**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.