Relaxation towards a nonequilibrium steady state: an unexpected spectral property of the asymmetric exclusion process

The asymmetric exclusion process (ASEP) is a stochastic model of a system of driven diffusive particles on a lattice, also known as a lattice gas. Variants of the model have been used to describe systems such as traffic flow and hyphal tip growth in fungi. The nonequilibrium steady state phase diagram has been completely characterised.

In a recent paper, de Gier and Essler use the Bethe Ansatz to solve the spectrum of the Markovian transition matrix, which encodes the dynamics of the model. The authors find a crossover between regions in parameter space with different dynamical scaling exponents. One such region is further divided, with the transition consisting of a discontinuity for finite system size in the eigenvalue corresponding to the first excited state, i.e. the longest-lived relaxational mode. This finding prompts a modification of the existing domain wall description of the dynamics. Whilst clearly a necessary 'fix', we do not find this modification enlightening with regards to the physical mechanisms underlaying the transition. More importantly we expect continuity of eigenvalues to hold at least within a particular limit, in contradiction to the Bethe Ansatz results.

I will therefore discuss our attempts at using Dynamic Monte Carlo simulation, exact diagonalisation and a perturbative approach to verify the results of their analysis and understand its implications for finite systems. What we are aiming for is a clear picture of the relaxational physics associated with this purported transition.