A concise, unified overview of the subject

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116 pages

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English

This is an academic textbook for a one-semester course in statistical physics at honours BSc level.

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Content

This is an academic textbook for a one-semester course in statistical physics at honours BSc level. It is in three parts and begins with a unified treatment of equilibrium systems, based on the concept of the statistical ensemble, in which the usual combinatorial calculation only has to be worked out once. In the second part, it deals with strongly interacting systems in terms of many-body theory, including the virial expansion and critical phenomena at the level of mean-field theory. The third, and last, part of the book is concerned with time-dependence; and, it begins with a classical treatment of the paradox posed by the `arrow of time'. This is the question of why macroscopic systems are irreversible when their constituent microscopic interactions are reversible in time. It then treats the derivation of transport equations, linear response theory, and quantum dynamics. Throughout the book, the emphasis is on a clear, concise exposition, with all steps being clearly explained.

- Introduction
- The isolated assembly
- Method of the most probable distribution
- Ensemble of assemblies: relationship between Gibbs and Boltzmann entropies

- Stationary ensembles
- Types of ensemble
- Variational method for the most probable distribution
- Canonical ensemble
- Compression of a perfect gas
- The Grand Canonical Ensemble (GCE)

- Examples of stationary ensembles
- Assembly of distinguishable particles
- Assembly of nonconserved, indistinguishable particles
- Conserved particles: general treatment for Bose-Einstein and Fermi-Dirac statistics
- The Classical Limit: Boltzmann Statistics

- The bedrock problem: strong interactions
- The interaction Hamiltonian
- Diagonal forms of the Hamiltonian
- Theory of specific heats of solids
- Quasi-particles and renormalization
- Perturbation theory for low densities
- The Debye-Hückel theory of the electron gas

- Phase transitions
- Critical exponents
- The ferro-paramagnetic transition
- The Weiss theory of ferromagnetism
- Macroscopic mean field theory: the Landau model for phase transitions
- Theoretical models
- The Ising model
- Mean-field theory with a variational principle
- Mean-field critical exponents for the Ising model

- Classical treatment of the Hamiltonian N-body assembly
- Hamilton’s equations and phase space
- Hamilton’s equations and 6N-dimensional phase space
- Liouville’s theorem for N particles in a box
- Probability density as a fluid
- Liouville’s equation: operator formalism
- The generalised H-theorem (due to Gibbs)
- Reduced probability distributions
- Basic cells in Γ space

- Derivation of transport equations
- BBGKY hierarchy (Born, Bogoliubov, Green, Kirkwood, Yvon)
- Equations for the reduced distribution functions
- The kinetic equation
- The Boltzmann equation
- The Boltzmann H-theorem
- Macroscopic balance equations

- Dynamics of Fluctuations
- Brownian motion and the Langevin equation
- Fluctuation-dissipation relations
- The response (or Green) function
- General derivation of the fluctuation-dissipation theorem

- Quantum dynamics
- Fermi’s master equation
- Applications of the master equation

- Consequences of time-reversal symmetry
- Detailed balance
- Dynamics of fluctuations
- Onsager’s theorem