Energy Conservation, Counting Statistics, and Return to Equilibrium

Abstract

We study a microscopic Hamiltonian model describing an N-level quantum system $${\mathcal{S}}$$ coupled to an infinitely extended thermal reservoir $${\mathcal{R}}$$ . Initially, the system $${\mathcal{S}}$$ is in an arbitrary state while the reservoir is in thermal equilibrium at inverse temperature $${\beta}$$ . Assuming that the coupled system $${\mathcal{S}+\mathcal{R}}$$ is mixing with respect to the joint thermal equilibrium state, we study the Full Counting Statistics (FCS) of the energy transfers $${\mathcal{S} \to \mathcal{R}}$$ and $${\mathcal{R} \to \mathcal{S}}$$ in the process of return to equilibrium. The first FCS describes the increase of the energy of the system $${\mathcal{S}}$$ . It is an atomic probability measure, denoted $${\mathbb{P}_{\mathcal{S},\lambda,t}}$$ , concentrated on the set of energy differences $${{\rm sp}(H_{\mathcal{S}})-{\rm sp}(H_{\mathcal{S}})}$$ ( $${H_{\mathcal{S}}}$$ is the Hamiltonian of $${\mathcal{S}}$$ , t is the length of the time interval during which the measurement of the energy transfer is performed, and $${\lambda}$$ is the strength of the interaction between $${\mathcal{S}}$$ and $${\mathcal{R}}$$ ). The second FCS, $${\mathbb{P}_{\mathcal{R},\lambda,t}}$$ , describes the decrease of the energy of the reservoir $${\mathcal{R}}$$ and is typically a continuous probability measure whose support is the whole real line. We study the large time limit $${t \rightarrow \infty}$$ of these two measures followed by the weak coupling limit $${\lambda \rightarrow 0}$$ and prove that the limiting measures coincide. This result strengthens the first law of thermodynamics for open quantum systems. The proofs are based on modular theory of operator algebras and on a representation of $${\mathbb{P}_{\mathcal{R},\lambda,t}}$$ by quantum transfer operators.