Conserved Mass Models and Particle Systems in One Dimension

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.