Numerical Estimation of the Current Large Deviation Function in the Asymmetric Simple Exclusion Process with Open Boundary Conditions

TL;DR

This paper numerically investigates the large deviation function of the total current in asymmetric simple exclusion processes with open boundaries, revealing new behaviors and convergence challenges not seen in exactly solvable models.

Contribution

It introduces numerical methods to estimate the large deviation function and uncovers novel behaviors, including system-size dependence and a cusp in the asymmetric case.

Findings

Different system-size dependences for even and odd parts of the generating function.

Total current definition is crucial for accurate Monte Carlo estimation.

A cusp appears in the large deviation function for the asymmetric process.

Abstract

We numerically study the large deviation function of the total current, which is the sum of local currents over all bonds, for the symmetric and asymmetric simple exclusion processes with open boundary conditions. We estimate the generating function by calculating the largest eigenvalue of the modified transition matrix and by population Monte Carlo simulation. As a result, we find a number of interesting behaviors not observed in the exactly solvable cases studied previously as follows. The even and odd parts of the generating function show different system-size dependences. Different definitions of the current lead to the same generating function in small systems. The use of the total current is important in the Monte Carlo estimation. Moreover, a cusp appears in the large deviation function for the asymmetric simple exclusion process. We also discuss the convergence property of the population Monte Carlo simulation and find that in a certain parameter region, the convergence is very slow and the gap between the largest and second largest eigenvalues of the modified transition matrix rapidly tends to vanish with the system size.