PH-M11 Statistical Mechanics
Phenomenology of thermodynamics; general definitions; temperature, pressure, energy and entropy; the three laws of thermodynamics.
The first and second law of thermodynamics in differential form; state functions and exact differentials; Legendre transforms; thermodynamics potentials; Maxwell's relations.
Hamiltonian evolution; the phase space Gamma; the density of states in Gamma space.
Postulate of equal a priory probability; Thermodynamic limit; the microcanonical ensemble; entropy in the microcanonical ensemble; derivation of thermodynamic quantities.
The canonical ensemble; the partition function and its relationship with the Helmotz free energy; derivation of thermodynamic quantities.
The ideal gas in the microcanonical approach: entropy, internal energy, temperature and equation of state.
The ideal gas in the canonical approach: partition function, internal energy, equation of state.
The grand-canonical ensemble: chemical potential, grand partition function, equation of state, number fluctuation, free energy, equivalence with the canonical ensemble.
More on ensembles: the quantum factors in the measure and the Gibbs paradox; the partition function as an integral over the energy and the density of states; statistical mechanics of the quantum harmonic oscillator.
Phase transitions and non-analyticity of Z and A; Ehrenfest classifications; first and second order phase transitions in general.
Phase diagram in general; phase diagram of water: critical lines, tricritical point, critical point, phase coexistence and phase separation.
Phase transitions and symmetry breaking; the order parameter and its behaviour for first and second order phase transitions.
The Van der Waals gas: equation of state, phase transitions, Maxwell construction, the critical point; universal form for the Van der Waals equation equation of state.
Ferromagnetic to paramagnetic transition; Curie temperature; spontaneous magnetisation.
The Ising model in d dimensions: Hamiltonian, Z(2) symmetry, spontaneous symmetry breaking; the magnetisation as an order parameter; the magnetic susceptibility.
Energy vs. Entropy arguments in Statistical Mechanics; absense of transition in 1d; the transfer matrix; exact solution in 1d.
Existence of a transition in 2d: Peyerls argument, domain wall translationally invariant along one direction.
The mean-field approximation; mean-field solution in 2d.
Behaviour of the magnetisation and the magnetic susceptibility near the critical point in the mean-field approximation.
Mean-field approximation: the free energy, the internal energy and the specific heat near the critical point. The magnetisation as a function of the magnetic field at criticality.
Comparison between the exact and the mean-field solution.
Correlation function, fluctuation-dissipation theorem and critical indices; the Ornstein-Zernike form of the correlation function; correlation length and its critical exponent; anomalous dimension.
Scaling of observables and relations among the exponents: derivation via dimensional analysis, scaling hypothesis and scale invariance.
The Landau form for the free energy and its derivation for the Ising model.
Mean-field from the Landau free energy; computation of the critical exponents; Gaussian model.
Universality; Ginzburg criterion; more on anomalous dimensions.
Renormalisation group: block-spin transformation; fixed point; scaling field; relevant and irrelevant variables; critical surface; universality; renormalisation group flow.