Transport in two dimensions. I the self-diffusion coefficient

The Mori formalism is used to study generalized transport coefficients in two dimensions. All finite multilinear products of the single particle density and momentum density comprise the set of the variables in the calculation of the self-diffusion coefficient. A self-consistent equation, which is a non-linear integral equation, is obtained for the leading asymptotic behavior of the generalized self-diffusion coefficient. An asymptotic solution is presented which for small wavevector (k) and frequency (s) behaves like $\text{In}{}^{\mathrm{<mtext>1</mtext><mtext>2</mtext>}}\text{(s + k}{}^2\text{D)}{}^{\mathrm{?1}}$, where D has the dimensions of a diffusion coefficient. The mean square and mean fourth displacements of a tagged particle are also calculated. The long time behavior of the momentum correlation function exhibits a tail of the form $\text{[t}\text{In}{}^{\mathrm{<mtext>1</mtext><mtext>2</mtext>}}\text{(t)]}{}^{\mathrm{?1}}$ whose coefficient is dependent of the intermolecular potential.