Microscopic Method for Calculating Memory Functions in Transport Theory

It is shown how the method of thermodynamic Green's functions can be used to approximate the memory function associated with the equilibrium-fluctuation function $S_c(\overrightarrow{\mathrm{r}}-{\overrightarrow{\mathrm{r}}}^{\prime },\overrightarrow{\mathrm{p}}{\overrightarrow{\mathrm{p}}}^{\prime },t-t^{\prime })\equiv 〈\left[f\right(\overrightarrow{\mathrm{r}}\overrightarrow{\mathrm{p}}t)-〈f(\overrightarrow{\mathrm{r}}\overrightarrow{\mathrm{p}}t\left)〉\right]\left[f\right({\overrightarrow{\mathrm{r}}}^{\prime }{\overrightarrow{\mathrm{p}}}^{\prime }{t}^{\prime })-〈f({\overrightarrow{\mathrm{r}}}^{\prime }{\overrightarrow{\mathrm{p}}}^{\prime }{t}^{\prime }\left)〉\right]〉$, where $f\left(\overrightarrow{\mathrm{r}}\overrightarrow{\mathrm{p}}t\right)$ is the phase-space distribution operator. We obtain an approximation for the memory function for a gas, in the low-density limit, that is valid for all distances and times, satisfied various relevant symmetry conditions and sum rules, reduces for long times and distances to the Boltzmann collision operator, and gives results completely consistent with the conservation laws governing the system. We also indicate how these methods can be extended to treat other types of systems.