Relaxation spectrum of the asymmetric exclusion process with open boundaries

We calculate numerically the exact relaxation spectrum of the totally asymm etric simple exclusion process (TASEP) with open boundary conditions on lat tices up to 16 sites. Ln the low- and high-density phases and along the non equilibrium first-order phase transition between these phases, but sufficie ntly far away from the second-order phase transition into the maximal-curre nt phase, the low-lying spectrum (corresponding to the longest relaxation t imes) 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 k inetics of first-order transitions. We conclude that the description of the domain wall motion in the TASEP in terms of this theory of boundary-induce d phase transitions is meaningful for very small systems of the order of te n lattice sites.