Relaxation spectrum of the asymmetric exclusion process with open boundaries

We calculate numerically the exact relaxation spectrum of the totally asymmetric simple exclusion process (TASEP) with open boundary conditions on lattices up to 16 sites. Ln the low- and high-density phases and along the nonequilibrium first-order phase transition between these phases, but sufficiently far away from the second-order phase transition into the maximal-current phase, the low-lying spectrum (corresponding to the longest relaxation times) agrees well with the spectrum of a biased random walker confined to a finite lattice of the same size. The hopping rates of this random walk are given by the hopping rates of a shock (a domain wall separating stationary low- and high-density regions), which are calculated in the framework of a recently developed non-equilibrium version of Zel'dovich's theory of the kinetics of first-order transitions. We conclude that the description of the domain wall motion in the TASEP in terms of this theory of boundary-induced phase transitions is meaningful for very small systems of the order of ten lattice sites.