On the Diffusion of Sticky Particles in 1D

Abstract: The 1D Ising model is the simplest Hamiltonian-based model in statistical mechanics. The sim- plest interacting particle process is the Symmetric Exclusion Process (SEP), a 1D lattice gas of particles that hop symmetrically and cannot overlap. Combining the two gives a model for sticky particle diffusion, SPM, which is described here. SPM dynamics are based on SEP with short-range interaction, allowing flow due to non-equilibrium boundary conditions. We prove that SPM is also a detailed-balance respecting, particle-conserving, Monte Carlo (MC) description of the Ising model. Neither the Ising model nor SEP have a phase transition in 1D, but the SPM exhibits a non-equilibrium transition from a diffusing to a blocked state as stickiness increases. This transition manifests in peaks in the MC density fluctuation, a change in the dependency of flow-rate on stickiness, and odd structure in the eigenspectrum of the transition rate matrix. We derive and solve a fully non-linear, analytic, mean-field solution, which has a crossover from a positive to a negative diffusion constant where the MC shows the transition. The negative diffusion constant in fact indicates a breakdown of the mean-field approximation, with close to zero flow and breaking into a two-phase mixture, and thus the mean field theory successfully predicts its own demise. We also present an analytic solution for the flow via direct analysis of the transition rate matrix. The simplicity of the model suggest a wide range of possible applications.