Burgers equation with finite particle correction of the asymmetric simple exclusion process derived from the derivative nonlinear Schr\"{o}dinger equation

We investigate the dynamics of the asymmetric simple exclusion process (ASEP) on a ring. The ASEP is equivalent to the derivative nonlinear Schr"{o}dinger equation (DNLS), which is integrable quantum field theory, in the continuous limit. We derive the Burgers equation with finite particle correction from the DNLS and numerically confirm that the obtained Burgers equation describes the dynamics of the ASEP at small numbers of particles better than the conventional Burgers equation.