Properties of the kinetic memory function in classical fluids

Abstract

The kinetic theory of gases and liquids is discussed in terms of the phase-space density $f\left(\overrightarrow{\mathrm{r}}\overrightarrow{\mathrm{p}}t\right)$ and its autocorrelation function. Associated with the latter is the memory function ${\Sigma }^{\mathrm{}\left(c\right)\mathrm{}}\left(\overrightarrow{\mathrm{k}}z\right)$ troduced in an earlier paper. Here, we analyze those properties of ${\Sigma }^{\mathrm{}\left(c\right)\mathrm{}}$, for a classical system of structureless interacting particles, which can be obtained without approximation. Apart from symmetry- and stability-related properties, and those that express the conservation laws, we give microscopic derivations of (i) a new sum rule for ${\Sigma }^{\mathrm{}\left(c\right)\mathrm{}}\left(\overrightarrow{\mathrm{k}}z\right)$ which involves only the static-pair-correlation function and (ii) two new relations between ${\Sigma }^{\mathrm{}\left(c\right)\mathrm{}}$ and thermodynamic derivatives, namely the specific heat $c_v$ and the pressure derivative ${\left(\frac{\partial p}{\partial T}\right)}_n$. These exact relations, if fulfilled by an approximate model for ${\Sigma }^{\mathrm{}\left(c\right)\mathrm{}}$, guarantee that the approach to equilibrium is described in a thermodynamically and dynamically consistent manner. It is also shown how, for such a model, the transport coefficients are obtained.