Another Return of 'Return to Equilibrium'

TL;DR

This paper proves the return to equilibrium property for certain quantum models involving atoms or spins interacting with radiation or heat baths, with bounds on interaction strength that are independent of temperature in some cases.

Contribution

It establishes return to equilibrium for quantum models with realistic infrared behavior, providing temperature-independent bounds for atom models and logarithmic bounds for spin-boson models.

Findings

Return to equilibrium holds for models with weak interactions.

Interaction strength bounds are independent of temperature for atom models.

For spin-boson models, bounds tend to zero as temperature approaches zero.

Abstract

The property of ``{\it return to equilibrium}'' is established for a class of quantum-mechanical models describing interactions of a (toy) atom with black-body radiation, or of a spin with a heat bath of scalar bosons, under the assumption that the interaction strength is {\it sufficiently weak}. For models describing the first class of systems, our upper bound on the interaction strength is {\it independent} of the temperature $T$, (with $0<T\leq T_0<\infty$), while, for the spin-boson model, it tends to zero logarithmically, as $T\to 0$. Our result holds for interaction form factors with physically realistic infrared behaviour. Three key ingredients of our analysis are: a suitable concrete form of the Araki-Woods representation of the radiation field, Mourre's positive commutator method combined with a recent virial theorem, and a norm bound on the difference between the equilibrium states of the interacting and the non-interacting system (which, for the system of an atom coupled to black-body radiation, is valid for {\it all} temperatures $T\geq 0$, assuming only that the interaction strength is sufficiently weak).