Zaitsev R.O. Introduction to Modern Statistical Physics:
A Set of Lectures

Contents

Introduction

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 C_V,N and the heat capacity for a given chemical potential C_V,\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 k_B = 1 and gj\muB = 1, where gJ is the gyromagnetic ratio and \mu B is the Bohr magneton.

El autor

Zaitsev R.O.

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.