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Perimeter Institute Lecture Notes on Statistical Physics part 0: Outline Version 1.6 9/12/09 LeoKadanof

Outline for

Fundamentals ofStatistical Physics

Leo P. Kadanoff

text:Statistical Physics,

Statics, Dynamics, RenormalizationLeo Kadanoff

I also referred often to Wikipedia and found itaccurate and helpful.

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1Saturday, September 12, 2009


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Course Outline

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part number text chapter number title length

(slides)

number of

lectures

1

2

3

4

56

7

8

Fundamentals of StatisticalPhysics

1 Once over Lightly 16 1

2 & 3 Basics 24 2

4 & 13 &15 Quantum Mechanics and Lattices 28 3

5 Diffusion and Hops 24 3

6 & 8 Momentum hops 27 39 Bose and Fermi 16 1

10 Phase Transitions & mean fields 40 2

11 & 12 & 13 &14 &15 Phase Transitions: Beyond mean

fields

42 2

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Part 1: Once over lightly

Concepts which specifically belong to statistical physics

Interesting Physical Science Advances have a Major Statistical ComponentProbabilities: One die

Quantum Stat MechClassical Stat Mech

ThermodynamicsRandom Walk

Brownian DynamicsFrom Quantum to Classical: The Ising model

Averages from Derivatives

Degenerate DistributionsThermodynamic PhasesPhase Transitions

Big Words

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Part 2: StatisticsProbabilities

Simple probabilities

averagesComposite probabilitiesIndependent events

simple and complexMany dice

Probability distributionsStatistical Mechanics

Hamiltonian descriptionStatistical Independence

one and manyStructural Invariance

Intensive and ExtensiveGaussian

Statistical Variables Integrals and Probabilities Statistical Distributions AveragesApproximate Gaussian integrals

Calculation of Averages and Fluctuations The Result Going Slowlysums and averages in classical mechanics

more sums and averageshomework

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art : n a attice: rom uantum to sing to

From Classical Stat Mech to QuantumAll of quantum mechanics

one dimensional lattice from classical to quantumSummaryfrom quantum to classical

the path integral for particles

The Linear Chain Ising model Transfer Matrix Dual Couplings Solution of One-Dimensional Ising ModelOn Quantum Chains Ground state averages Statistical Correlations Average magnetizations

Correlations Correlation Length Bloch WallRenormalization for 1D Ising

Block TransformNew Coupling

2D IsingHigh Temperature Expansion

Low Temperature ExpansionDuality

Specific HeatBlock Transform

Physical quantitiesFlows

Homework5



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Part 4: HoppingFrom Discrete to Continuous

Hopping on a Lattice

Notation: Even and OddFrom one step to many

An example

Continuous Langevin equationAn integration

Gaussian Properties Generating Function A probabilityDiscrete

A probabilitycalculation

we get an answer!fourier transform formulation

binomial theoremHigher Dimension

picturescontinued

probability density One dimensionCurrent

Diffusion equation

Higher DimensionYou cannot go back

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Part 5: Time Dependence

Brownian MotionDefine Situation

Calculate momentumCalculate VarianceCalculate Probability Distribution

Probability Distribution in Classical MechanicsStatistical and Hamiltonian Dynamics

Probability Distributions in Dynamical Systemstime dependence of dynamical systems

calculation set up

calculation continuedcalculation concludedPoisson bracket

generalizing stat mechtime dependence in Hamiltonian systems

one-particle distributionClassical Mechanics

Brownian motion again: toward a unique solutionfriction

collisionscalculation set up

calculation continuedcalculation concluded

a unique probability distributionSummary

Homework

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Perimeter Institute Lecture Notes on Statistical Physics part 0: Outline Version 1.6 9/12/09 LeoKadanof 8

Part 5: Time Dependence,continued

Boltzmann EquationScattering

forwardbackward

all together

Symmetriesdetailed balance-local equilibriumconservation of particle number

H-theoremsign of dissipation termdS/dt

Homework

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Part 6: Bosons and FermionsSecond Quantization

Second Quantization vs Classical

Quantum DescriptionOne Mode

Independent ExcitationsExtreme limit for fermions

Extreme limit for bosonsWaves

Waves= Special Bosonsphotons in cavity

Conserved particles

conserved fermions in a boxconserved bosons in a boxbose transition

Dynamicsfermions

Boltzmann equation for fermionsbosons

References

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Part 7: More is the Same

IssuesAbstract

phases of matterphase diagram water

basic physics in phase transitions Gibbs definition; Ehrenfest classificationMagnetic Phase Diagrama qualitative change in Behavior

from Gibbs to singularities

Mean Field Theorymore is the same

for Ising model Mean Field Theory is Only Partially RightCalculate resultssimplified phase diagram Graph of Order Parameterbehavior in neighborhood of critical point

go after magnetizationgo after energy

go after correlationscritical opalescence

basic equationcorrelation function

susceptability

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Part 7: More is the Same,continued

Mean field theory of fluidsfrom van der Waals to Weiss

van der Waals first effectvan der Waals second effect Many Different Phase Transitions

After van der Waals

Landau Mean Field Theory order parameter generalized generalized mean field scheme recent example why minimizevery M to vary Fh=0, no space variationjump like square root of Tc-T

Summaryorder parameter and free energy were basic

Look ahead:

a worry

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P 8 B d M Fi ld Th


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Part 8: Beyond Mean Field Theory:Less is the Same too

PeriodizationWorries about Mean Field Theory

fluid experiment 1

fluid experiment 2earlier hintstheoretical workuniversalityturbulencemore rumbles

Next Steps

toward the revolutionadditional phenomenology

Block Spinless is the samerenormalization for Ising modelother renormalizationsfields relevant, irrelevant, ...

a worthwhile phenomenologyThe Wilson Revolution

particle RG before Wilsonthe renormalization revolutioncrucial ideas for the revolutionoutcome of revolutionconceptual advances

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Part 8: Beyond Mean ... continued

After the Revolution2d XY modelRG point of view is absorbed into particle physicsCoulomb gasFeigenbaum and routes to chaosoperator product expansionfield theorysummary

Conformal Field TheoryPolyakov, Virasoro algebraFriedan, Qiu, Shenkercorrelation functionsmany exact calculationsquantum gravity

SLE

Schrammcritical shapesensemble of critical shapesensemble of critical shapes

SummaryReferences

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13Saturday September 12 2009

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