Squeezing stationary distributions of stochastic chemical reaction systems

TL;DR

This paper introduces a novel analytical method for computing stationary distributions of stochastic chemical reaction systems by leveraging an analogy with quantum squeezed states, extending the class of systems with known distributions.

Contribution

It develops a new approach using second quantization and squeezed states to find stationary distributions for reaction networks lacking complex-balanced steady states.

Findings

Analytic expressions for stationary distributions of transformed reaction networks.

Extension of product-form Poisson distributions to squeezed states.

New insights into non-complex-balanced reaction systems.

Abstract

Stochastic modeling of chemical reaction systems based on master equations has been an indispensable tool in physical sciences. In the long-time limit, the properties of these systems are characterized by stationary distributions of chemical master equations. In this paper, we describe a novel method for computing stationary distributions analytically, based on a parallel formalism between stochastic chemical reaction systems and second quantization. Anderson, Craciun, and Kurtz showed that, when the rate equation for a reaction network admits a complex-balanced steady-state solution, the corresponding stochastic reaction system has a stationary distribution of a product form of Poisson distributions. In a formulation of stochastic reaction systems using the language of second quantization initiated by Doi, product-form Poisson distributions correspond to coherent states. Pursuing this analogy further, we study the counterpart of squeezed states in stochastic reaction systems. Under the action of a squeeze operator, the time-evolution operator of the chemical master equation is transformed, and the resulting system describes a different reaction network, which does not admit a complex-balanced steady state. A squeezed coherent state gives the stationary distribution of the transformed network, for which analytic expression is obtained.