The extension of classical dynamics for unstable Hamiltonian systems

Abstract

Classical dynamics can be formulated in terms of trajectories or in terms of statistical ensembles whose time evolution is described by the Liouville equation. It is shown that for the class of large nonintegrable Poincaré systems (LPS), the two descriptions are not equivalent. Practically all dynamical systems studied in statistical mechanics belong to this class. The basic step is the extension of the Liouville operator L H outside the Hilbert space to functions singular in their Fourier transformation. This function space plays an important role in statistical mechanics as functions of the Hamiltonian, and therefore equilibrium distribution functions belong to this class. Physically, these functions correspond to situations characterized by “persistent interactions” as they are realized in macroscopic physics. Persistent interactions are introduced in contrast to “transient interactions” studied in quantum mechanics by the S-matrix approach (asymptotically free in and out states). The eigenvalue problem for the Liouville operator L H is solved in this generalized function space for LPS. We obtain a complex, irreducible spectral representation. Complex means that the eigenvalues are complex numbers, whose imaginary parts refer to the various irreversible processes such as relaxation times, diffusion.… Irreducible means that these representations cannot be implemented by trajectory theory. As the result, the dynamical group of evolution splits into two semigroups. Moreover, the laws of classical dynamics take a new form as they have to be formulated on the statistical level. They express “possibilities” and no more “certitudes”. Two examples of typical classical systems, i.e., interacting particles and anharmonic lattices are studied.