Exact diffusion constant for one-dimensional asymmetric exclusion models

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Abstract
The one-dimensional fully asymmetric exclusion model, which describes a system of particles hopping in a preferred direction with hard core interactions, is considered on a ring of size N. The steady state of this system is known (all configurations have equal weight), which allows for easy computation of the average velocity of a particle in the steady state. Here an exact expression for the diffusion constant of a particle is obtained for arbitrary number of particles and system size, by using a matrix formulation. Two limits of infinite system size N are discussed: firstly, when the number of particles remains finite as N to infinity the diffusion constant remains dependent on the exact number of particles due to correlations between successive collisions; secondly, when the density p of particles is non-zero (i.e. when the number of particles is equal to N rho as N to infinity ) the diffusion constant scales as N-12/. The exponent -1/2 is related to the dynamic exponent z = 3/2 of the KPZ equation in (1+1) dimensions.