Finite lattice kinetic equations for bosons, fermions, and discrete NLS

TL;DR

This paper develops finite lattice kinetic equations for bosons, fermions, and discrete NLS, providing an approximate evolution of covariance functions by truncating cumulant hierarchies, with results on well-posedness, decay, and robustness.

Contribution

It introduces a unified approach to derive and analyze finite lattice kinetic equations for multiple quantum models, including error estimates and robustness results.

Findings

Solutions decay over time independently of coupling strength

The equations are well-posed up to finite kinetic times on large lattices

Solutions are insensitive to the approximation of the energy conservation delta function

Abstract

We introduce and study finite lattice kinetic equations for bosons, fermions, and discrete NLS. For each model this closed evolution equation provides an approximate description for the evolution of the appropriate covariance function in the system. It is obtained by truncating the cumulant hierarchy and dropping the higher order cumulants in the usual manner. To have such a reference solution should simplify controlling the full hierarchy and thus allow estimating the error from the truncation. The harmonic part is given by nearest neighbour hopping, with arbitrary symmetric interaction potential of coupling strength $\lambda>0$. We consider the well-posedness of the resulting evolution equation up to finite kinetic times on a finite but large enough lattice. We obtain decay of the solutions and upper bounds that are independent of $\lambda$ and depend on the lattice size only via some Sobolev type norms of the interaction potential and initial data. We prove that the solutions are not sensitive to how the energy conservation delta function is approximated.