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arXiv · 2610.31374

Symmetric and antisymmetric cumulant hierarchies and Wick polynomials of operators

Jani Lukkarinen·Sakari Pirnes
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Abstract

There has been much progress in understanding mathematical foundations of kinetic theory in recent years, in particular, for the original model of a rarefied gas of classical particles with repulsive interactions, as well as for weakly nonlinear wave evolutions related to wave turbulence. These results are mainly based on refined analysis of evolution hierarchies of moments of the relevant random variables, or of related quantities such as of cumulants or of correlation errors. In this work, we extend two of such basic tools, cumulants and Wick polynomials, to a setting where the random variables no longer need to commute, with the goal of handling expectations of creation and annihilation operators on a Fock space, both in the bosonic and fermionic case. We call these constructions symmetric and antisymmetric cumulants and Wick polynomials, respectively. As an application, we derive a hierarchy of evolution equations satisfied by such cumulants, usually called truncated correlation functions in this context, for a gas of bosons or fermions hopping on a finite lattice and interacting via a two-body potential. With sufficient control over solutions to these hierarchies, one could control the magnitude of the error terms related to an approximation of the state of the quantum system by a finite lattice kinetic equation studied in our earlier work, closely connected to standard bosonic and fermionic Boltzmann--Nordheim equations.

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Jani Lukkarinen, Sakari Pirnes. 2026-10-22. Symmetric and antisymmetric cumulant hierarchies and Wick polynomials of operators. https://arxiv.org/abs/2610.31374

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