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

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.