The work contains a critical analysis of the methods and results currently used and applied in the theory of condensed matter. A unified mathematical apparatus has been developed - the method of functional integration which is equally applicable to all main distributions of statistical physics: microcanonical, canonical, and grand canonical ensembles. Within the framework of this method, an exact factorization of the configuration integral with respect to atomic coordinates was performed and a connection between the microcanonical and canonical ensembles was established. It is shown that interatomic interactions can be eliminated by renormalizing external random fields, and random external fields can be eliminated by renormalizing interatomic potentials. A new formulation of problems of equilibrium statistical mechanics as a dynamic field theory with a Hamiltonian depending on temperature is proposed.
The work contains a critical analysis of the methods and results currently used and applied in the theory of condensed matter. A unified mathematical apparatus has been developed - the method of functional integration which is equally applicable to all main distributions of statistical physics: microcanonical, canonical, and grand canonical ensembles. Within the framework of this method, an exact factorization of the configuration integral with respect to atomic coordinates was performed and a connection between the microcanonical and canonical ensembles was established. It is shown that interatomic interactions can be eliminated by renormalizing external random fields, and random external fields can be eliminated by renormalizing interatomic potentials. A new formulation of problems of equilibrium statistical mechanics as a dynamic field theory with a Hamiltonian depending on temperature is proposed.
Providing a broad review of many techniques and their application to condensed matter systems, this book begins with a review of thermodynamics and statistical mechanics, before moving onto real and imaginary time path integrals and the link between Euclidean quantum mechanics and statistical mechanics. A detailed study of the Ising, gauge-Ising and XY models is included. The renormalization group is developed and applied to critical phenomena, Fermi liquid theory and the renormalization of field theories. Next, the book explores bosonization and its applications to one-dimensional fermionic systems and the correlation functions of homogeneous and random-bond Ising models. It concludes with Bohm–Pines and Chern–Simons theories applied to the quantum Hall effect. Introducing the reader to a variety of techniques, it opens up vast areas of condensed matter theory for both graduate students and researchers in theoretical, statistical and condensed matter physics.
Providing a broad review of many techniques and their application to condensed matter systems, this book begins with a review of thermodynamics and statistical mechanics, before moving onto real and imaginary time path integrals and the link between Euclidean quantum mechanics and statistical mechanics. A detailed study of the Ising, gauge-Ising and XY models is included. The renormalization group is developed and applied to critical phenomena, Fermi liquid theory and the renormalization of field theories. Next, the book explores bosonization and its applications to one-dimensional fermionic systems and the correlation functions of homogeneous and random-bond Ising models. It concludes with Bohm-Pines and Chern-Simons theories applied to the quantum Hall effect. Introducing the reader to a variety of techniques, it opens up vast areas of condensed matter theory for both graduate students and researchers in theoretical, statistical and condensed matter physics.
Classical statistical mechanics tries to derive the macroscopic properties of matter starting from the mechanical laws that rule the behaviour of single particles. To describe equilibrium states, only observables accounting for correlations among states are considered, and only systems consisting of a large number of particles are taken into account. The observables are described by continuous functions on phase space, and the states are represented by linear assignments of a number to each observable. Within the framework of the canonical ensemble, one deals with mechanical systems in thermal equilibrium with a thermal reservoir, and the equilibrium state is reached as a result of the interaction with the external world. The external world may be really external, or equally well, the unrecognized internal degrees of freedom, like in the calculation of viscosity or thermal conductivity. If really only the external world is the cause we may expect some surface dependence while an internal unrecognized degree of freedom would have a volume effect, unless the interactions are long range. The microcanonical ensemble is instead introduced to study isolated mechanical systems, and the equilibrium is viewed as a temporal average, rather than as a limit. Attention is then focused on partition functions, the theorem of equipartition of energy and an elementary theory of specific heats. In the second part, the Planck derivation of the law of black-body radiation is analysed, presenting in chronological order the Kirchhoff theorem, the Stefan law, the Wien displacement law, the Rayleigh–Jeans formula and the Planck hypothesis. Further topics discussed are the Einstein and Debye quantum models for specific heats of solids. These topics prepare the ground for the introduction of quantum statistical mechanics. The third part is, in fact, devoted to the analysis of identical particles in quantum mechanics.
Thermodynamics is used to describe the bulk properties of matter in or near equilibrium. Many scientists, notably Boyle, Carnot, Clausius, Gay-Lussac, Gibbs, Joule, Kelvin, and Rumford, contributed to the development of the field over three centuries. Quantities such as mass, pressure, energy, and so on are readily defined and measured. Classical statistical mechanics attempts to understand thermodynamics by the application of classical mechanics to the microscopic particles making up the system. Great progress in this field was made by physicists like Boltzmann and Maxwell. Temperature, entropy, particle number, and chemical potential are thus understandable in terms of the microscopic nature of matter. Classical mechanics is inadequate in many circumstances however, and ultimately must be replaced by quantum mechanics. In fact, the ultraviolet catastrophe encountered by the application of classical mechanics and electromagnetism to blackbody radiation was one of the problems that led to the development of quantum theory. The development of quantum statistical mechanics was achieved by a number of twentieth century physicists, most notably Planck, Einstein, Fermi, and Bose. The purpose of this chapter is to give a mini-review of the basic concepts of quantum statistical mechanics as applied to noninteracting systems of particles. This will set the stage for the functional integral representation of the partition function, which is a cornerstone of modern relativistic quantum field theory and the quantum statistical mechanics of interacting particles and fields. Ensembles One normally encounters three types of ensemble in equilibrium statistical mechanics. The microcanonical ensemble is used to describe an isolated system that has a fixed energy E , a fixed particle number N , and a fixed volume V .
The sophistication of modern tools used in the study of statistical mechanics and field theory is often an obstacle to the easy understanding of new important current results reported in journals. The main purpose of this book is to introduce the reader to the methods of the fluctuation (field) theory of phase transitions and critical phenomena so as to provide a good source for research. The introductory contents are concerned with ideas of description, thermodynamic stability theory related to phase transitions, major experimental facts, basic models and their relationships. Special attention is paid to the mean field approximation and to the Landau expansion for simple and complex models of critical and multicritical phenomena. An instructive representation of the modern perturbation theory and the method of the renormalization group is developed for field models of phase transitions. The essential influence of the fluctuations on the critical behaviour is established together with the theory of correlation functions, Gaussian approximation, the Ginzburg criterion, [symbol]- and 1/n- expansions as practical realizations of the renormalization group ideas. Applications of the theory to concrete aspects of condensed matter physics are considered: quantum effects, Bose condensation, crystal anisotropy, superconductors and liquid crystals, effects of disorder of type randomly distributed quenched impurities and random fields. This volume can be used as an advanced University course book for students with a basic knowledge of statistical physics and quantum mechanics. It could be considered as a complementary text to a standard University course on statistical physics
We develop a statistical mean-field theory of finite quantum systems in thermal equilibrium. Our formulation employs the canonical ensemble of statistical mechanics, and it enables us to analytically determine the occupation number distributions of interacting particles obeying Bose–Einstein or Fermi–Dirac statistics. We have also developed a numerical procedure that enables us to obtain a universal scaled occupation number distribution that, for a given total number of interacting particles in a finite system, makes it possible to determine the occupation number distribution for any temperature. The developed mean-field theory is applicable to a wide range of atomic, nuclear and condensed matter systems for which finite-size effects can play an important role. In particular, the present approach makes it possible to formulate a finite temperature mean-field theory for a specific ion in a dense plasma.