The current distribution of the multiparticle hopping asymmetric diffusion model

TL;DR

This paper analyzes the multiparticle hopping asymmetric diffusion model (MADM) on the integer lattice, deriving explicit transition probabilities and particle position distributions using Bethe ansatz, and discusses its relation to PushASEP.

Contribution

It provides explicit formulas for transition probabilities and particle position distributions in MADM, connecting it to PushASEP, which advances understanding of multi-particle asymmetric diffusion models.

Findings

Transition probability expressed as sum of contour integrals

Distribution of the m-th particle position derived

Connection between MADM and PushASEP discussed

Abstract

In this paper we treat the \textit{multiparticle hopping asymmetric diffusion model} (MADM) on $\mathbb{Z}$ introduced by Sasamoto and Wadati in 1998. The transition probability of the MADM with $N$ particles is provided by using the Bethe ansatz. The transition probability is expressed as the sum of $N$-dimensional contour integrals of which contours are circles centered at the origin with restrictions on their radii. By using the transition probability we find $\mathbb{P}(x_m(t) =x)$, the probability that the $m$th particle from the left is at $x$ at time $t$. The probability $\mathbb{P}(x_m(t) =x)$ is expressed as the sum of $|S|$-dimensional contour integrals over all $S \subset \{1,...,N\}$ with $|S| \geq m$, and is used to give the current distribution of the system. The mapping between the MADM and the pushing asymmetric simple exclusion process (PushASEP) is discussed.