m in a Gaussian random potential is investigated. The potential is characterized by two parameters: $\eta $ whose square is the variance of the potential energy) and L (correlation length). Relevance to the case of a polycrystalline non-degenerate semiconductor is suggested. The autocorrelation function of the potential is taken Gaussian, W(r$^{\prime}$ - r$^{\prime \prime}$ = exp [- (r$^{\prime}$ - r$^{\prime \prime}$)$^{2}$/L$^{2}$]. The averaged canonical density matrix $\langle $C$_{\beta}$(r, r$_{0}$)$\rangle $ is calculated by the use of Feynman's path-integral formulation. Using the replacement W(r$^{\prime}$ - r$^{\prime \prime}$)$^{2}\sim $ 1 - (r$^{\prime}$ - r$^{\prime \prime}$)$^{2}$/L$^{2}$ (after discussing conditions under which it is possible), we derive an explicit formula for $\langle $C$_{\beta}$(r, r$_{0}$)$\rangle $. A characteristic frequency $\omega _{\text{G}}$ = ($\eta $/L) (2/mk$_{\text{B}}$T)$^{\frac{1}{2}}$ is found and interpreted as being due to itinerant oscillators. A characteristic mass m$_{\text{G}}$ = m$\frac{1}{2}\beta $h$\omega _{\text{G}}$ coth ($\frac{1}{2}\beta $h$\omega _{\text{G}}$), $\beta $ = 1/k$_{\text{B}}$T is found; the ratio (m$_{\text{G}}$ - m)/m is interpreted as a measure of localization of the electrons in the random potential under consideration. A formula for the energy-level density is discussed with respect to conditions of applicability of the quasi-classical approximation. The function $\langle $C$_{\beta}$(r, r$_{0}$)$\rangle $ is a simple expression if the condition {$\hslash \eta $/L$\surd $(2m)}$^{\frac{2}{3}}$ $\ll $ k$_{\text{B}}$T is satisfied; moreover, if $\eta \ll $ k$_{\text{B}}$T, then the 'quantal distribution function' (averaged Wigner function) is shown to equal the Maxwell distribution function f(E$_{k}$) = f(0) exp (- $\beta $E$_{k}$) where E$_{k}$ is the kinetic energy, E$_{k}$ = $\hslash ^{2}$k$^{2}$/2m, and f(0) is practically independent of $\eta $. Additional conditions of validity of the Boltzmann-equation theory are thus found.