INTRODUCTION 
Lecture I. MICROCANONICAL AND CANONICAL
ENSEMBLES 
§ 1. The microcanonical ensemble 
 § 1.1. Description of the microcanonical ensemble 
 § 1.2. Adiabatic process 
 § 1.3. A twolevel system 
 § 1.4. A system of oscillators 
§ 2. The canonical ensemble 
 § 2.1. The specific heat of a twolevel system 
 § 2.2. The specific heat of a system of oscillators 
§ 3. Correlation corrections in a Coulomb gas 
References 
Lecture II. THE GRAND CANONICAL,
(p  T)  and (mu  p  T)ENSEMBLES 
§ 1. The grand canonical ensemble 
 § 1.1. Description of the grand canonical ensemble 
 § 1.2. The specific heat at a given mu 
 § 1.3. Expansion in powers of density 
§ 2. The (p  T)ensemble 
§ 3. The (mup  T)ensemble 
References 
Lecture III. THERMODYNAMIC FLUCTUATIONS

Introduction 
§ 1. Fluctuations in the (p  T)ensemble
at constant number of particles 
 § 1.1. Description of the (p  T)ensemble 
 § 1.2. The fluctuation probability distribution
in the (p  T)ensemble 
 § 1.3. Fluctuations at constant number of particles 
 § 1.4. Fluctuations of energy at constant number of particles 
§ 2. Fluctuations of energy in the canonical
ensemble 
§ 3. Fluctuations in the grand canonical ensemble 
 § 3.1. Fluctuations of energy in the grand canonical
ensemble 
 § 3.2. The fluctuation probability distribution
in the (mu  T)ensemble 
 § 3.3. Fluctuations in the (mu  T)ensemble 
Problems 
§ 4. Fluctuations in the (mu  p  T)ensemble 
 § 4.1. The fluctuation probability distribution
in the (mu  p  T)ensemble 
 § 4.2. Fluctuations at constant number of particles 
 § 4.3. Fluctuations at constant volume 
 § 4.4. Fluctuations at constant temperature 
Problems 
References 
Lecture IV. IDEAL GAS AT LOW
TEMPERATURES 
§ 1. Properties of Fermi and Bose gases 
§ 2. Ideal Fermi gas 
 § 2.1. The ground state (T = 0) 
 § 2.2. Lowtemperature expansion (T << epsilon_{f}) 
Problem 
 § 2.3. Hightemperature expansion (T >> epsilon_{f}) 
Problem 
 § 2.4. The magnetic susceptibility. Weak fields 
 § 2.5. The magnetic susceptibility. Strong fields 
References 
Lecture V. IDEAL BOSE GAS 
§ 1. General properties of Bose gases 
 § 1.1. BoseEinstein condensation 
 § 1.2. Hightemperature expansion (T >> T_{0}) 
 § 1.3. The ideal Bose gas in the neighborhood of T = T_{0} 
Problems 
§ 2. Thermodynamics of blackbody radiation 
§ 3. Thermodynamics of perfect lattice vibrations 
 § 3.1. Vibrational energy spectrum of a perfect lattice 
 § 3.2. The specific heat of a perfect lattice; the Debye
theory 
 § 3.3. The equation of state of a perfect lattice 
References 
Lecture VI. REPRESENTATION OF SECOND
QUANTIZATION 
§ 1. Second quantization for a system of electrons 
 § 1.1. Permutation relations 
 § 1.2. Definition of the density operator 
 § 1.3. Definition of a singleparticle potential energy 
 § 1.4. Definition of a twoparticle potential energy 
§ 2. Quantization of a phonon field 
 § 2.1. Transition to the Matsubara representation 
 § 2.2. Calculation of average values 
 § 2.2. Criterion of melting 
§ 3. Second quantization for a system of Bose
particles 
Problem 
§ 4. Notion of quasiparticles 
Problem 
References 
Lecture VII. NONIDEAL BOSE GAS AT LOW
TEMPERATURES 
§ 1. Phenomenological theory of superfluidity 
§ 2. Theory of nonideal Bose gas 
 § 2.1. Energy spectrum of elementary excitations 
 § 2.2. The distribution function of overcondensate particles 
 § 2.3. Lowtemperature properties of slightly nonideal
Bose gas 
§ 3. Density of normal and superfluid phases 
References 
Lecture VIII. NONIDEAL FERMI GAS WITH
A WEAK ATTRACTION 
§ 1. Introduction 
§ 2. The BardeenCooperSchrieffer (BCS) theory 
 § 2.1. The BCS model  Bogolyubov's version 
§ 3. The excitation spectrum 
Problem 
§ 4. Temperature dependence of the energy gap 
 § 4.1. The energy gap in the neighborhood
of the transition point 
 § 4.2. The energy gap in the neighborhood of T = 0 
§ 5. Thermodynamics of a superconductor 
 § 5.1. Superconductors at low temperatures 
 § 5.2. Superconductors in the neighborhood
of the transition point 
 § 5.3. Thermodynamic critical magnetic field 
§ 6. Density of normal and superconducting phases 
 § 6.1. Superconductors at low temperatures 
 § 6.2. Superconductors near the transition point 
§ 7. The GinzburgLandau theory 
 § 7.1. Fundamental equations 
 § 7.2. Calculation of the coefficients in the Ginzburg  Landau equationss 
Problem 
 § 7.3. Boundary conditions for the GinzburgLandau equations 
Problem 
 § 7.4. Critical magnetic fields 
Problem 
 § 7.5. Fluctuation correction to the GinzburgLandau
equations 
References 
Lecture IX. SECOND ORDER PHASE
TRANSITIONS 
§ 1. Phenomenological theory of the second order
phase transitions 
 § 1.1. Ehrenfest's equations 
§ 2. The Weiss meanfield theory 
§ 3. Correlation corrections near the transition
point 
§ 4. Theory of ferroelectrics of the displacement
type 
 § 4.1. Vibrational spectrum of ion crystals. Ferroelectric
instability 
 § 4.2. Spontaneous polarization and vibrational
energy of critical fluctuations below the transition point 
 § 4.3. Thermodynamic quantities near the point of
ferroelectric instability 
Problem 
References 
Lecture X. CALCULATION OF CRITICAL
EXPONENTS 
Introduction 
§ 1. The OrnsteinZernicke theory 
§ 2. Phase transitions in the (4  epsilon)dimensional space 
 § 2.1. Effective Hamiltonian 
 § 2.2. Zero Green's function 
 § 2.3. Universality hypothesis and diagrammatic technique 
 § 2.4. Parquet and nonparquet diagrams 
 § 2.5. Summation of parquet diagrams 
 § 2.6. The Sudakov equations 
 § 2.7. The solution of Sudakov's equations 
 § 2.8. Angular vertex part 
 § 2.9. Parquet equation for an angular vertex part 
 § 2.10. Oneparticle Green's function 
 § 2.11. The anomalous specific heat at T > T_{c} 
 § 2.12. Critical indices in zero magnetic field 
§ 3. ncomponent isotropic model 
 § 3.1. Critical indices in zero magnetic field 
 § 3.2. Critical indices in strong magnetic fields 
§ 4. Critical indices for epsilon = 1 and epsilon = 2 
Problems 
References 
Lecture XI. PERTURBATION THEORY AT LOW
TEMPERATURES 
§ 1. Thermodynamical perturbation theory 
Problem 
§ 2. The Wick theorem 
§ 3. First order of perturbation theory 
 § 3.1. Exchange interaction 
§ 4. Second order of perturbation theory 
§ 5. Diagrammatic technique 
§ 6. High density approximation 
References 
Lecture XII. THE THEORY OF
SUPERCONDUCTIVITY 
§ 1. Singleparticle Green's functions 
 § 1.1. The interaction representation 
 § 1.2. Averaging with H_{0} 
 § 1.3. Diagrammatic technique 
 § 1.4. Anomalous Green's functions 
§ 2. The Josephson effect 
 § 2.1. The tunnelling Hamiltonian 
 § 2.2. Stationary superconducting current 
§ 3. Superconductor in a weak magnetic field 
 § 3.1. General relations 
 § 3.2. The London penetration depth 
 § 3.3. Nonlocal equations. The Pippard case 
§ 4. Spin susceptibility of a superconductor 
§ 5. The Gor'kov equations 
 § 5.1. The Gor'kov equations near T_{c} 
 § 5.2. The linearized Gor'kov equation 
 § 5.3. Superconductor in a strong magnetic field 
 § 5.4. The GinzburgLandau equations 
References 
APPENDICES 
§ 1. The sum of inverse squares 
§ 2. The sum of inverse fourth powers 
§ 3. The Poisson formula 
§ 4. The EulerMaclaurin formula 
§ 5. The generalized EulerMaklorane formula 
§ 6. Second quantization 
§ 7. Certain definite integrals 
 § 7.1. Integrals which are reduced to the Euler Gammafunction 
 § 7.2. Integrals which are reduced to the Riemann
dzetafunction 
 § 7.3. Integral which is reduced to the Euler constant 
§ 8. The HopfWiener method 
§ 9. The Dyson equations 
§ 10. Thermodynamic Ward identity 
References 
BIBLIOGRAPHY 
Manuals and monographs 