Efficient Numeric Computation of a Phase Diagram in Biased Diffusion of Two Species

A lattice gas with equal numbers of oppositely charged particles, diffusing under the influence of a uniform electric field and an excluded volume condition undergoes an order-disorder phase transition, controlled by the particle density and the field strength. This transition may be continuous (second order) or continuous (first order). Results from previous discrete simulations are shown, and a theoretical continuum model is developed. As this is a nonequilibrium system, there is no associated free energy to determine the location of a first order transition. Instead, the model equations for this system are evolved in time numerically, and the locus of this transition is determined via the presence of a stable state with coexisting regions of order and disorder. The Crank-Nicholson, nonlinear Gauss-Seidel, and GMRES algorithms used to solve the model equations are discussed. Performance enhancements and limits on convergence are considered.