Ruprecht Karls Universität Heidelberg

Mathematical Physics

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Return to Equilibrium (RTE)

Consider a small system consisting of a few degrees of freedom in contact with a large system (infinite number of degrees of freedom), i.e. a heat bath. If the system is sufficiently ergodic, the small system is expected to thermalize under influence of the heat bath and its initial condition should get forgotten. At present, our mathematical abilities to prove such irreversible behaviour are very modest.This should be contrasted with static properties of equilibrium states, which can be successfully studied within the Gibbs formalism of statistical mechanics.

A simple model where these questions can be phrased is the Spin-Boson model: the spin is the small system and the boson field (possibly at positive temperature) is the heat bath. In the Spin-Boson model, the irreversible properties of the spin have been studied successfully by master equations such as the Lindblad equation, or, even simpler, the Pauli master equation. However, master equations are derived in a limit where the coupling strength between the spin and the bosons vanishes and as such they provide merely a suggestion of what happens at finite, but small, coupling strength.
Our recent result ( roughly states that, whenever the Markov approximation is well-defined and it exhibits RTE, the system at small coupling strength exhibits RTE as well. The study of RTE at arbitrary coupling remains a beautiful open problem in mathematical physics.

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