THE EXTENSION OF CLASSICAL DYNAMICS FOR UNSTABLE HAMILTONIAN-SYSTEMS

Classical dynamics can be formulated in terms of trajectories or in te rms of statistical ensembles whose time evolution is described by the Liouville equation. It is shown that for the class of large nonintegra ble Poincare systems (LPS), the two descriptions are not equivalent. P ractically all dynamical systems studied in statistical mechanics belo ng to this class. The basic step is the extension of the Liouville ope rator L-H outside the Hilbert space to functions singular in their Fou rier transformation. This function space plays an important role in st atistical mechanics as functions of the Hamiltonian, and therefore equ ilibrium distribution functions belong to this class. Physically, thes e functions correspond to situations characterized by ''persistent int eractions'' as they are realized in macroscopic physics. Persistent in teractions are introduced in contrast to ''transient interactions'' st udied in quantum mechanics by the S-matrix approach (asymptotically fr ee in and out states). The eigenvalue problem for the Liouville operat or L-H is solved in this generalized function space for LPS. We obtain a complex, irreducible spectral representation. Complex means that th e 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 sp lits into two semigroups. Moreover, the laws of classical dynamics tak e a new form as they have to be formulated on the statistical level. T hey express ''possibilities'' and no more ''certitudes''. Two examples of typical classical systems, i.e., interacting particles and anharmo nic lattices are studied.