Fluctuations of Quantum Lattice Systems under Boltzmann Evolution

TL;DR

This paper investigates the behavior of quantum lattice systems under Boltzmann evolution, revealing how initial state properties influence fluctuation dynamics and the emergence of irreversibility.

Contribution

It demonstrates that initial states with space clustering enable fluctuation algebra construction, while evolved states restrict fluctuations to local scales, clarifying fluctuation behavior under Boltzmann evolution.

Findings

Initial states exhibit space clustering necessary for fluctuation algebra.

Evolved states permit only local fluctuations, not mesoscopic.

Irreversible dynamics emerge under specific initial conditions.

Abstract

For quantum lattice systems a Boltzmann-type evolution arrises according to results of Hugenholtz in the limit of N-scaled time evolution together with an interaction scaled as N^-1/2. According ti Illner-Neunzert this passage to an irreversible dynamics can only happen when the initial state satisfies stronger assumptions than the evolved state. We show that the initial state satisfies space clustering as it is necessary for the construction of a fluctuation algebra whereas the evolved state allows fluctuations only in an appropriate local scaling but not on a mesoscopic level.