# Statistical Mechanics

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

- 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

- 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

- 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

- 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

- 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