Symmetric and antisymmetric cumulant hierarchies and Wick polynomials of operators
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.
Related
- Symmetric Grassmann Formulas: Monotone Dimension-Defect Hierarchies
- Cyclotomy of symmetric polynomials
- Degree-one Pieri rule for non-symmetric Jack polynomials
- Power sums of critical points of gap polynomials for symmetric and pseudo-symmetric numerical semigroups
- Matrix orthogonal polynomials related to diffference operators for L-orthogonal polynomials
BibTeX
@misc{lukkarinen2026symmetric,
title = {{Symmetric and antisymmetric cumulant hierarchies and Wick polynomials of operators}},
author = {Jani Lukkarinen and Sakari Pirnes},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.31374},
eprint = {2610.31374},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.31374},
}
RIS
TY - UNPB
TI - Symmetric and antisymmetric cumulant hierarchies and Wick polynomials of operators
AU - Jani Lukkarinen
AU - Sakari Pirnes
PY - 2026
DA - 2026/10/22
PB - arXiv
DO - 10.48550/arxiv.2610.31374
UR - https://arxiv.org/abs/2610.31374
AB - 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.
ER -