Statistical Mechanics Made Simple: A Guide For Students And Researchers - Hardcover

Mattis, Daniel C

Hardcover

ISBN 10: 9812381651 ISBN 13: 9789812381651

Publisher: World Scientific Publishing Co Pte Ltd, 2003

Synopsis

This book is an elaboration of the author's lecture notes in a graduate course in statistical physics and thermodynamics, augmented by some material suitable for self-teaching as well as for undergraduate study. The first 4 or 5 chapters are suitable for an undergraduate course for engineers and physicists in Thermodynamics and Statistical Physics and include detailed study of the various ensembles and their connections to applied thermodynamics. The Debye law of specific heats and reasons for deviations from the Debye formulas are covered, as are the Einstein theories of Brownian motion, black-body radiation and specific heat of solids. Van der Waals gases and the reason for the apparent failure of his Law of Corresponding States are discussed.The last 5 chapters treat topics of recent interest to researchers, including: the Ising and Potts models, spin waves in ferromagnetic and anti-ferromagnetic media, sound propagation in non-ideal gases and the decay of sound waves, introduction to the understanding of glasses and spin glasses, superfluidity and superconductivity.The selection of material is wide-ranging and the mathematics for handling it completely self-contained, ranging from counting (probability theory) to quantum field theory as used in the study of fermions, bosons and as an adjunct in the solutions of the equations of classical diffusion-reaction theory. In addition to the standard material found in most recent books on statistical physics the constellation of topics covered in this text includes numerous original items:• Generalization of “negative temperature” to interacting spins• Derivation of Gibbs' factor from first principles• Exact free energy of interacting particles in 1D (e.g., classical and quantum Tonk's gas)• Introduction to virial expansions, Equations of State, Correlation Functions and “critical exponents”• Superfluidity in ideal and non-ideal fluids (both Bogolubov and Feynman theories)• Superconductivity: thermodynamical approach and the BCS theory• Derivation of “Central Limit Theorem” and its applications• Boltzmann's “H-Theorem” and the nonlinear Boltzmann equation• Exact solution of nonlinear Boltzmann Equation for electrons in time-dependent electric field and the derivation of Joule heating, transport parameters in crossed electric and magnetic fields, etc.• Frequency spectrum and decay of sound waves in gases• Exact evaluation of free energy and thermodynamic properties of the two-dimensional Ising model in regular and fully frustrated (spin-glass like) lattices• The “zipper” model of crystal fracture or polymer coagulation ― calculation of Tc• Potts model in 2D: duality and Tc• “Doi's theory” of diffusion-limited chemical reactions with some exact results ― including the evaluation of statistical fluctuations in radioactive decay• Thermodynamic Green Functions and their applications to fermions and bosons with an example drawn from random matrix theory and much more.

About this title

Review

A valuable addition to any statistical physicist's library. -- American Journal of Physics

An admirable piece of work by an outstanding expert in the field. -- Physics Today

... a rather original contribution to the statistical mechanical literature. -- Mathematical Reviews

Synopsis

This text is an elaboration of the author's lecture notes on a graduate course in thermodynamics and statistical mechanics. The original notes supplemented topics lacking in traditional texts. In its present augmented version the text may be used as the sole or primary text in a one-semester course in thermodynamics/statistical mechanics or as an adjunct text in a two-semester course. Statistical mechanics is the application of physics or chemistry at finite temperature T and can encompass as many topics involving these disciplines as one wishes. This work is intended to respond to the curiosity of the reader about fundamental principles. It shows in detail how one solves the problems that arise in connection with these principles. There are some 50 problems scattered throughout the text and a similar number of illustrations. All the mathematics is self-contained, including the development of field-theoretic methods in the later, more difficult, chapters. The emphasis is not just on the topics, but on the mathematics used to understand them and on the methods of solution. The volume starts by answering the following questions: Where does thermodynamics "come from"?

What is temperature and how might one achieve negative temperatures (metastable states) for interacting spins? What are the various free energies and how do they differ? What is an equation of state and what is the nature of thermodynamic phase transitions? Why does the modern theory of critical-point phenomena disagree with van der Waals' original Law of Corresponding States? The text also includes a number of nonstandard topics, such as the exact construction of the thermodynamic properties in one-dimensional systems and the generalization to "transfermatrices" with which to solve for the free energy and correlations in translationally invariant systems of all sorts. It provides the exact solution to a model Boltzmann equation for the electrical conductivity of a metal or semiconductor -thereby pinpointing the sources of Joule heating. There are sections on the propagation and dissipation of sound waves in a fluid, obtained by finding the solutions to the same linearized Boltzmann equation in the representation of eigenstates of the two-body collision operator.

In fact, the use of operators and operator algebra to solve coupled linear equations is a main characteristic of this book. One obvious application is to superfluidity and superconductivity. In addition to the Bogoliubov treatment of the weakly interacting Bose gas, the author also demonstrates the Feynman-Cohen semiphenomenological theory. He considers superconductivity both thermodynamically and from the microscopic point of view of BCS theory. He uses the quantum field theory of Doi to model the "master equation" for diffusion-limited chemical reactions and uses the solutions to calculate the various rates and other properties. He discusses the Mermin-Wagner-Hohenberg theorem, which precludes long-range order in low-dimensional systems. Finally, he derives the thermodynamic Green functions and their equations of motion and apply them to glassy and amorphous materials. This allows a fairly detailed resolution of the well-known conundrum: Why does the T3 heat capacity of crystalline materials suddenly become linear in T when applied to glass, and what happens to the propagation of long-wavelength sound in the glassy material?

Edition details

• Publisher: World Scientific Publishing Co Pte Ltd
• Publication date: 2003
• Language: English
• ISBN 10: 9812381651
• ISBN 13: 9789812381651
• Binding: Hardcover
• Number of pages: 268