Wick polynomials and time-evolution of cumulants

TL;DR

This paper introduces a combinatorial approach to defining Wick polynomials for random variables, enabling a hierarchy of equations for cumulant time-evolution, and applies these methods to derive and simplify the Boltzmann-Peierls equation in nonlinear Schrödinger dynamics.

Contribution

It presents a novel combinatorial definition of Wick polynomials that simplifies cumulant expansions and their application to kinetic equations in nonlinear wave systems.

Findings

Wick polynomials remove internal contractions in cumulant expansions.

Hierarchy of equations for cumulant time-evolution is derived.

Simplified derivation of the Boltzmann-Peierls equation for DNLS.

Abstract

We show how Wick polynomials of random variables can be defined combinatorially as the unique choice which removes all "internal contractions" from the related cumulant expansions, also in a non-Gaussian case. We discuss how an expansion in terms of the Wick polynomials can be used for derivation of a hierarchy of equations for the time-evolution of cumulants. These methods are then applied to simplify the formal derivation of the Boltzmann-Peierls equation in the kinetic scaling limit of the discrete nonlinear Schr\"{o}dinger equation (DNLS) with suitable random initial data. We also present a reformulation of the standard perturbation expansion using cumulants which could simplify the problem of a rigorous derivation of the Boltzmann-Peierls equation by separating the analysis of the solutions to the Boltzmann-Peierls equation from the analysis of the corrections. This latter scheme is general and not tied to the DNLS evolution equations.