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

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 Schrödinger 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.

42 pages, 1 figure

Cited by in corpus (11)

• Modeling Translation in Protein Synthesis with TASEP: A Tutorial and Recent Developments
• Inhomogeneous exclusion processes with extended objects: The effect of defect locations
• Exact solutions of exactly integrable quantum chains by a matrix product ansatz
• Generalization of the matrix product ansatz for integrable chains
• Exactly solvable interacting vertex models
• Asymmetric exclusion model with impurities
• Exactly Solvable Interacting Spin-Ice Vertex Model
• Asymmetric exclusion model with several kinds of impurities
• Simulations of driven and reconstituting lattice gases
• Exact field-driven interface dynamics in the two-dimensional stochastic Ising model with helicoidal boundary conditions
• The spectral gap and the dynamical critical exponent of an exact solvable probabilistic cellular automaton