  Within the general formalism of quantum theory irreversibility and the arrow
of time in the evolution of various physical systems are studied. Irreversible
behavior often manifests itself in the guise of entropy production. This
motivates me to begin this paper with a brief review of quantum-mechanical
entropy, a subject that Elliott Lieb has made outstanding contributions to,
followed by an enumeration of examples of irreversible behavior and of an arrow
of time analyzed in later sections. Subsequently, a derivation of the laws of
thermodynamics from (quantum) statistical mechanics, and, in particular, of the
Second Law of thermodynamics, in the forms given to it by Clausius and Carnot,
is presented. In a third part, results on diffusive (Brownian) motion of a
quantum particle interacting with a quasi-free quantum-mechanical heat bath are
reviewed. This is followed by an outline of a theory of friction by emission of
Cherenkov radiation of sound waves in a system consisting of a particle moving
through a Bose- Einstein condensate and interacting with it. In what may be the
most important section of this paper, the fundamental arrow of time inherent in
Quantum Mechanics is discussed.
