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Statistical Mechanics: Entropy, Order Parameters and Complexity by James Sethna

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352pp

Textbook Information

ISBN-13: 9780198566779
Edition Description: New Edition
Edition Number: 1
Pub. Date: April 2006
Publisher: Oxford University Press

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Product Details

Pub. Date: April 2006
Publisher: Oxford University Press
Format: Textbook Paperback, 352pp

Series: Oxford Master Series in Physics Series, #14
ISBN-13: 9780198566779
ISBN: 0198566778
Edition Number: 1
Edition Description: New Edition

Synopsis

In each generation, scientists must redefine their fields: abstracting, simplifying and distilling the previous standard topics to make room for new advances and methods. Sethna's book takes this step for statistical mechanics--a field rooted in physics and chemistry whose ideas and methods are now central to information theory, complexity, and modern biology. Aimed at advanced undergraduates and early graduate students in all of these fields, Sethna limits his main presentation to the topics that future mathematicians and biologists, as well as physicists and chemists, will find fascinating and central to their work. The amazing breadth of the field is reflected in the author's large supply of carefully crafted exercises, each an introduction to a whole field of study: everything from chaos through information theory to life at the end of the universe.

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Biography

Prof. James P. Sethna is Professor of Physics, Laboratory of Atomic and Solid State Physics, Cornell University, Ithaca, NY.

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Table of Contents

	List of figures	xv
1	What is statistical mechanics?	1
	Exercises	4
1.1	Quantum dice	4
1.2	Probability distributions	5
1.3	Waiting times	6
1.4	Stirling's approximation	7
1.5	Stirling and asymptotic series	7
1.6	Random matrix theory	8
1.7	Six degrees of separation	9
1.8	Satisfactory map colorings	12
2	Random walks and emergent properties	15
2.1	Random walk examples: universality and scale invariance	15
2.2	The diffusion equation	19
2.3	Currents and external forces	20
2.4	Solving the diffusion equation	22
2.4.1	Fourier	23
2.4.2	Green	23
	Exercises	25
2.1	Random walks in grade space	25
2.2	Photon diffusion in the Sun	26
2.3	Molecular motors and random walks	26
2.4	Perfume walk	27
2.5	Generating random walks	28
2.6	Fourier and Green	28
2.7	Periodic diffusion	29
2.8	Thermal diffusion	30
2.9	Frying pan	30
2.10	Polymers and random walks	30
2.11	Stocks, volatility, and diversification	31
2.12	Computational finance: pricing derivatives	32
2.13	Building a percolation network	33
3	Temperature and equilibrium	37
3.1	The microcanonical ensemble	37
3.2	The microcanonical ideal gas	39
3.2.1	Configuration space	39
3.2.2	Momentum space	41
3.3	What is temperature?	44
3.4	Pressure and chemical potential	47
3.4.1	Advanced topic: pressure in mechanics and statistical mechanics	48
3.5	Entropy, the ideal gas, and phase-space refinements	51
	Exercises	53
3.1	Temperature and energy	54
3.2	Large and very large numbers	54
3.3	Escape velocity	54
3.4	Pressure computation	54
3.5	Hard sphere gas	55
3.6	Connecting two macroscopic systems	55
3.7	Gas mixture	56
3.8	Microcanonical energy fluctuations	56
3.9	Gauss and Poisson	57
3.10	Triple product relation	58
3.11	Maxwell relations	58
3.12	Solving differential equations: the pendulum	58
4	Phase-space dynamics and ergodicity	63
4.1	Liouville's theorem	63
4.2	Ergodicity	65
	Exercises	69
4.1	Equilibration	69
4.2	Liouville vs. the damped pendulum	70
4.3	Invariant measures	70
4.4	Jupiter! and the KAM theorem	72
5	Entropy	77
5.1	Entropy as irreversibility: engines and the heat death of the Universe	77
5.2	Entropy as disorder	81
5.2.1	Entropy of mixing: Maxwell's demon and osmotic pressure	82
5.2.2	Residual entropy of glasses: the roads not taken	83
5.3	Entropy as ignorance: information and memory	85
5.3.1	Non-equilibrium entropy	86
5.3.2	Information entropy	87
	Exercises	90
5.1	Life and the heat death of the Universe	91
5.2	Burning information and Maxwellian demons	91
5.3	Reversible computation	93
5.4	Black hole thermodynamics	93
5.5	Pressure-volume diagram	94
5.6	Carnot refrigerator	95
5.7	Does entropy increase?	95
5.8	The Arnol'd cat map	95
5.9	Chaos, Lyapunov, and entropy increase	96
5.10	Entropy increases: diffusion	97
5.11	Entropy of glasses	97
5.12	Rubber band	98
5.13	How many shuffles?	99
5.14	Information entropy	100
5.15	Shannon entropy	100
5.16	Fractal dimensions	101
5.17	Deriving entropy	102
6	Free energies	105
6.1	The canonical ensemble	106
6.2	Uncoupled systems and canonical ensembles	109
6.3	Grand canonical ensemble	112
6.4	What is thermodynamics?	113
6.5	Mechanics: friction and fluctuations	117
6.6	Chemical equilibrium and reaction rates	118
6.7	Free energy density for the ideal gas	121
	Exercises	123
6.1	Exponential atmosphere	124
6.2	Two-state system	125
6.3	Negative temperature	125
6.4	Molecular motors and free energies	126
6.5	Laplace	127
6.6	Lagrange	128
6.7	Legendre	128
6.8	Euler	128
6.9	Gibbs-Duhem	129
6.10	Clausius-Clapeyron	129
6.11	Barrier crossing	129
6.12	Michaelis-Menten and Hill	131
6.13	Pollen and hard squares	132
6.14	Statistical mechanics and statistics	133
7	Quantum statistical mechanics	135
7.1	Mixed states and density matrices	135
7.1.1	Advanced topic: density matrices	136
7.2	Quantum harmonic oscillator	139
7.3	Bose and Fermi statistics	140
7.4	Non-interacting bosons and fermions	141
7.5	Maxwell-Boltzmann 'quantum' statistics	144
7.6	Black-body radiation and Bose condensation	146
7.6.1	Free particles in a box	146
7.6.2	Black-body radiation	147
7.6.3	Bose condensation	148
7.7	Metals and the Fermi gas	150
	Exercises	151
7.1	Ensembles and quantum statistics	151
7.2	Phonons and photons are bosons	152
7.3	Phase-space units and the zero of entropy	153
7.4	Does entropy increase in quantum systems?	153
7.5	Photon density matrices	154
7.6	Spin density matrix	154
7.7	Light emission and absorption	154
7.8	Einstein's A and B	155
7.9	Bosons are gregarious: superfluids and lasers	156
7.10	Crystal defects	157
7.11	Phonons on a string	157
7.12	Semiconductors	157
7.13	Bose condensation in a band	158
7.14	Bose condensation: the experiment	158
7.15	The photon-dominated Universe	159
7.16	White dwarfs, neutron stars, and black holes	161
8	Calculation and computation	163
8.1	The Ising model	163
8.1.1	Magnetism	164
8.1.2	Binary alloys	165
8.1.3	Liquids, gases, and the critical point	166
8.1.4	How to solve the Ising model	166
8.2	Markov chains	167
8.3	What is a phase? Perturbation theory	171
	Exercises	174
8.1	The Ising model	174
8.2	Ising fluctuations and susceptibilities	174
8.3	Waiting for Godot, and Markov	175
8.4	Red and green bacteria	175
8.5	Detailed balance	176
8.6	Metropolis	176
8.7	Implementing Ising	176
8.8	Wolff	177
8.9	Implementing Wolff	177
8.10	Stochastic cells	178
8.11	The repressilator	179
8.12	Entropy increases! Markov chains	182
8.13	Hysteresis and avalanches	182
8.14	Hysteresis algorithms	185
8.15	NP-completeness and kSAT	186
9	Order parameters, broken symmetry, and topology	191
9.1	Identify the broken symmetry	192
9.2	Define the order parameter	192
9.3	Examine the elementary excitations	196
9.4	Classify the topological defects	198
	Exercises	203
9.1	Topological defects in nematic liquid crystals	203
9.2	Topological defects in the XY model	204
9.3	Defect energetics and total divergence terms	205
9.4	Domain walls in magnets	206
9.5	Landau theory for the Ising model	206
9.6	Symmetries and wave equations	209
9.7	Superfluid order and vortices	210
9.8	Superfluids: density matrices and ODLRO	211
10	Correlations, response, and dissipation	215
10.1	Correlation functions: motivation	215
10.2	Experimental probes of correlations	217
10.3	Equal-time correlations in the ideal gas	218
10.4	Onsager's regression hypothesis and time correlations	220
10.5	Susceptibility and linear response	222
10.6	Dissipation and the imaginary part	223
10.7	Static susceptibility	224
10.8	The fluctuation-dissipation theorem	227
10.9	Causality and Kramers-Kronig	229
	Exercises	231
10.1	Microwave background radiation	231
10.2	Pair distributions and molecular dynamics	233
10.3	Damped oscillator	235
10.4	Spin	236
10.5	Telegraph noise in nanojunctions	236
10.6	Fluctuation-dissipation: Ising	237
10.7	Noise and Langevin equations	238
10.8	Magnetic dynamics	238
10.9	Quasiparticle poles and Goldstone's theorem	239
11	Abrupt phase transitions	241
11.1	Stable and metastable phases	241
11.2	Maxwell construction	243
11.3	Nucleation: critical droplet theory	244
11.4	Morphology of abrupt transitions	246
11.4.1	Coarsening	246
11.4.2	Martensites	250
11.4.3	Dendritic growth	250
	Exercises	251
11.1	Maxwell and van der Waals	251
11.2	The van der Waals critical point	252
11.3	Interfaces and van der Waals	252
11.4	Nucleation in the Ising model	253
11.5	Nucleation of dislocation pairs	254
11.6	Coarsening in the Ising model	255
11.7	Origami microstructure	255
11.8	Minimizing sequences and microstructure	258
11.9	Snowflakes and linear stability	259
12	Continuous phase transitions	263
12.1	Universality	265
12.2	Scale invariance	272
12.3	Examples of critical points	277
12.3.1	Equilibrium criticality: energy versus entropy	278
12.3.2	Quantum criticality: zero-point fluctuations versus energy	278
12.3.3	Dynamical systems and the onset of chaos	279
12.3.4	Glassy systems: random but frozen	280
12.3.5	Perspectives	281
	Exercises	282
12.1	Ising self-similarity	282
12.2	Scaling and corrections to scaling	282
12.3	Scaling and coarsening	282
12.4	Bifurcation theory	283
12.5	Mean-field theory	284
12.6	The onset of lasing	284
12.7	Renormalization-group trajectories	285
12.8	Superconductivity and the renormalization group	286
12.9	Period doubling	288
12.10	The renormalization group and the central limit theorem: short	291
12.11	The renormalization group and the central limit theorem: long	291
12.12	Percolation and universality	293
12.13	Hysteresis and avalanches: scaling	296
A	Appendix: Fourier methods	299
A.1	Fourier conventions	299
A.2	Derivatives, convolutions, and correlations	302
A.3	Fourier methods and function space	303
A.4	Fourier and translational symmetry	305
	Exercises	307
A.1	Sound wave	307
A.2	Fourier cosines	307
A.3	Double sinusoid	307
A.4	Fourier Gaussians	308
A.5	Uncertainty	309
A.6	Fourier relationships	309
A.7	Aliasing and windowing	310
A.8	White noise	311
A.9	Fourier matching	311
A.10	Gibbs phenomenon	311
	References	313
	Index	323

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