Statistical Mechanics

Graduate course about Statistical Mechanics, edited by Leonardo Pacciani, student at the master in physics at University of Padova, Italy.

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Review of thermodynamics

• Introduction
• State variables
• Entropy
• Thermodynamic potentials
• Gibbs-Duhem and Maxwell relations
• Response functions
• Phase transitions and phase diagrams
• Phase coexistence and general properties of phase transitions
• Gibbs phase rule
• Order parameters
• Classification of phase transitions
• Critical exponents and universality
• Inequalities between critical exponents

Brownian motion

• Random walks and diffusion
• General properties of random walks
• The diffusion equation
• Fokker-Planck and Langevin's equations

Ensemble theory

• A bridge between the microscopic and the macroscopic
• The microcanonical ensemble
• The monoatomic ideal gas
• Statistics and thermodynamics
• Some remarks and refinements in the definition of entropy
• Some explicit computations for the monoatomic ideal gas
• The foundations of statistical mechanics
• The canonical ensemble
• The canonical and the microcanonical ensemble
• Helmholtz free energy
• The canonical ensemble at work
• The equipartition theorem
• The grand canonical ensemble
• Fluctuations in the grand canonical ensemble
• Grand potential

Entropy and its meanings

• Introduction
• Heat engines and entropy
• Time and entropy
• Entropy as disorder
• Mixing of gases
• Entropy as ignorance: information entropy
• Shannon's entropy
• Conclusions: the most general definition of entropy
• Entropy and the arrow of time
• A niftier framework for the statistical ensembles

Statistical mechanics of phase transitions

• Introduction
• Statistical mechanics and phase transitions
• Long range correlations
• Finite size effects on phase transitions
• Models in statistical mechanics
• The Ising model - Introduction
• Analytic properties of the Ising model
• Absence of phase transitions for finite systems
• Ising model and fluids
• Ising model and the ideal gas
• Ising model and binary alloys
• Ising model and neural networks
• Final remark on equivalence in statistical mechanics
• Ising model in one dimension
• Bulk free energy, thermodynamic limit and absence of phase transitions
• Irrelevance of boundary conditions
• Absence of spontaneous magnetization
• The transfer matrix method
• A slightly trickier system: the Heisenberg model
• The Ising model in higher dimensions
• Additional remarks on the Ising model
• The role of dimensionality
• The role of symmetry
• The role of interaction range

Mean field theories

• Introduction
• Weiss mean field theory for the Ising model
• Critical exponents of Weiss mean field theory for the Ising model
• Hubbard-Stratonovich mean field theory for the Ising model
• Variational methods
• Bragg-Williams approximation for the Ising model
• Bragg-Williams approximation for the Potts model
• Mean field theory for the Blume-Emery-Griffiths model
• Mean field theories for fluids
• Van der Waals equation
• Mean field theories for weakly interacting systems
• Introduction to Landau theory
• Assumptions of the Landau theory
• Landau theory for the Ising model
• First-order phase transitions in Landau theory
• Liquid crystals
• Introduction: Ginzburg criterion
• Functional partition function and coarse graining
• Coarse graining procedure for the Ising model

Scaling theory

• Introduction
• Widom's static scaling theory
• Relations between critical exponents
• Rescaled state equation
• Kadanoff's scaling and correlation lengths
• Kadanoff's argument for the Ising model

The Renormalization Group

• Introduction
• Basic ideas of the Renormalization Group
• Singular behaviour in the Renormalization Group
• Fixed points of the Renormalization Group flow
• Renormalization Group flow near a fixed point
• Global properties of the Renormalization Group flow
• Universality in the Renormalization Group
• The origins of scaling and critical behaviour
• Irrelevant variables
• Renormalization Group in coordinate space
• Decimation to a third of spins for a one-dimensional Ising model with H=0
• Decimation to a half of spins for a one-dimensional Ising model with H not 0
• Decimation in dimensions higher than one: proliferation of the interactions

Appendices

• Volume of a hypersphere
• The saddle point approximation
• Stirling's approximation
• Introduction
• Liouville's theorem
• Ergodicity
• Homogeneous functions of one or more variables
• Generalized homogeneous functions

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