QUANTUM STATISTICAL MECHANICS

Prof. Lev Pitaevskii

PROGRAMME

1. THE GIBBS DISTRIBUTION IN CLASSICAL STATISTICAL MECHANICS
(Applications)
Deviation of a gas from the ideal state in classical statistical
mechanics. LL. 74, KH 10.1.
 The Van der Waals equation. The critical point. The Maxwell's rule. LL 76, 83, 84.
2. THE GIBBS DISTRIBUTION IN QUANTUM STATISTICAL MECHANICS\\
The canonical distribution in quantum statistical mechanics. The Gibbs distribution for a variable number of particles. (The Grand canonical distribution.) The thermodynamic potential W. Derivation of the thermodynamic relations from the grand canonical distribution. KH 8.3; LL 79, 31, 35, 36.
The ideal Fermi and Bose gases. Occupation numbers. The Fermi distribution.The Bose distribution. A degenerate Fermi gas. Black-body radiation. (Bose gas of photons.) A degenerate Bose-gas. Bose-Einstein condensation. KH 8.6, 11.1, 12.1, 12.3; LL 53, 54, 56, 57, 58, 62, 63.
Solids at high and low temperatures. Phonons. Debaye's interpolation
formula. KH 12.2; LL 64, 65, 66.
Entropy in quantum statistical physics. The third law of thermodynamics. KH 8.4; LL. 7, 23.
Fluctuations of energy in the canonical distribution. Fluctuations of
number of particles in the grand canonical distribution. Connection
between fluctuations and response on an external force in classical
statistical mechanics. KH 7.2, 7.4.
3. PHASE TRANSITIONS OF THE SECOND ORDER
The notion of symmetry. Order parameter and phase transition of the second order. Relations between the discontinuities of different quantities. Effect of an external field on the phase transition of the second order. Fluctuations of the order parameter. Correlation function and correlation radius. Critical indexes. LL 142, 143,
144, 146, 148.
Phase transition into superfluid state. Order parameter for
this transition. Superfluidity and vortex lines. Phase transitions in
two-dimensional systems. Dislocations. KH 12.5, 13.5; LP 26, 29.

4. THE FLUCTUATION-DISSIPATION THEOREM
The generalized susceptibility. Its analytical properties.
Dispersion relations. Expression for the susceptibility through
matrix elements.Correlation function of fluctuations. Fluctuation-dissipation theorem. Fluctuation-dissipation theorem for more then one variable. LL. 123, 126, 122, 124, 125.

REFERENCE TEXTS
KH - K. Huang, Statistical Mechanics (Meccanica Statistica)
LL5 - L.D. Landau, E.M. Lifshitz, Statistical Physics, Part. 1
LP - E.M. Lifshitz, L.P. Pitaevskii, Statistical Physics, Part. 2

Numbers mean numbers of relevant sections of the books.