Part 1 Overview

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Perimeter Institute Lecture Notes on Statistical Physics: part I: OverviewVersion 1.7 9/13/09 LeoKadanof

Fundamentals ofStatistical PhysicsLeo P. Kadanoff

University of Chicago, USA

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text:Statistical Physics,

Statics, Dynamics, RenormalizationLeo Kadanoff

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


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Course Outline

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part number text chapter number title length(slides)

number oflectures

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 meanfields

42 2

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Perimeter Institute Lecture Notes on Statistical Physics: part I: OverviewVersion 1.7 9/13/09 LeoKadanof3

Part I: Once over lightly

Concepts which specifically belong to statistical physicsInteresting Physical Science Advances have a Major Statistical ComponentProbabilities: One dieQuantum Stat Mech

Classical Stat MechAverages from DerivativesThermodynamicsFrom Quantum to Classical: The Ising modelDegenerate DistributionsThermodynamic PhasesPhase TransitionsRandom WalkBrownian DynamicsBig Words


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Perimeter Institute Lecture Notes on Statistical Physics: part I: OverviewVersion 1.7 9/13/09 LeoKadanof4

Where do we come from?

Undergraduate Institution:

41


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Concepts which specifically belong to statistical physics:Not in quantum mechanics or in Classical Mechanics

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Temperature


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Interesting Physical Science Advances have a MajorStatistical Component

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Bekenstein-Hawking: entropy of black holes

Fluctuation spectrum of 3 degree kelvin background radiation

Bells theorem: statistics of quantum measurements

source of complexity in the universe

probabilities of hearing from civilizations elsewhere in universe

Why do markets crash?

Time Reversal Invariance: Nature of Irreversability

Probabilities of major earth-asteroid collision

Probabilistic interpretation of quantum mechanics and of wave functions.

Is our universe likely?


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Part 2. Start with Probabilities: Dice

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fair dice --> all probabilities are equal -->

average number on a throw =< >=

= 3.5

general rule: To calculate the average of anyfunction f() that gives the probability that what

will come out will be , you use the formula < f() >= f()

Do we understand what this formula means?? How would we describe a loaded die? Anaverage from a loaded die? If I told you that =2 was twice as likely as all the other values,

and these others were all equally likely, what would be the probability? What would we havefor the average throw on the die?

number of times turns up = N; total number of events N

probability of choosing a side with number = =N/Ni.1

total probability =1 --> =1

i.2

i.3

for all values of=1/6

r = relative probability of event . e.g. for fair dice r = const z= r = r/z


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Part 2. Start with Probabilities: Dice

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fair dice --> all probabilities are equal -->

average number on a throw =< >=

= 3.5

general rule: To calculate the average of anyfunction f() that gives the probability that what

will come out will be , you use the formula < f() >= f()

number of times turns up = N; total number of events N

probability of choosing a side with number = =N/Ni.1

total probability =1 --> =1

i.2

i.3

for all values of=1/6

r = relative probability of event . e.g. for fair dice r = const z= r = r/z


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Part 3: Lattices

stable fixed point

unstable fixed point

024 3

432 60

iterations

iterations

flow


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Part 4: Random Walks & Diffusion

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http://particlezoo.files.wordpress.com/2008/09/randomwalk.png
http://particlezoo.files.wordpress.com/2008/09/randomwalk.pnghttp://particlezoo.files.wordpress.com/2008/09/randomwalk.png


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Part 5 : Statistics of Motion

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tf(p,r,t) + (p/m) .r f(p,r,t) -r U(r,t) .p f(p,r,t) = effects of collisions

p

q

p

q

Albert Einstein (1905) explained this dancing by many, many collisions with molecules in fluid

dp/dt=......+ (t)-p/

p=(px, py, pz) = (x,y,z)

(t) is a Gaussian random variable resulting from random kicks produced by collisions. Since

the kicks have random directions =0. Different collisions are assumed to bestatistically independent

=(t-s)j,k


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Part 6: Bose & Fermi

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particle statistics, i.e. the symmetry properties of the particles wave functions, havea major role in determining the behavior of many interesting physical systems. Thisis especially true when the system is degenerate, i.e. there is a sufficiently highdensity of identical particles so that there could be a substantial overlap of the wavefunctions involved. Important degenerate systems include:

for fermions: the electrons in atoms

for conserved bosons:

for non-conserved bosons


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Part 7: Phase Transitions and Mean Fields

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phases of matter:

http://azahar.files.wordpress.com/2008/12/snowflake_.jpg

which symmetries of nature have beenlost in the snowflake?

are they really lost?
http://azahar.files.wordpress.com/2008/12/snowflake_.jpghttp://azahar.files.wordpress.com/2008/12/snowflake_.jpg


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Part 8: After Mean Fields: Big Words

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Universality:In appropriate limits, very different systems can have essentially identical properties

Renormalization

Take advantage of scale invariance and universality to produce a theory of phasetransitions.

Scale InvarianceSystems look the same at different spatial scales


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A start:

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Ising system has as its basic variable a spin, z which takes on the values 1.

We shall use the abbreviation, for this spin.

The behavior of a physical system is described by its Hamiltonian. If we put thisspin in a magnetic field in the z-direction it has a Hamiltonian H=- Bz ,

Statistical Mechanics is defined by a probability. Here the probability is

() =(1/z) exp[-H/(kBT)]=(1/z) exp[-H/(kBT)]= (1/z) exp[ Bz/(kBT)]

We describe this by using the abbreviation, h, for the parameters in the probability

() = (1/z) exp(h ) h= Bz /(kBT)

normalization: total probability =1= (1) + (-1)= (1/z) exp(h)+ (1/z) exp(-h)

therefore z= exp(h)+ exp(-h)=2 cosh h

averageX = =()X

therefore < > = (1)1 + (-1)(-1)= 1/(2 cosh h) {exp (h)-exp(-h)}

= (2 sinh h)/(2 cosh h)= tanh h


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Averages from Derivatives

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z= exp(h) = 2coshh

d(lnz) / dh= exp(h)

/ z=< >= tanhh

note how the second derivative gives the mean squared fluctuationshomework: Read Chapters 1 and 2 in textbook.

show that -d(ln Z) /d = = and d2(ln Z) /d2 =< (-)2> and

All derivatives of the log of the partition function are thermodynamic functions of

some kinds. As I shall say below, we expect simple behavior from the log of Z butnot Z itself. The derivatives described above are respectively called themagnetization, M= and the magnetic susceptibility,, = dM/dH. The analogous

first derivative with respect to is minus the energy. The next derivative withrespect to is proportional to the specific heat, or heat capacity, another traditionalthermodynamic quantity. The derivative of partition function with respect tovolume is the pressure.

d2(lnz) / (dh)2 = ( < >)2 exp(h)

/ z=< ( < >)2 >

=1 < >2=1 (tanh h)2

If lnZ= const + N ln + N ln T, with being the volume, find the average pressureand its fluctuations.


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End Survey: Start More Intensive/Extensive Discussions

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Do you know what intensive and extensive mean in statistical physics?

Part 1 Overview

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Transcript of Part 1 Overview