PH-M11 Statistical Mechanics

Syllabus

1. Phenomenology of thermodynamics; general definitions; temperature, pressure, energy and entropy; the three laws of thermodynamics.

2. The first and second law of thermodynamics in differential form; state functions and exact differentials; Legendre transforms; thermodynamics potentials; Maxwell's relations.

3. Hamiltonian evolution; the phase space Gamma; the density of states in Gamma space.

4. Postulate of equal a priory probability; Thermodynamic limit; the microcanonical ensemble; entropy in the microcanonical ensemble; derivation of thermodynamic quantities.

5. The canonical ensemble; the partition function and its relationship with the Helmotz free energy; derivation of thermodynamic quantities.

6. The ideal gas in the microcanonical approach: entropy, internal energy, temperature and equation of state.

7. The ideal gas in the canonical approach: partition function, internal energy, equation of state.

8. The grand-canonical ensemble: chemical potential, grand partition function, equation of state, number fluctuation, free energy, equivalence with the canonical ensemble.

9. 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.

10. Phase transitions and non-analyticity of Z and A; Ehrenfest classifications; first and second order phase transitions in general.

11. Phase diagram in general; phase diagram of water: critical lines, tricritical point, critical point, phase coexistence and phase separation.

12. Phase transitions and symmetry breaking; the order parameter and its behaviour for first and second order phase transitions.

13. 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.

14. Ferromagnetic to paramagnetic transition; Curie temperature; spontaneous magnetisation.

15. The Ising model in d dimensions: Hamiltonian, Z(2) symmetry, spontaneous symmetry breaking; the magnetisation as an order parameter; the magnetic susceptibility.

16. Energy vs. Entropy arguments in Statistical Mechanics; absense of transition in 1d; the transfer matrix; exact solution in 1d.

17. Existence of a transition in 2d: Peyerls argument, domain wall translationally invariant along one direction.

18. The mean-field approximation; mean-field solution in 2d.

19. Behaviour of the magnetisation and the magnetic susceptibility near the critical point in the mean-field approximation.

20. 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.

21. Comparison between the exact and the mean-field solution.

22. Correlation function, fluctuation-dissipation theorem and critical indices; the Ornstein-Zernike form of the correlation function; correlation length and its critical exponent; anomalous dimension.

23. Scaling of observables and relations among the exponents: derivation via dimensional analysis, scaling hypothesis and scale invariance.

24. The Landau form for the free energy and its derivation for the Ising model.

25. Mean-field from the Landau free energy; computation of the critical exponents; Gaussian model.

26. Universality; Ginzburg criterion; more on anomalous dimensions.

27. Renormalisation group: block-spin transformation; fixed point; scaling field; relevant and irrelevant variables; critical surface; universality; renormalisation group flow.