Eikonal solutions for moment hierarchies of Chemical Reaction Networks in the limits of large particle number

TL;DR

This paper develops a novel eikonal approach for analyzing moment hierarchies in chemical reaction networks, linking Hamiltonian dynamics with factorial moments in large-system limits.

Contribution

It introduces a direct method to derive eikonals from moment hierarchies, connecting fixed points to boundary conditions and clarifying the role of canonical transformations.

Findings

Steady-state conditions reduce to mappings between eikonals and factorial moments.

Boundary values are anchored in interior fixed points of the Hamiltonian system.

Clarifies the relation between coherent-state and number fields in Doi-Peliti theory.

Abstract

Trajectory-based methods are well-developed to approximate steady-state probability distributions for stochastic processes in large-system limits. The trajectories are solutions to equations of motion of Hamiltonian dynamical systems, and are known as eikonals. They also express the leading flow lines along which probability currents balance. The existing eikonal methods for discrete-state processes including chemical reaction networks are based on the Liouville operator that evolves generating functions of the underlying probability distribution. We have previously derived a representation for the generators of such processes that acts directly in the hierarchy of moments of the distribution, rather than on the distribution itself or on its generating function. We show here how in the large-system limit the steady-state condition for that generator reduces to a mapping from eikonals to the ratios of neighboring factorial moments, as a function of the order $k$ of these moments. The construction shows that the boundary values for the moment hierarchy, and thus its whole solution, are anchored in the interior fixed points of the Hamiltonian system, a result familiar from Freidlin-Wenztell theory. The direct derivation of eikonals from the moment representation further illustrates the relation between coherent-state and number fields in Doi-Peliti theory, clarifying the role of canonical transformations in that theory.