Quantum and classical results are often presented as being dependent upon separate postulates as if the two are distinct and unrelated, and there is little attempt to show how the quantum implies the classical. The transformation to classical phase space gives researchers access to a range of algorithms derived from classical statistical mechanics that promise results on much more favourable numerical terms. Quantum Statistical Mechanics in Classical Phase Space offers not just a new computational approach to condensed matter systems, but also a unique conceptual framework for understanding the quantum world and collective molecular behaviour. A formally exact transformation, this revolutionary approach goes beyond the quantum perturbation of classical condensed matter to applications that lie deep in the quantum regime. It offers scalable computational algorithms and tractable approximations tailored to specific systems. Concrete examples serve to validate the general approach and demonstrate new insights. For example, the computer simulations and analysis of the λ-transition in liquid helium provide a new molecular-level explanation of Bose-Einstein condensation and a quantitative theory for superfluid flow. The intriguing classical phase space formulation in this book offers students and researchers a range of new computational algorithms and analytic approaches. It offers not just an efficient computational approach to quantum condensed matter systems, but also an exciting perspective on how the classical world that we observe emerges from the quantum mechanics that govern the behaviour of atoms and molecules. The applications, examples, and physical insights foreshadow new discoveries in quantum condensed matter systems.
Key Features
- A computationally efficient approach to quantum condensed-matter many-particle systems based on a formally exact transformation of quantum statistical mechanics to classical phase space.
- Shows how the observed classical world emerges from the underlying quantum mechanics of atoms and molecules.
- Describes successive quantum corrections to classical statistical mechanics.
- Derives a number of expansions that account for the non-commutativity of quantum position and momentum operators when transformed to classical phase space.
- Tests various computer simulation algorithms against benchmark results for model condensed matter systems.
Author(s): Phil Attard
Publisher: IOP Publishing
Year: 2021
Language: English
Pages: 348
City: Bristol
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
