 1525 Jani Lukkarinen, Matteo Marcozzi
 Wick polynomials and timeevolution of cumulants
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Mar 19, 15

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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 nonGaussian case.
We discuss how an expansion in terms of the Wick polynomials can be used for
derivation of a hierarchy of equations for the timeevolution of cumulants.
These methods are then applied to simplify the formal derivation of
the BoltzmannPeierls 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 BoltzmannPeierls equation
by separating the analysis of the solutions to the
BoltzmannPeierls equation from the analysis of the corrections. This latter
scheme is general and not tied to the DNLS evolution equations.
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