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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 Schödinger 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.

Published by AIP Publishing.

Received 09 December 2015
Accepted 26 July 2016
Published online 11 August 2016

Acknowledgments:
We thank Giovanni Gallavotti, Dario Gasbarra, Antti Kupiainen, Peng Mei, Alessia Nota, Sergio Simonella, and Herbert Spohn for useful discussions on the topic. The research of J. Lukkarinen and M. Marcozzi has been supported by the Academy of Finland via the Centre of Excellence in Analysis and Dynamics Research (Project No. 271983) and from an Academy Project (Project No. 258302).

Article outline:

I. INTRODUCTION
II. SETUP AND NOTATIONS
III. COMBINATORIAL DEFINITION AND PROPERTIES OF THE WICK POLYNOMIALS
A. Basic properties of the Wick polynomials
IV. CUMULANTS AND WICK POLYNOMIALS AS DYNAMICAL VARIABLES
V. KINETIC THEORY OF THE DISCRETE NLS EQUATION REVISITED
A. Translation invariant initial measures
1. Heuristic derivation of the Boltzmann-Peierls equation
2. Decay of field time-correlations
VI. DISCUSSION ABOUT FURTHER APPLICATIONS
A. Limitations of the direct renormalization procedure: Inhomogeneous DNLS
B. Kinetic theory beyond kinetic time scales?

/content/aip/journal/jmp/57/8/10.1063/1.4960556

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J. Lukkarinen,

M. Marcozzi, and

A. Nota, “

Summability of joint cumulants of nonindependent lattice fields,” e-print

arXiv:1601.08163 (

2016).

http://aip.metastore.ingenta.com/content/aip/journal/jmp/57/8/10.1063/1.4960556

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2016-08-11

2016-10-22

### 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 Schödinger 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.

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