Statistical Mechanics for States with Complex Eigenvalues and Quasi-stable Semiclassical Systems

TL;DR

This paper develops a statistical mechanics framework for unstable states with complex eigenvalues, revealing new entropy, observable time scales, and stable quasi-stable systems in higher dimensions, exemplified by parabolic barriers.

Contribution

It introduces a novel entropy for complex eigenstates and demonstrates the existence of stable and quasi-stable semiclassical systems with observable flows.

Findings

Existence of stable and quasi-stable systems in higher dimensions.

Introduction of a new entropy related to imaginary eigenvalues.

Construction of examples using parabolic potential barriers.

Abstract

Statistical mechanics for states with complex eigenvalues, which are described by Gel'fand triplet and represent unstable states like resonances, are discussed on the basis of principle of equal ${\it a priori}$ probability. A new entropy corresponding to the freedom for the imaginary eigenvalues appears in the theory. In equilibriums it induces a new physical observable which can be identified as a common time scale. It is remarkable that in spaces with more than 2 dimensions we find out existence of stable and quasi-stable systems, even though all constituents are unstable. In such systems all constituents are connected by stationary flows which are generally observable and then we can say that they are semiclassical systems. Examples for such semiclassical systems are constructed in parabolic potential barriers. The flexible structure of the systems is also pointed out.