Wick polynomials and time-evolution of cumulants

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 Schrödinger 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.

References in corpus (4)

• Heat transport in low dimensions: introduction and phenomenology
• Energy Transport in Weakly Anharmonic Chains
• Kinetic theory of phonons in weakly anharmonic particle chains
• Summability of joint cumulants of nonindependent lattice fields

Cited by in corpus (7)

• Renormalization of Generalized KPZ equation
• A central limit theorem for the KPZ equation
• Hopf-algebraic deformations of products and Wick polynomials
• Kinetic theory of phonons in weakly anharmonic particle chains
• Thermalization of oscillator chains with onsite anharmonicity and comparison with kinetic theory
• Summability of joint cumulants of nonindependent lattice fields
• Random selection of factors preserves the correlation structure in a linear factor model to a high degree