Gibbs Ensembles, Thermodynamic Limit, KMS-Condition

Abstract

In chapter IV we focused attention on states which, with respect to local observations, differ from the vacuum essentially only in some finite space region at a given time. This is an appropriate idealization for elementary particle physics but not for the study of properties of bulk matter, the “many body problem” (many ∼ 10²⁴). There the simplification leading to the deduction of laws of thermodynamics and hydrodynamics from statistical physics is achieved by the idealization that matter fills all space with finite density and, instead of the vacuum, we have as the simplest states of interest the thermodynamic equilibrium states. To avoid misinterpretation: of course the realistic material systems in which we are interested have finite extension and the standard approach in statistical mechanics starts with a system of N particles enclosed in a box of volume 𝒱 with total energy E. But in order to give concepts like “temperature“, “phase transition” an unambiguous meaning the thermodynamic limit N → ∞, 𝒱 → ∞, E → ∞ with N/𝒱 and E/𝒱 finite must be performed (or is implicitly understood). By basing the theory on the algebra 𝔄 of local observables with its net structure we have the advantage that the box is dispensable. Equilibrium states in the thermodynamic limit are good states over 𝔄 and can be directly characterized. In the terminology of thermodynamics the elements of 𝔄 are “intensive quantities”.