Quantum Statistical Mechanics in Classical Phase Space

PRELIMS.pdf
	Author biography
		Phil Attard
CH001.pdf
	Chapter 1 Introduction
		1.1 Why phase space?
		1.2 Why not direct quantum methods?
		1.3 Advantages and challenges of phase space
			1.3.1 Quantization
			1.3.2 Superposition
			1.3.3 Non-commutation
			1.3.4 Symmetrization
			1.3.5 Non-localization
		1.4 Old applications, new perspectives
			1.4.1 λ-Transition and superfluidity
		References
CH002.pdf
	Chapter 2 Wave packet formulation
		2.1 Introduction
		2.2 Wave packets as eigenfunctions in the classical limit
			2.2.1 Definition
			2.2.2 Eigenfunctions
		2.3 Wave packet symmetrization and overlap
			2.3.1 Pair transposition
			2.3.2 Symmetrization and occupancy of single-particle states
		2.4 Statistical averages in phase space
			2.4.1 Partition function
			2.4.2 Averages
			2.4.3 Phase space
			2.4.4 Symmetrization factor in the classical limit
		References
CH003.pdf
	Chapter 3 Symmetrization factor and permutation loop expansion
		3.1 Introduction
		3.2 Partition function
		3.3 Symmetrization and occupancy for multi-particle states
			3.3.1 Single-particle states
			3.3.2 Multi-particle states
		3.4 Symmetrization expansion of the partition function
			3.4.1 Localization of permuted states
			3.4.2 Loop expansion of the permutation series
			3.4.3 Exact expansion for single-particle states
			3.4.4 Approximate localization for multi-particle states
		References
CH004.pdf
	Chapter 4 Applications with single-particle states
		4.1 Ideal gas
			4.1.1 Single-particle states
			4.1.2 Classical phase space
		4.2 Independent harmonic oscillators
			4.2.1 General single-particle energy states
			4.2.2 Simple harmonic oscillators
			4.2.3 Conventional derivation and interpretation
		4.3 Occupancy of single-particle states
			4.3.1 Loop grand potential and derivatives
			4.3.2 Average occupation number
			4.3.3 Occupancy fluctuations
			4.3.4 Occupancy probability distribution
		4.4 Ideal fermions
			4.4.1 Loop grand potential, energy, number
			4.4.2 Heat capacity, free energy, and entropy
			4.4.3 Fermi energy
		References
CH005.pdf
	Chapter 5 The λ-transition and superfluidity in liquid helium
		5.1 Introduction
		5.2 Ideal gas approach to the λ-transition
			5.2.1 Loop forms of the grand potential and number
			5.2.2 Total grand potential and number
			5.2.3 Heat capacity
			5.2.4 Critique of the ideal gas model
		5.3 Ideal gas: exact enumeration
			5.3.1 Numerical results
		5.4 The λ-transition for interacting bosons
			5.4.1 Momentum integrals
			5.4.2 Monte Carlo algorithm
			5.4.3 Results for liquid helium
		5.5 Interactions on the far side
			5.5.1 Factorization and the heat capacity
			5.5.2 Permutation entropy
			5.5.3 Kinky loops
		5.6 Permutation loops, the λ-transition, and superfluidity
			5.6.1 The λ-transition
			5.6.2 Superfluidity
		References
CH006.pdf
	Chapter 6 Further applications
		6.1 Vibrational heat capacity of solids
		6.2 One-dimensional harmonic crystal
			6.2.1 Model
			6.2.2 Eigenvalues and eigenvectors
			6.2.3 Normal modes
			6.2.4 Quantum mechanics
			6.2.5 Numerical results
		6.3 Loop Markov superposition approximation
			6.3.1 Temperature derivatives
			6.3.2 Average bond length in a loop
			6.3.3 Tests
		6.4 Symmetrization for spin-position factorization
			6.4.1 Exact form
			6.4.2 Approximate form
			6.4.3 Comparison of exact and approximate forms
			6.4.4 Simple example, N = 2
		References
CH007.pdf
	Chapter 7 Phase space formalism for the partition function and averages
		7.1 Partition function in classical phase space
			7.1.1 Grand partition function
			7.1.2 Statistical averages
			7.1.3 Energy
		7.2 Loop expansion, grand potential and average energy
			7.2.1 Symmetrization loops
			7.2.2 Grand potential
			7.2.3 Energy
			7.2.4 Average number
			7.2.5 Factorization of averages
			7.2.6 Explicit comparison for the ideal gas
		7.3 Multi-particle density
			7.3.1 Average energy factorized
		7.4 Virial pressure
		References
CH008.pdf
	Chapter 8 High temperature expansions for the commutation function
		8.1 Preliminary definitions
			8.1.1 History
			8.1.2 Definition
			8.1.3 Classical limit
			8.1.4 Extensivity
		8.2 Expansion 1
			8.2.1 Derivation of expansion
			8.2.2 An error in Kirkwood
			8.2.3 Position configuration weight density
			8.2.4 Average kinetic energy
		8.3 Expansion 2
		8.4 Expansion 3
			8.4.1 Second order analysis
			8.4.2 Higher order analysis
		8.5 Fluctuation expansion
			8.5.1 Hamiltonian fluctuations as a phase function
			8.5.2 Exponentiated fluctuation series
			8.5.3 Explicit form for the fluctuations
			8.5.4 Recursion relation
			8.5.5 Advantages and disadvantages of the fluctuation expansion
		8.6 Numerical results
			8.6.1 Interacting Lennard–Jones particles
			8.6.2 Simple harmonic oscillator
		References
CH009.pdf
	Chapter 9 Nested commutator expansion for the commutation function
		9.1 Introduction
		9.2 Commutator factorization of exponentials
			9.2.1 Second order
			9.2.2 Third order
			9.2.3 Fourth order
		9.3 Maxwell–Boltzmann operator factorized
			9.3.1 Formal and algorithmic expressions
			9.3.2 Commutation function operators
		9.4 Temperature derivative of the commutation function operator
			9.4.1 Nested commutator form
			9.4.2 Expanded operator form
		9.5 Evaluation of the commutation function
			9.5.1 Expectation values
			9.5.2 Gradients of the singlet and pair potential
			9.5.3 Central pair potential
		9.6 Results for the one-dimensional harmonic crystal
			9.6.1 Expectation value of high temperature expansion operators
			9.6.2 Harmonic crystal potential and gradients
			9.6.3 Simulation results
		References
CH010.pdf
	Chapter 10 Local state expansion for the commutation function
		10.1 Effective local field and operator
			10.1.1 Recapitulation of phase space formulation
			10.1.2 Effective local field
			10.1.3 Expansion of the Maxwell–Boltzmann operator
			10.1.4 Singlet, first order commutation function
		10.2 Higher order local fields
			10.2.1 Singlet, second order
			10.2.2 Singlet, modified second order
			10.2.3 Singlet, third order
			10.2.4 Pair, first order
		10.3 Harmonic local field
			10.3.1 Singlet harmonic field approximation
			10.3.2 Cluster harmonic field approximation
		10.4 Gross–Pitaevskii mean field Schrödinger equation
		10.5 Numerical results in one-dimension
			10.5.1 Harmonic crystal
			10.5.2 Lennard–Jones fluid
		References
CH011.pdf
	Chapter 11 Many-body expansion for the commutation function
		11.1 Commutation function
			11.1.1 Background
			11.1.2 Many-body expansion
			11.1.3 Pair term
			11.1.4 Linear solution
			11.1.5 Non-linear solution
			11.1.6 Singlet plus pair potential
			11.1.7 Three-body term
			11.1.8 Effective Mayer f-functions
			11.1.9 Ursell clusters and Mayer f-functions
		11.2 Symmetrization function
		11.3 Generalized Mayer f-function
		11.4 Numerical analysis
			11.4.1 Two-body commutation function
			11.4.2 Core asymptote
			11.4.3 Three-body commutation function
			11.4.4 Results
		11.5 Ursell clusters, Lee–Yang theory, classical phase space
			11.5.1 Ursell cluster theory
			11.5.2 Identities for the Maxwell–Boltzmann operator
			11.5.3 Exponential series expansion
			11.5.4 Exponential expansion for the phase space weight
		References
CH012.pdf
	Chapter 12 Density matrix and partition function
		12.1 Introduction
		12.2 Quantum statistical average
			12.2.1 Expectation value
			12.2.2 Statistical average
		12.3 Uniform weight density of wave space
			12.3.1 Equal state probability hypothesis
			12.3.2 Trajectory uniformity
			12.3.3 Time average and hypersurface density
		12.4 Canonical equilibrium system
			12.4.1 Entropy of energy states
			12.4.2 Wave function entanglement
			12.4.3 Expectation values and wave function collapse
			12.4.4 Maxwell–Boltzmann probability operator
			12.4.5 Environmental selection
			12.4.6 Symmetrization
		References