  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.
