Statistical mechanics applied to quantum mechanical systems has become one of the most important streams of quantum mechanics. This encyclopedia is intended to provide a ûavor of the physical ideas underlying some of the concepts of Quantum Statistical Mechanics that will be studied in the schools devoted to Quantum Systems. It provides a contemporary understanding of irreversibility, particularly in quantum systems. The encyclopedia focuses on the foundations of non-equilibrium statistical mechanics with emphasis on quantum mechanics and contains alternative views of quantum statistical mechanics, and topics of current interest for advanced graduate level study and research. Quantum entanglements are treated in application to quantum computation and the difficulties arising from de-coherence. The authors make these subjects accessible through carefully worked examples illustrating the technical aspects of the subject, and intuitive explanations of what is going on behind the mathematics. A unique and useful resource, this book is a valuable tool for anyone seeking to apply the principles of quantum mechanics to the development of practical technical solutions.