Quantum Statistical Mechanics: Equilibrium and non-equilibrium theory from first principles

Title
Copyright
Contents
Preface
Author biography
1 Probability operator and statistical averages
	1.1 Expectation, density operator and averages
		1.1.1 Expectation value
		1.1.2 Density operator
		1.1.3 Statistical average
	1.2 Uniform weight density of wave space
		1.2.1 Probability flux and trajectory uniformity
		1.2.2 Time average on the hypersurface
	1.3 Canonical equilibrium system
		1.3.1 Entropy of energy states
		1.3.2 Wave function entanglement
		1.3.3 Expectation values and wave function collapse
		1.3.4 Statistical average and probability operator
	1.4 Environmental selection
	1.5 Wave function collapse and the classical universe
		1.5.1 Mechanism of statistical collapse
		1.5.2 Probabilistic nature of the wave function
		1.5.3 Quantum interference is uniquely non-classical
		1.5.4 Classical phase space and Hamilton’s equations
	References
2 Examples and applications: equilibrium
	2.1 Bosons, fermions and wave function symmetry
	2.2 Ideal quantum gas
		2.2.1 Leading classical term
		2.2.2 First quantum correction
		2.2.3 Quantum correction as a potential of mean force
	2.3 State occupancy by ideal particles
		2.3.1 Bosons
		2.3.2 Fermions
		2.3.3 Classical particles
	2.4 Thermodynamics and statistical mechanics of ideal particles
	2.5 Classical ideal gas
	2.6 Ideal Bose gas
		2.6.1 Black body radiation
		2.6.2 Heat capacity of solids
	2.7 Ideal Fermi gas
	2.8 Simple harmonic oscillator
	References
3 Probability in quantum systems
	3.1 Formulation of probability
		3.1.1 States
		3.1.2 Weight
		3.1.3 Probability
		3.1.4 Entropy
		3.1.5 Averages
	3.2 Transitions
		3.2.1 Transition weight operator
		3.2.2 Time correlation function
		3.2.3 First reduction condition
		3.2.4 Second reduction condition
		3.2.5 Parity and reversibility
	3.3 Non-equilibrium probability
	References
4 Time propagator for an open quantum system
	4.1 Adiabatic time propagator
		4.1.1 Constant Hamiltonian operator
		4.1.2 Time-varying Hamiltonian operator
		4.1.3 Adiabatic Heisenberg picture
		4.1.4 Adiabatic Liouville operator
		4.1.5 Adiabatic Pauli master equation
	4.2 Stochastic time propagator
		4.2.1 General features
		4.2.2 Variance of the stochastic operator
		4.2.3 A quantum fluctuation-dissipation theorem
		4.2.4 Properties of the stochastic time propagator
		4.2.5 Heisenberg picture
		4.2.6 Stochastic Liouville super-operator
		4.2.7 Kubo cumulant expansion
		4.2.8 Liouville equation
	4.3 Kraus representation and Lindblad equation
		4.3.1 Kraus operators for a sub-system and reservoir
		4.3.2 Lindblad equation
	4.4 Caldeira–Leggett model
		4.4.1 Fundamentals
		4.4.2 Generalised Langevin equation
	4.5 Time correlation function
		4.5.1 Equilibrium time correlation function
		4.5.2 Unitary and evolution conditions
	4.6 Transition probability
		4.6.1 Unconditional transition probability operator
		4.6.2 Conditional transition probability operator
	4.7 Microscopic reversibility
	References
5 Evolution of the canonical equilibrium system
	5.1 Transitions between entropy states
		5.1.1 Random phase for transitions
		5.1.2 Mean and stochastic parts of the state transition
		5.1.3 Form of the time propagator. 1
		5.1.4 Time correlation function
		5.1.5 Stochastic and dissipative operators
	5.2 Second entropy for transitions
		5.2.1 Fluctuation form
		5.2.2 Small time expansion for most likely transition
		5.2.3 Stochastic, dissipative equation of motion
		5.2.4 Stationarity of the state probability
		5.2.5 Form of the time propagator. 2
	5.3 Trajectory in wave space
		5.3.1 Stochastic dissipative Schrödinger equation
		5.3.2 Average on the trajectory
	5.4 Time derivative of entropy operator
	References
6 Probability operator for non-equilibrium systems
	6.1 Entropy operator for a trajectory
		6.1.1 Wave space formulation
		6.1.2 Propagator formulation
	6.2 Point entropy operator
		6.2.1 Reduction of the trajectory entropy
		6.2.2 Form and interpretation of the point entropy
		6.2.3 Propagator formulation
	6.3 Non-equilibrium probability operator
		6.3.1 Operator and average
		6.3.2 Time derivative
	6.4 Approximations for the dynamic entropy operator
		6.4.1 Odd projection of the dynamic entropy operator
		6.4.2 Adiabatic approximation for steady state thermodynamic systems
	6.5 Perturbation of the non-equilibrium probability operator
	6.6 Linear response theory
		6.6.1 Unitary transformations
		6.6.2 Interaction picture
		6.6.3 Probability operator
		6.6.4 Susceptibility
		6.6.5 Dissipation
	References
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