Coherent-state path integral versus coarse-grained effective stochastic equation of motion: From reaction diffusion to stochastic sandpiles

TL;DR

This paper compares the coherent-state path integral and a coarse-grained stochastic equation of motion for non-equilibrium processes, highlighting the difficulties of the former and proposing a practical alternative for stochastic sandpiles.

Contribution

It introduces a real-noise coarse-grained stochastic equation of motion as a practical alternative to the complex coherent-state path integral in non-equilibrium systems.

Findings

Coherent-state path integral contains counter-intuitive quartic vertices.

Imaginary noise in the stochastic equation causes convergence issues.

The coarse-grained stochastic equation with real noise is effective for stochastic sandpiles.

Abstract

We derive and study two different formalisms used for non-equilibrium processes: The coherent-state path integral, and an effective, coarse-grained stochastic equation of motion. We first study the coherent-state path integral and the corresponding field theory, using the annihilation process $A+A\to A$ as an example. The field theory contains counter-intuitive quartic vertices. We show how they can be interpreted in terms of a first-passage problem. Reformulating the coherent-state path integral as a stochastic equation of motion, the noise generically becomes imaginary. This renders it not only difficult to interpret, but leads to convergence problems at finite times. We then show how alternatively an effective coarse-grained stochastic equation of motion with real noise can be constructed. The procedure is similar in spirit to the derivation of the mean-field approximation for the Ising model, and the ensuing construction of its effective field theory. We finally apply our findings to stochastic Manna sandpiles. We show that the coherent-state path integral is inappropriate, or at least inconvenient. As an alternative, we derive and solve its mean-field approximation, which we then use to construct a coarse-grained stochastic equation of motion with real noise.