On the Brownian gas: a field theory with a Poissonian ground state

TL;DR

This paper develops a field theory for non-interacting Brownian particles, revealing a Poissonian ground state and emphasizing the importance of particle-based constraints over coarse-graining, with exact correlation functions computed.

Contribution

It introduces a field theory with a particle-originated interaction term, exactly computes all n-point functions, and confirms the fluctuation-dissipation theorem in this context.

Findings

Exact n-point density correlation functions computed.

Poissonian nature of the ground state established.

Time-reversal symmetry enforces fluctuation-dissipation relation.

Abstract

As a first step towards a successful field theory of Brownian particles in interaction, we study exactly the non-interacting case, its combinatorics and its non-linear time-reversal symmetry. Even though the particles do not interact, the field theory contains an interaction term: the vertex is the hallmark of the original particle nature of the gas and it enforces the constraint of a strictly positive density field, as opposed to a Gaussian free field. We compute exactly all the n-point density correlation functions, determine non-perturbatively the Poissonian nature of the ground state and emphasize the futility of any coarse-graining assumption for the derivation of the field theory. We finally verify explicitly, on the n-point functions, the fluctuation-dissipation theorem implied by the time-reversal symmetry of the action.