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Statistical Mechanics:
Entropy, Order Parameters
and Complexity
James P. Sethna, Physics, Cornell University, Ithaca, NY
c
April 19, 2005
B
b
Q
F
Alcohol
E
Two-Phase Mix
E+ E
Oil
Water
Electronic version of text available at
http://www.physics.cornell.edu/sethna/StatMech/book.pdf
Contents
1
Why Study Statistical Mechanics?
3
Exercises ..............................
7
1.1
QuantumDice.....................
7
1.2
Probability Distributions. . . . . . . . . . . . . . .
8
1.3
Waitingtimes. ....................
8
1.4
Stirling’s Approximation and Asymptotic Series. .
9
1.5
RandomMatrixTheory................ 10
2
Random Walks and Emergent Properties 13
2.1 Random Walk Examples: Universality and Scale Invariance 13
2.2 TheDiffusionEquation ................... 17
2.3 CurrentsandExternalForces................. 19
2.4 SolvingtheDiffusionEquation ............... 21
2.4.1 Fourier ........................ 21
2.4.2 Green ......................... 22
Exercises .............................. 23
2.1
Random walks in Grade Space. . . . . . . . . . . .
24
2.2
Photon diffusion in the Sun. . . . . . . . . . . . . .
24
2.3
Ratchet and Molecular Motors. . . . . . . . . . . .
24
2.4
Solving Diffusion: Fourier and Green.
. . . . . . .
26
2.5
Solving the Diffusion Equation. . . . . . . . . . . .
26
2.6
FryingPan ...................... 26
2.7
ThermalDiffusion................... 27
2.8
PolymersandRandomWalks. ........... 27
3 Temperature and Equilibrium 29
3.1 The Microcanonical Ensemble . . . . . . . . . . . . . . . . 29
3.2 The Microcanonical Ideal Gas . . . . . . . . . . . . . . . . 31
3.2.1 Configuration Space . . . . . . . . . . . . . . . . . 32
3.2.2 MomentumSpace .................. 33
3.3 WhatisTemperature?.................... 37
3.4 Pressure and Chemical Potential . . . . . . . . . . . . . . 40
3.5 Entropy, the Ideal Gas, and Phase Space Refinements . . 44
Exercises .............................. 46
3.1
EscapeVelocity. ................... 47
3.2
TemperatureandEnergy. .............. 47
i
ii
CONTENTS
3.3
Hard Sphere Gas . . . . . . . . . . . . . . . . . . .
47
3.4
Connecting Two MacroscopicSystems........ 47
3.5
GaussandPoisson................... 48
3.6
MicrocanonicalThermodynamics.......... 49
3.7
Microcanonical Energy Fluctuations. . . . . . . . .
50
4
Phase Space Dynamics and Ergodicity 51
4.1 Liouville’s Theorem . . . . . . . . . . . . . . . . . . . . . 51
4.2 Ergodicity........................... 54
Exercises .............................. 58
4.1
The Damped Pendulum vs. Liouville’s Theorem. .
58
4.2
Jupiter! and the KAM Theorem
. . . . . . . . . .
58
4.3
InvariantMeasures. ................. 60
5
Entropy 63
5.1 Entropy as Irreversibility: Engines and Heat Death . . . . 63
5.2 EntropyasDisorder ..................... 67
5.2.1 Mixing: Maxwell’s Demon and Osmotic Pressure . 67
5.2.2 Residual Entropy of Glasses: The Roads Not Taken 69
5.3 Entropy as Ignorance: Information and Memory . . . . . 71
5.3.1 Nonequilibrium Entropy . . . . . . . . . . . . . . . 72
5.3.2 InformationEntropy................. 73
Exercises .............................. 76
5.1
Life and the Heat Death of the Universe.
. . . . .
77
5.2
P-VDiagram...................... 77
5.3
CarnotRefrigerator.................. 78
5.4
DoesEntropyIncrease? ............... 78
5.5
EntropyIncreases:Diffusion. ............ 80
5.6
Informationentropy.................. 80
5.7
Shannonentropy.................... 80
5.8
EntropyofGlasses................... 81
5.9
RubberBand...................... 82
5.10
DerivingEntropy. .................. 83
5.11
Chaos, Lyapunov, and Entropy Increase. . . . . . .
84
5.12
BlackHoleThermodynamics............. 84
5.13
FractalDimensions. ................. 85
6
Free Energies 87
6.1 The Canonical Ensemble . . . . . . . . . . . . . . . . . . . 88
6.2 Uncoupled Systems and Canonical Ensembles . . . . . . . 92
6.3 GrandCanonicalEnsemble ................. 95
6.4 WhatisThermodynamics? ................. 96
6.5 Mechanics:FrictionandFluctuations............100
6.6 Chemical Equilibrium and Reaction Rates . . . . . . . . . 101
6.7 Free Energy Density for the Ideal Gas . . . . . . . . . . . 104
Exercises ..............................106
6.1
Two–statesystem. ..................107
6.2
BarrierCrossing....................107
To be pub. Oxford UP,
∼
Fall’05
www.physics.cornell.edu/sethna/StatMech/
CONTENTS
iii
6.3
Statistical Mechanics and Statistics. . . . . . . . . 108
6.4
Euler, Gibbs-Duhem, and Clausius-Clapeyron.
. . 109
6.5
NegativeTemperature.................110
6.6
Laplace.........................110
6.7
Lagrange........................111
6.8
Legendre. .......................111
6.9
Molecular Motors: Which Free Energy? . . . . . . 111
6.10
Michaelis-Menten and Hill . . . . . . . . . . . . . . 112
6.11
PollenandHardSquares. ..............113
7
Quantum Statistical Mechanics 115
7.1 Mixed States and Density Matrices.............115
7.2 Quantum Harmonic Oscillator . . . . . . . . . . . . . . . . 120
7.3 BoseandFermiStatistics ..................120
7.4 Non-Interacting Bosons and Fermions . . . . . . . . . . . 121
7.5 Maxwell-Boltzmann “Quantum” Statistics . . . . . . . . . 125
7.6 Black Body Radiation and Bose Condensation . . . . . . 127
7.6.1 Free Particles in a Periodic Box . . . . . . . . . . . 127
7.6.2 BlackBodyRadiation ................128
7.6.3 Bose Condensation . . . . . . . . . . . . . . . . . . 129
7.7 MetalsandtheFermiGas..................131
Exercises ..............................132
7.1
Phase Space Units and the Zero of Entropy. . . . . 133
7.2
Does Entropy Increase in Quantum Systems? . . . 133
7.3
Phonons on a String. . . . . . . . . . . . . . . . . . 134
7.4
CrystalDefects. ...................134
7.5
DensityMatrices....................134
7.6
Ensembles and Statistics: 3 Particles, 2 Levels. . . 135
7.7
Bosons are Gregarious: Superfluids and Lasers
. . 135
7.8
Einstein’sAandB..................136
7.9
Phonons and Photons are Bosons.
. . . . . . . . . 137
7.10
Bose Condensation in a Band.
. . . . . . . . . . . 138
7.11
Bose Condensation in a Parabolic Potential. . . . . 138
7.12
Light Emission and Absorption. . . . . . . . . . . . 139
7.13
Fermions in Semiconductors.
. . . . . . . . . . . . 140
7.14
White Dwarves, Neutron Stars, and Black Holes. . 141
8 Calculation and Computation 143
8.1 What is a Phase? Perturbation theory. . . . . . . . . . . . 143
8.2 TheIsingModel .......................146
8.2.1 Magnetism ......................146
8.2.2 BinaryAlloys.....................147
8.2.3 Lattice Gas and the Critical Point . . . . . . . . . 148
8.2.4 HowtoSolvetheIsingModel. ...........149
8.3 MarkovChains ........................150
Exercises ..............................154
8.1
TheIsingModel....................154
8.2
Coin Flips and Markov Chains. . . . . . . . . . . . 155
c
James P. Sethna, April 19, 2005
Entropy, Order Parameters, and Complexity
iv
CONTENTS
8.3
RedandGreenBacteria...............155
8.4
DetailedBalance....................156
8.5
Heat Bath, Metropolis, and Wolff.
. . . . . . . . . 156
8.6
StochasticCells. ...................157
8.7
TheRepressilator. ..................159
8.8
Entropy Increases! Markov chains. . . . . . . . . . 161
8.9
Solving ODE’s: The Pendulum . . . . . . . . . . . 162
8.10
SmallWorldNetworks. ...............165
8.11
Building a Percolation Network.
. . . . . . . . . . 167
8.12
Hysteresis Model: Computational Methods. . . . . 169
9 Order Parameters, Broken Symmetry, and Topology 171
9.1 IdentifytheBrokenSymmetry ...............172
9.2 DefinetheOrderParameter.................172
9.3 ExaminetheElementaryExcitations............176
9.4 ClassifytheTopologicalDefects...............178
Exercises ..............................183
9.1
Topological Defects in the XY Model. . . . . . . . 183
9.2
Topological Defects in Nematic Liquid Crystals.
. 184
9.3
Defect Energetics and Total Divergence Terms. . . 184
9.4
Superfluid Order and Vortices.
. . . . . . . . . . . 184
9.5
Landau Theory for the Ising model.
. . . . . . . . 186
9.6
BlochwallsinMagnets. ...............190
9.7
Superfluids: Density Matrices and ODLRO. . . . . 190
10 Correlations, Response, and Dissipation 195
10.1 Correlation Functions: Motivation . . . . . . . . . . . . . 195
10.2ExperimentalProbesofCorrelations ............197
10.3 Equal–Time Correlations in the Ideal Gas . . . . . . . . . 198
10.4 Onsager’s Regression Hypothesis and Time Correlations . 200
10.5 Susceptibility and the Fluctuation–Dissipation Theorem . 203
10.5.1 Dissipation and the imaginary part
χ
(
ω
).....204
10.5.2 Static susceptibility
χ
0
(
k
)..............205
10.5.3
χ
(
r
,t
)andFluctuation–Dissipation.........207
10.6 Causality and Kramers Kr¨onig ...............210
Exercises ..............................212
10.1
Fluctuations in Damped Oscillators. . . . . . . . . 212
10.2
Telegraph Noise and RNA Unfolding.
. . . . . . . 213
10.3
Telegraph Noise in Nanojunctions. . . . . . . . . . 214
10.4
Coarse-Grained Magnetic Dynamics. . . . . . . . . 214
10.5
Noise and Langevin equations.
. . . . . . . . . . . 216
10.6
Fluctuations, Correlations, and Response: Ising . . 216
10.7
Spin Correlation Functions and Susceptibilities. . . 217
11 Abrupt Phase Transitions 219
11.1MaxwellConstruction.....................220
11.2 Nucleation: Critical Droplet Theory. . . . . . . . . . . . . 221
11.3 Morphology of abrupt transitions.
. . . . . . . . . . . . . 223
To be pub. Oxford UP,
∼
Fall’05
www.physics.cornell.edu/sethna/StatMech/
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