A Graphical representation of the grand canonical partition function

TL;DR

This paper develops a graphical framework for representing solutions to a stochastic PDE on a lattice, including the grand canonical partition function, using rooted trees and Borel summability, with special cases analyzed for Lévy noise.

Contribution

It introduces a novel graphical representation of the grand canonical partition function and solutions to a stochastic PDE, including the use of formal power series and Borel summability techniques.

Findings

Graphical representation of the solution and its logarithm.

Application of Borel sum for summability of formal solutions.

Simplified solution representations for Lévy noise cases.

Abstract

In this paper we consider a stochastic partial differential equation defined on a Lattice $L_\delta$ with coefficients of non-linearity with degree $p$. An analytic solution in the sense of formal power series is given. The obtained series can be re-expressed in terms of rooted trees with two types of leaves. Under the use of the so-called Cole-Hopf transformation and for the particular case $p=2$, one thus get the generalized Burger equation. A graphical representation of the solution and its logarithm is done in this paper. A discussion of the summability of the previous formal solutions is done in this paper using Borel sum. A graphical calculus of the correlation function is given. The special case when the noise is of L\'evy type gives a simplified representations of the solution of the generalized Burger equation. From the previous results we recall a graphical representation of the grand canonical partition function.