Quantum vs Classical Birth and Death Processes; Exactly Solvable Examples

TL;DR

This paper introduces a quantum approach to birth and death processes, providing exact solutions and eigenvalues, which could enhance simulation capabilities of these stochastic systems.

Contribution

It presents a novel coinless quantisation method for classical birth and death processes, linking them to exactly solvable quantum systems with analytical eigenvalues.

Findings

Quantum and classical systems share eigenvalues and eigenvectors are related.

Exactly solvable models with orthogonal polynomial eigenvectors.

Many examples exhibit integer eigenvalues, indicating periodicity.

Abstract

A coinless quantisation procedure of continuous and discrete time Birth and Death (BD) processes is presented. The quantum Hamiltonian H is derived by similarity transforming the matrix L describing the BD equation in terms of the square root of the stationary (reversible) distribution. The quantum and classical systems share the entire eigenvalues and the eigenvectors are related one to one. When the birth rate B(x) and the death rate D(x) are chosen to be the coefficients of the difference equation governing the orthogonal polynomials of Askey scheme, the quantum system is exactly solvable. The eigenvectors are the orthogonal polynomials themselves and the eigenvalues are given analytically. Many examples are periodic since their eigenvalues are all integers, or all integers for integer parameters. The situation is very similar to the exactly solvable one dimensional quantum mechanical systems. These exactly solvable Markov chains contain many adjustable free parameters which could be helpful for various simulation purposes.