Conserved Mass Models and Particle Systems in One Dimension

Abstract

In this paper we study analytically a simple one dimensional model of mass transport. We introduce a parameter \(p\) that interpolates between continuous time dynamics (\(p\to 0\) limit) and discrete parallel update dynamics (\(p=1\)). For each \(p\), we study the model with (i) both continuous and discrete masses and (ii) both symmetric and asymmetric transport of masses. In the asymmetric continuous mass model, the two limits \(p=1\) and \(p\to 0\) reduce respectively to the \(q\)-model of force fluctuations in bead packs [S.N. Coppersmith et. al., Phys. Rev. E. {\bf 53}, 4673 (1996)] and the recently studied asymmetric random average process [J. Krug and J. Garcia, cond-mat/9909034]. We calculate the steady state mass distribution function \(P(m)\) assuming product measure and show that it has an algebraic tail for small \(m\), \(P(m)\sim m^{-\beta}\) where the exponent \(\beta\) depends continuously on \(p\). For the asymmetric case we find \(\beta(p)=(1-p)/(2-p)\) for \(0\leq p <1\) and \(\beta(1)=-1\) and for the symmetric case, \(\beta(p)=(2-p)^2/(8-5p+p^2)\) for all \(0\leq p\leq 1\). We discuss the conditions under which the product measure ansatz is exact. We also calculate exactly the steady state mass-mass correlation function and show that while it decouples in the asymmetric model, in the symmetric case it has a nontrivial spatial oscillation with an amplitude decaying exponentially with distance.