# Large deviations and dynamical phase transitions in stochastic chemical networks

## TL;DR

This paper develops a large deviations framework for stochastic chemical networks, revealing dynamical phase transitions and providing a path integral and Hamiltonian formalism to analyze system trajectories and rare events.

## Contribution

It introduces a novel large deviations approach for chemical reaction networks, linking microscopic models to a Hamiltonian formalism and uncovering phase transitions.

## Key findings

- Chemical networks exhibit first-order dynamical phase transitions near zero bias.
- The Hamiltonian formalism identifies most likely system trajectories and phase behavior.
- Multistability and ergodicity breaking are analyzed in large-volume limits.

## Abstract

Chemical reaction networks offer a natural nonlinear generalisation of linear Markov jump processes on a finite state-space. In this paper, we analyse the dynamical large deviations of such models, starting from their microscopic version, the chemical master equation. By taking a large-volume limit, we show that those systems can be described by a path integral formalism over a Lagrangian functional of concentrations and chemical fluxes. This Lagrangian is dual to a Hamiltonian, whose trajectories correspond to the most likely evolution of the system given its boundary conditions. The same can be done for a system biased on time-averaged concentrations and currents, yielding a biased Hamiltonian whose trajectories are optimal paths conditioned on those observables. The appropriate boundary conditions turn out to be mixed, so that, in the long time limit, those trajectories converge to well-defined attractors. We are then able to identify the largest value that the Hamiltonian takes over those attractors with the scaled cumulant generating function of our observables, providing a non-linear equivalent to the well-known Donsker-Varadhan formula for jump processes. On that basis, we prove that chemical reaction networks that are deterministically multistable generically undergo first-order dynamical phase transitions in the vicinity of zero bias. We illustrate that fact through a simple bistable model called the Schl\"ogl model, as well as multistable and unstable generalisations of it, and we make a few surprising observations regarding the stability of deterministic fixed points, and the breaking of ergodicity in the large-volume limit.