Introduction to Statistical Physics

by Salinas, Silvio R. A.

• ISBN13: 9780387951195
• ISBN10: 0387951199
• Format: Hardcover
• Copyright: 2001-04-01
• Publisher: Springer Verlag

Summary

Intended for beginning graduate students or advanced undergraduates, this text covers the statistical basis of thermodynamics, including examples from solid-state physics. It also treats some topics of more recent interest such as phase transitions and non-equilibrium phenomena. The presentation introducesmodern concepts, such as the thermodynamic limit and equivalence of Gibbs ensembles, and uses simple models (ideal gas, Einstein solid, simple paramagnet) and many examples to make the mathematical ideas clear. Frequently used mathematical methods are discussed in detail and reviews in an appendix. The book begins with a review of statistical methods and classical thermodynamics, making it suitable for students from a variety of backgrounds. Statistical mechanics is formulated in the microcanonical ensemble; some simple arguments and many examples are used to construct th canonical and grand-canonical ensembles. The discussion of quantum statistical mechanics includes Bose and Fermi ideal gases, the Bose-Einstein condensation, blackbody radiation, phonons and magnons. The van der Waals and Curoe-Weiss phenomenological models are used to illustrate the classical theories of phase transitions and critical phenomena; modern developments are intorducted with discussions of the Ising model, scaling theory, and renormalization-group ideas. The book concludes withy two chapters on nonequilibrium phenomena: one using Boltzmann's kinetic approach, and the other based on stochastic methods.Exercises at the end of each chapter are an integral part of the course, clarifying and extending topics discussed in the text. Hints and solutions can be found on the author's web site.

Table of Contents

Preface	v
Introduction to Statistical Methods	1	(18)
The random walk in one dimension	2	(2)
Mean values and standard deviations	4	(2)
Gaussian limit of the binomial distribution	6	(3)
Distribution of several random variables. Continuous distributions	9	(3)
Probability distribution for the generalized random walk in one dimension. The Gaussian limit	12	(7)
Exercises	15	(4)
Statistical Description of a Physical System	19	(20)
Specification of the microscopic states of a quantum system	20	(5)
Specification of the microscopic state of a classical system of particles	25	(4)
Ergodic hyphotesis and fundamental postulate of statistical mechanics	29	(4)
Formulation of statistical mechanics for quantum systems	33	(6)
Exercises	35	(4)
Overview of Classical Thermodynamics	39	(22)
Postulates of equilibrium thermodynamics	39	(2)
Intensive parameters of thermodynamics	41	(3)
Equilibrium between two thermodynamic systems	44	(3)
The Euler and Gibbs-Duhem relations	47	(1)
Thermodynamic derivatives of physical interest	47	(1)
Thermodynamic potentials	48	(4)
The Maxwell relations	52	(4)
Variational principles of thermodynamics	56	(5)
Exercises	59	(2)
Microcanonical Ensemble	61	(24)
Thermal interaction between two microscopic systems	62	(3)
Thermal and mechanical interaction between two systems	65	(2)
Connection between the microcanonical ensemble and thermodynamics	67	(12)
Classical monatomic ideal gas	79	(6)
Exercises	82	(3)
Canonical Ensemble	85	(18)
Ideal paramagnet of spin 1/2	91	(2)
Solid of Einstein	93	(2)
Particles with two energy levels	95	(2)
The Boltzmann gas	97	(6)
Exercises	98	(5)
The Classical Gas in the Canonical Formalism	103	(18)
Ideal classical monatomic gas	105	(2)
The Maxwell-Boltzmann distribution	107	(1)
The theorem of equipartition of energy	108	(1)
The classical monatomic gas of particles	109	(4)
The thermodynamic limit of a continuum system	113	(8)
Exercises	117	(4)
The Grand Canonical and Pressure Ensembles	121	(20)
The pressure ensemble	122	(5)
The grand canonical ensemble	127	(14)
Exercises	137	(4)
The Ideal Quantum Gas	141	(20)
Orbitals of a free particle	143	(3)
Formulation of the statistical problem	146	(3)
Classical limit	149	(5)
Diluted gas of diatomic molecules	154	(7)
Exercises	157	(4)
The Ideal Fermi Gas	161	(26)
Completely degenerate ideal Fermi gas	164	(2)
The degenerate ideal Fermi gas (T < TF)	166	(5)
Pauli paramagnetism	171	(5)
Landau diamagnetism	176	(11)
Exercises	182	(5)
Free Bosons: Bose--Einstein Condensation; Photon Gas	187	(24)
Bose--Einstein condensation	188	(11)
Photon gas. Planck statistics	199	(12)
Exercises	208	(3)
Phonons and Magnons	211	(24)
Crystalline phonons	211	(9)
Ferromagnetic magnons	220	(9)
Sketch of a theory of superfluidity	229	(6)
Exercises	232	(3)
Phase Transitions and Critical Phenomena: Classical Theories	235	(22)
Simple fluids. Van der Waals equation	236	(8)
The simple uniaxial ferromagnet. The Curie--Weiss equation	244	(7)
The Landau phenomenology	251	(6)
Exercises	254	(3)
The Ising Model	257	(20)
Exact solution in one dimension	260	(3)
Mean-field approximation for the Ising model	263	(3)
The Curie--Weiss model	266	(2)
The Bethe--Peierls approximation	268	(3)
Exact results on the square lattice	271	(6)
Exercises	273	(4)
Scaling Theories and the Renormalization Group	277	(28)
Scaling theory of the thermodynamic potentials	277	(4)
Scaling of the critical correlations	281	(2)
The Kadanoff construction	283	(2)
Renormalization of the ferromagnetic Ising chain	285	(3)
Renormalization of the Ising model on the square lattice	288	(3)
General scheme of application of the renormalization group	291	(4)
Renormalization group for the Ising ferromagnet on the triangular lattice	295	(10)
Exercises	301	(4)
Nonequilibrium Phenomena: I. Kinetic Methods	305	(26)
Boltzmann's kinetic method	306	(12)
The BBGKY hierarchy	318	(13)
Exercises	326	(5)
Nonequilibrium phenomena: II. Stochastic Methods	331	(26)
Brownian motion. The Langevin equation	332	(5)
The Fokker-Planck equation	337	(3)
The master equation	340	(4)
The kinetic Ising model: Glauber's dynamics	344	(8)
The Monte Carlo method	352	(5)
Exercises	354	(3)
Appendices	357	(14)
A.1 Striling's asymptotic series	357	(2)
A.2 Gaussian integrals	359	(1)
A.3 Dirac's delta function	360	(2)
A.4 Volume of a hypersphere	362	(1)
A.5 Jacobian transformations	363	(2)
A.6 The saddle-point method	365	(3)
A.7 Numerical constants	368	(3)
Bibliography	371	(4)
Index	375