On the Diffusion of Sticky Particles in 1D
Abstract
The 1D Ising model is the simplest Hamiltonianbased 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 shortrange interaction, allowing flow due to nonequilibrium boundary conditions. We prove that SPM is also a detailedbalance respecting, particleconserving, 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 nonequilibrium 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 flowrate on stickiness, and odd structure in the eigenspectrum of the transition rate matrix. We derive and solve a fully nonlinear, analytic, meanfield 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 meanfield approximation, with close to zero flow and breaking into a twophase 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.
 Publication:

arXiv eprints
 Pub Date:
 March 2018
 arXiv:
 arXiv:1803.09712
 Bibcode:
 2018arXiv180309712H
 Keywords:

 Condensed Matter  Statistical Mechanics;
 Physics  Computational Physics
 EPrint:
 10 pages, 10 figures