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Cover Zaitsev R.O. Introduction to Modern Statistical Physics: A Set of Lectures
Id: 74084
24.9 EUR

Introduction to Modern Statistical Physics: A Set of Lectures

URSS. 416 pp. (English). Paperback. ISBN 978-5-484-01040-0.


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.

High-temperature corrections to the thermodynamic potential are calculated by employing ring diagrams and also by expanding in powers of the gas parameter.

The universality hypothesis gives a possibility to calculate critical exponents in the framework of perturbation theory. Summation of the most strongly diverging diagrams yields parquet equations. Solving these equations, the singular specific heat and magnetic susceptibility are determined in the (4--epsilon)-dimensional space.

The gradient-invariant microscopic equations which describe a superconducting state are derived. Using these equations, the Ginsburg---Landau equations together with the microscopic boundary conditions are obtained. The critical magnetic field of forming superconducting nuclei is calculated.

The tunneling effects between two superconductors (the Josephson effect) and also between a superconductor and a normal metal are studied. The microscopic description of the Meissner effect, calculation of the spin susceptibility in a superconductor (the Knight shift) are given.

The book is intended for researchers, graduate and undergraduate students studying the theory of condensed matter.




§ 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

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


§ 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
§ 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


§ 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)
 § 2.3. High-temperature expansion (T >> epsilonf)
 § 2.4. The magnetic susceptibility. Weak fields
 § 2.5. The magnetic susceptibility. Strong fields


§ 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
§ 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


§ 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
§ 4. Notion of quasiparticles


§ 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


§ 1. Introduction
§ 2. The Bardeen-Cooper-Schrieffer (BCS) theory
 § 2.1. The BCS model -- Bogolyubov's version
§ 3. The excitation spectrum
§ 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
 § 7.3. Boundary conditions for the Ginzburg-Landau equations
 § 7.4. Critical magnetic fields
 § 7.5. Fluctuation correction to the Ginzburg-Landau equations


§ 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


§ 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


§ 1. Thermodynamical perturbation theory
§ 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


§ 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
§ 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
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.

In the book we used the energy system of units in which the Boltzman constant kB = 1 and gjmuB = 1, where gJ is the gyromagnetic ratio and mu B is the Bohr magneton.

 About the author

Rogdai Olegovich Zaitsev was born in Moscow in 1938. He graduated from the physical faculty of the Moscow State University in 1961; worked in the Kurchatov Institute of Nuclear Energy from 1965 to 2008. Now he works in the Moscow Institute of Physics and Technology as a professor of theoretical physics.

His scientific interests are devoted to theories of solid state, superconductivity, ferromagnetism, and the non-linear theory of noise.

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