Nonanalyticity of Transport Coefficients and the Complete Density Expansion of Momentum Correlation Functions

The complete "formal" density expansion of any momentum autocorrelation function—at any frequency—is derived from the density expansion of the generalized master equation. (The sth term in this expansion is explicitly defined in terms of the time-displacement operator of $s+1\mathrm{}$ particles and involves the dynamics of $s+1\mathrm{}$ isolated particles.) The entire derivation consists of only a few algebraic steps, and is valid for noncentral (polar) pair forces as well as central forces. It is then shown that the third and higher order terms in the zero-frequency limit of the density expansion diverge—although the first two terms and the entire sum coverage (the density expansion "breaks down"). This suggests that transport coefficients are not analytic functions of the density. It is suggested that a partial resummation (renormalization)—analogous to that used in the electron-gas problem—be used to calculate the nonanalytic behavior of transport coefficients.