Dynamics of Systems with Large Number of Degrees of Freedom and Generalized Transformation Theory

A generalized transformation theory which leads to a non-Hamiltonian description of dynamics is introduced. The transformation is such that all averages of observables remain invariant. However, the time evolution of the density matrix can no longer be expressed in terms of a commutator with the Hamiltonian. Therefore such transformations are not canonical in the usual sense. An explicit „two components” representation of the equations of motion is given which has the following properties: ( a ) each of the components satisfies a separate equation of motion, and ( b ) one component satisfies a kinetic equation of a generalized Boltzmann type. We obtain, therefore, the most remarkable result that the relation between dynamics and statistical mechanics (or thermodynamics) takes a specially transparent and simple form: thermodynamics appears in a precise sense as the random phase approximation of dynamics. Other problems such as the meaning of diagonalization of the Hamiltonian and definition of excitations will be treated in a forthcoming paper.