Squeezing stationary distributions of stochastic chemical reaction systems

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.