Kinetic theories for lattice models of liquids and polymers

We develop a theoretical formalism to investigate the time correlation functions of density fluctuations from equilibrium of a simple lattice model of dynamics using a method originally developed by Andersen to study atomic liquids. The lattice model of interest consists of a simple square or cubic lattice on which each particle occupies a site but two particles are forbidden to occupy the same site. This excluded volume lattice gas is governed by stochastic dynamics in which a particle can move to a nearest neighbor site if it is not occupied by a particle. We derive two formally exact diagrammatic expressions for the time correlation function of density fluctuations in this lattice system. We use these diagrammatic series to develop a series of approximate kinetic theories for the time correlation function of a two component system in which one component is present in trace amounts.; To test these approximate theories, we use a continuous time algorithm to simulate the stochastic dynamics of the excluded volume lattice gas and calculate the incoherent intermediate scattering function, the spatial Fourier transform of the time correlation function of the trace species. We perform simulations for low, intermediate, and high concentrations of particles on both simple square and cubic lattices. We can then directly compare the results of the incoherent intermediate scattering function with the predictions of our approximate kinetic theories for a range of wavevectors. In general, the approximate theories most accurately predict the results of computer simulation for the low concentration and the higher dimension. We find the largest deviation of the theory from the simulation results at the high concentration for small wavevectors in two dimensions. Our formalism for the excluded volume lattice gas provides a means of constructing more accurate kinetic theories in these regimes. Moreover, the theoretical methods used here can be generalized to more complex lattice models such as a lattice gas with attractive interactions or a bond fluctuation model of a polymer, and the present study of a simple model provides a foundation upon which one can make these generalizations.