This book establishes the foundations of non-equilibrium quantum statistical mechanics in order to support students and academics in developing and building their understanding. The formal theory is derived from first principles by mathematical analysis, with concrete physical interpretations and worked examples throughout. It explains the central role of entropy; its relation to the probability operator and the generalisation to transitions, as well as providing first principles derivation of the von Neumann trace form, the Maxwell-Boltzmann form and the Schrödinger equation.

1 Probability Operator and Statistical Averages
2 Examples and Applications: Equilibrium
3 Probability in Quantum Systems
4 Time Propagator for an Open Quantum System
5 Evolution of the Canonical Equilibrium System
6 Probability Operator for Non-Equilibrium Systems
A Probability Densities and the Statistical Average
B Stochastic State Transitions for a Non-Equilibrium System
C Entropy Eigenfunctions, State Transitions, and Phase Space

This book establishes the foundations of non-equilibrium quantum statistical mechanics in order to support students and academics in developing and building their understanding. The formal theory is derived from first principles by mathematical analysis, with concrete physical interpretations and worked examples throughout. It explains the central role of entropy; its relation to the probability operator and the generalisation to transitions, as well as providing first principles derivation of the von Neumann trace form, the Maxwell-Boltzmann form and the Schrödinger equation.