The Exact Solution of the Asymmetric Exclusion Problem With Particles of Arbitrary Size: Matrix Product Ansatz

TL;DR

This paper presents an exact solution to the asymmetric exclusion problem with particles of arbitrary size using a matrix product ansatz, linking it to quantum Hamiltonian diagonalization and integrability via algebraic relations.

Contribution

It introduces a matrix product ansatz for solving the asymmetric exclusion problem with particles of arbitrary size and multiple classes, establishing its integrability through algebraic relations.

Findings

Exact solution for particles of arbitrary size and class hierarchy

Matrix product ansatz expresses eigenfunctions as matrix products

Associativity of algebra implies Yang-Baxter relations

Abstract

The exact solution of the asymmetric exclusion problem and several of its generalizations is obtained by a matrix product {\it ansatz}. Due to the similarity of the master equation and the Schr\"odinger equation at imaginary times the solution of these problems reduces to the diagonalization of a one dimensional quantum Hamiltonian. We present initially the solution of the problem when an arbitrary mixture of molecules, each of then having an arbitrary size ($s=0,1,2, ...$) in units of lattice spacing, diffuses asymmetrically on the lattice. The solution of the more general problem where we have | the diffusion of particles belonging to $N$ distinct class of particles ($c=1, ..., N$), with hierarchical order, and arbitrary sizes is also solved. Our matrix product {\it ansatz} asserts that the amplitudes of an arbitrary eigenfunction of the associated quantum Hamiltonian can be expressed by a product of matrices. The algebraic properties of the matrices defining the {\it ansatz} depend on the particular associated Hamiltonian. The absence of contradictions in the algebraic relations defining the algebra ensures the exact integrability of the model. In the case of particles distributed in $N>2$ classes, the associativity of the above algebra implies the Yang-Baxter relations of the exact integrable model.