Exact Mean - Field Solutions of the Asymmetric Random Average Process 3 2 . A Functional Equation of Exact Mean - Field

We consider the asymmetric random average process (ARAP) with continuous mass variables and parallel discrete time dynamics studied recently by Krug/Garcia and Rajesh/Majumdar [both Jrl. Stat. Phys. 99 (2000)]. The model is defined by an arbitrary state-independent fraction density function φ(r) with support on the unit interval. We examine the exactness of mean-field steady-state mass distributions in dependence of φ and identify as a conjecture based on high order calculations the class M of density functions yielding product measure solutions. Additionally the exact form of the associated mass distributions P (m) is derived. Using these results we show examplary the exactness of the mean-field ansatz for monomial fraction densities φ(r) = (n − 1)r n−2 with n ≥ 2. For verification we calculate the mass distributions P (m) explicitly and prove directly that product measure holds. Furthermore we show that even in cases where the steady state is not given by a product measure very accurate approximants can be found in the class M.