| INTRODUCTION |
| Lecture I. M ICROCANONICAL AND CANONICAL
ENSEMBLES |
| § 1. The microcanonical ensemble |
| | § 1.1. Description of the microcanonical ensemble |
| | § 1.2. Adiabatic process |
| | § 1.3. A two-level system |
| | § 1.4. A system of oscillators |
| § 2. The canonical ensemble |
| | § 2.1. The specific heat of a two-level 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 (\mu-p – 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. Low-temperature expansion (T << \epsilonf) |
| Problem |
| | § 2.3. High-temperature expansion (T >> \epsilonf) |
| 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. Bose-Einstein condensation |
| | § 1.2. High-temperature expansion (T >> T0) |
| | § 1.3. The ideal Bose gas in the neighborhood of T = T0 |
| Problems |
| § 2. Thermodynamics of black-body 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 single-particle potential energy |
| | § 1.4. Definition of a two-particle 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. NON-IDEAL BOSE GAS AT LOW
TEMPERATURES |
| § 1. Phenomenological theory of superfluidity |
| § 2. Theory of non-ideal Bose gas |
| | § 2.1. Energy spectrum of elementary excitations |
| | § 2.2. The distribution function of over-condensate particles |
| | § 2.3. Low-temperature properties of slightly non-ideal
Bose gas |
| § 3. Density of normal and superfluid phases |
| References |
| Lecture VIII . NON-IDEAL FERMI GAS WITH
A WEAK ATTRACTION |
| § 1. Introduction |
| § 2. The Bardeen-Cooper-Schrieffer (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 Ginzburg-Landau theory |
| | § 7.1. Fundamental equations |
| | § 7.2. Calculation of the coefficients in the Ginzburg – Landau equationss |
| Problem |
| | § 7.3. Boundary conditions for the Ginzburg-Landau equations |
| Problem |
| | § 7.4. Critical magnetic fields |
| Problem |
| | § 7.5. Fluctuation correction to the Ginzburg-Landau
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 mean-field 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. C ALCULATION OF CRITICAL
EXPONENTS |
| Introduction |
| § 1. The Ornstein-Zernicke 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 non-parquet 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. One-particle Green's function |
| | § 2.11. The anomalous specific heat at T > Tc |
| | § 2.12. Critical indices in zero magnetic field |
| § 3. n-component 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. Single-particle Green's functions |
| | § 1.1. The interaction representation |
| | § 1.2. Averaging with H0 |
| | § 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. Non-local equations. The Pippard case |
| § 4. Spin susceptibility of a superconductor |
| § 5. The Gor'kov equations |
| | § 5.1. The Gor'kov equations near Tc |
| | § 5.2. The linearized Gor'kov equation |
| | § 5.3. Superconductor in a strong magnetic field |
| | § 5.4. The Ginzburg-Landau equations |
| References |
| APPENDICES |
| § 1. The sum of inverse squares |
| § 2. The sum of inverse fourth powers |
| § 3. The Poisson formula |
| § 4. The Euler-Maclaurin formula |
| § 5. The generalized Euler-Maklorane formula |
| § 6. Second quantization |
| § 7. Certain definite integrals |
| | § 7.1. Integrals which are reduced to the Euler \Gamma-function |
| | § 7.2. Integrals which are reduced to the Riemann
\dzeta-function |
| | § 7.3. Integral which is reduced to the Euler constant |
| § 8. The Hopf-Wiener method |
| § 9. The Dyson equations |
| § 10. Thermodynamic Ward identity |
| References |
| BIBLIOGRAPHY |
| Manuals and monographs |
The book is a set of lectures which the author has been reading
in Moscow Institute of Physical Technology since 2000. The aim of
this book is to add certain new topics to the material of the famous
textbook "Statistical physics" by L.D.Landau and E.M.Lifshitz.
Without them it is not possible to solve modern theoretical problems.
The basic ideas by Boltzmann and Gibbs are given in the first
two Lectures together with the calculation of high temperature corrections
to to the free energy by the ring diagram method (high
density approximation) and by expanding in the gas parameter
(low density approximation). Actually this is the basis of thermodynamic
perturbation theory. Lecture II is concluded with the
consideration of the so-called open ensembles which exist at a given
pressure and at a given number of particles or at a given chemical
potential.
Lecture III is devoted to thermodynamic fluctuations in all ensembles
apart from the microcanonical one. The suggested method
allows to analyze all possible fluctuations without using uncertain
notions such as "minimal work", "non-equilibrium entropy", and
so on.
Lecture IV is devoted to an ideal Fermi gas which is considered
in the big canonical ensemble using quantum occupation numbers.
The difference between the heat capacity for a given number of
particles CV,N and the heat capacity for a given chemical potential
CV,\mu is emphasized. The magnetic properties of an electron gas are
studied at low and high magnetic fields (the De Haas–Van Alphen
effect).
The traditional course of statistical physics is finished by Lecture
V which is devoted to an ideal Bose gas, black-body radiation,
and a harmonic ideal lattice.
The second quantization method, which is the basis of a theory
of quantum non-ideal gases, is considered in Lecture VI. While
studying quantum lattice oscillations at low temperatures, the Matsubara
representation is used.
Lecture VII contains theories of non-ideal Bose gas. The Landau
phenomenological superfluidity theory is accompanied by Bogolubov
's method for studying a non-ideal Bose gas at low temperatures.
Lecture VIII is devoted to superconductivity of ideal metals.
The basic equations obtained by Bogolubov's method together with
a suggestion that it is possible to average over self-consistent field
allow to calculate the thermodynamic properties of a superconductor.
The same approach is used to derive the Ginzburg–Landau
equations, which describe properties of superconductors of the second
kind.
Lecture IX is devoted to the phenomenological Landau theory
of second order phase transitions. While considering simple models
such as isotropic ferromagnet and cubic ferroelectric of the displacement
type, it becomes clear that the mean-field theory gives results
supporting the conclusions of phenomenological theory. Corrections
to the mean-field theory results are calculated by summation
of ring diagrams. They determine the conditions of applicability of
the mean-field theory. Applying the summation of ring diagrams to
phenomena near the critical point, we have obtained certain results
of the Ornstein–Zernicke theory.
Lecture X describes fluctuation theory of phase transitions. The
universality hypothesis allows to calculate certain critical exponents
in the framework of perturbation theory. Summation of the most
strongly diverging diagrams leads to the Sudakov equations. The
solution of these equations determines the singular heat capacity
and the spin susceptibility in the (4 – \epsilon)-dimensional space. The
same method is used to calculate the spontaneous magnetic moment
and the heat capacity in the ordered phases. The Wilson hypothesis
allows us to pass to the limit \epsilon –> 1 and determine all critical
exponents in the 3-dimensional space.
Lecture XI gives details of thermodynamic perturbation theory
which works at all temperatures. The Wick theorem for the electron
field operators is proven. The calculation of low temperature corrections
to the the ground state energy shows the role of exchange
and correlation effects in a degenerate electron gas. The screening
radius of Coulomb potential for all temperatures and electron
concentrations is calculated by summation of ring diagrams.
Lecture XII develops a general description of superconductivity
based on the Green function method. Two Green's functions, normal
and anomalous, are considered. Such an approach allows to
construct a theory of tunneling effect between two different superconductors
(the Josephson effect), give the microscopic description
of the Meissner effect, and calculate the spin susceptibility of
a superconductor (the Knight shift).
The gradient-invariant system of equations for the normal and
anomalous Green's functions (the Gor'kov equations) gives a nonlinear
system of equations for the vector potential and the wave
function of a Cooper pair. This system of equations leads to the
Ginzburg–Landau equations with microscopic boundary conditions
for the interface between two different superconductors and interface
between a normal metal and a superconductor. It is also shown
that the Gor'kov equations determine conditions for forming superconducting
nuclei inside a normal metal placed in a strong external
magnetic field.
Original and simple methods of obtaining certain mathematical
formulae are given in Appendices. The basic formulae of second
quantization for non-relativistic electrons are derived in detail.
The Hoph–Wiener method is applied for considering effects on
the boundary between a superconductor and a normal metal. The
Dyson equation and Ward identity are also derived.
Zaitsev R.O.