Relaxation,amplification, and the KMS conditions

We formulate the dynamics of an oscillator, $\text{gS}$, harmonically coupled to an infinite quasifree bose system Σ, the initial states of Σ and Σ being mutually uncorrelated and the initial state of Σ being quasifree, primary, and stationary. It is shown that, subject to certain regularity conditions, the oscillator $\text{gS}$ will either relax towards a terminal state, whose value is independent of the initial state of the oscillator, or else will become amplified, according to whether the energy quantum for $\text{gS}$ is positive or negative. In the former case, the weak-coupling limit for the terminal state of $\text{gS}$ is a Gibbs state, corresponding to a temperature which takes the same value for all $\text{Σ-}\text{gS}$ couplings of the prescribed class, if and only if the initial state of Σ satisfies the KMS conditions for that temperature. We argue that this result provides an operational justification for identifying equilibrium conditions with KMS conditions, at least for the class of stationary states of Σ that we consider.