Quantum statistical mechanics : equilibrium and non equilibrium theory from first principles 2 Examples and applications : equilibrium 2.1 Bosons fermions and wave function symmetry 2.2 Ideal quantum gas 2.3 State occupancy by ideal particles 2.4 Thermodynamics and statistical mechanics of ideal particles 2.5 Classical ideal gas 2.6 Ideal Bose gas 2.7 Ideal Fermi gas 2.8 Simple harmonic oscillator 5 Evolution of the canonical equilibrium system 5.1 Transitions between entropy states 5.2 Second entropy for transitions 5.3 Trajectory in wave space 5.4 Time derivative of entropy operator Appendices. A. Probability densities and the statistical average B. Stochastic state transitions for a non equilibrium system C. Entropy eigenfunctions state transitions and phase space 3 Probability in quantum systems 3.1 Formulation of probability 3.2 Transitions 3.3 Non equilibrium probability 4 Time propagator for an open quantum system 4.1 Adiabatic time propagator 4.2 Stochastic time propagator 4.3 Kraus representation and Lindblad equation 4.4 Caldeira Leggett model 4.5 Time correlation function 4.6 Transition probability 4.7 Microscopic reversibility Preface Author biography 1 Probability operator and statistical averages 1.1 Expectation density operator and averages 1.2 Uniform weight density of wave space 1.3 Canonical equilibrium system 1.4 Environmental selection 1.5 Wave function collapse and the classical universe 6 Probability operator for non equilibrium systems 6.1 Entropy operator for a trajectory 6.2 Point entropy operator 6.3 Non equilibrium probability operator 6.4 Approximations for the dynamic entropy operator 6.5 Perturbation of the non equilibrium probability operator 6.6 Linear response theory 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