Additivity of the entropy and definition of the temperature for quantum systems

Abstract

A closed quantum systemL is considered which is described by a microcanonical ensemble.L consists of two rather weakly interacting subsystemsL ₁,L ₂. In a rigorous way, the additivity of the entropy is proved by deriving an expression for the entropy density ofL in terms of the entropy densities ofL ₁ andL ₂. “Rigorous” implies that the thermodynamic limit is taken. In the second part, it is shown how a microcanonical state$\omega _\varepsilon (\Lambda ) = \mathop {\lim }\limits_{\Lambda \to \infty } \frac{{Tr\delta ^\Delta (H(\Lambda ) - E)A}}{{Tr\delta ^\Delta (H(\Lambda ) - E)}}$ of the composite system — provided this limit exists — gives rise to a “canonical” stateω ^β, when restricted toL ₁, providedL ₁ is very “small” as compared toL ₂;ω ^β is defined as a limit of Gibbs states. This yields a definition of the equilibrium temperature β⁻¹.