Part 2 Basics of Statistical Physics1

7/29/2019 Part 2 Basics of Statistical Physics1

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Perimeter Institute Lecture Notes on Statistical Physics part 2: Basics of Statistical Physics Version 1.7 9/13/09 LeoKadanof1

Part 2: Basics of Statistical PhysicsProbabilities Simple probabilities

averages

Composite probabilitiesIndependent eventssimple and complexMany diceProbability distributions

Statistical MechanicsHamiltonian descriptionAverages from derivativesone and many

Structural InvarianceIntensive and Extensive

GaussianStatistical Variables

Integrals and Probabilities Statistical Distributions Averages

Gaussian random variableApproximate Gaussian integrals

Calculation of Averages and Fluctuations The Result

Going Slowlysums and averages in classical mechanicsmore sums and averageshomework


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Perimeter Institute Lecture Notes on Statistical Physics part 2: Basics of Statistical Physics Version 1.7 9/13/09 LeoKadanof

Simple probabilities (reprise)

2

total probability =1 --> =1

ii.2

cubic dice 6 sides: fair dice --> all probabilities are equal -->

r=1--> z=6--> for all values of=1/6

mutually exclusive events described by =1,2,3,......

probability of getting a side with number is =N/N ii.1number of times turns up=N; total number of events N

N= N

relative probability: relative chance that will turn up =r, e.g. fair dice have r =constant

from r to z= r

normalize (=fix up size) : =r/z


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Averages (reprise)

3

average number on a throw = < >=

= 3.5

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 relative probability? What wouldwe have for the average throw on the die?

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()

ii.3

=1/6

=? = ? This last quantity is called the variance of

average number on a throw= =

(

N

)/N


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Composite Probabilities

4

and are two different kinds of events might describe the temperature on January 1, computed as N /N

might describe the precipitation on December 31, with probabilities

Both kinds of events are complete

The prime indicates that the two probabilities are quite different from one another.

=1

=1

Let be the probability that both will happen. The technical term for

this is ajointprobability. The joint probability satisfies

,

,,

=1

(|) is the probability that event occurs if that we know that event has or willoccur. This quantity is called a conditional probability. It obeys(|) = ,/

Something must happen, implies that

( | )=

1


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Independent Events

5

Physically two events are independent if the outcome of onedoes not affect the outcome of the other. It is a mutual relation,ifis independent of then is independent of.

This can then be stated in terms of conditional probabilities. If (|) isindependent of then we say and are statistically independent. After alittle algebraic manipulation, it follows that the joint probability ,obeys

, =

equivalently, two events are statistically independent, if the number of times bothshow up is expressed in terms of the number of times each one individually shows upas

This can be generalized to the statement that a series of m different events arestatistically independent if the joint probabilities of the outcomes of all these events issimply the product of all the m individual probabilities.

N, = NN

/N

The word uncorrelated is also used to describe statistically independent quantities.


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Simple and Complex

definition: simple outcome: can happen only one way: like 2 coming up when a die in thrown

definition: complex outcome: can happen several ways: like 7 coming up when two dice arethrown.

One should calculate probability of complex outcome as a sum of probabilities of simpleoutcomes.

If the simple outcomes are equally likely, probability of complex outcome is the number ofdifferent simple outcomes times the probability of a single simple outcome. There is lots ofcounting in statistical mechanics. The number of ways that something can happen is oftendenoted by the symbol W. Entropy is given by

Entropy S=k ln W , where k=kB is Boltzmanns constant.


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Many Dice

7

Given two dice independent of one another: one fair and the other one unfair. Howdoes one describe the probabilities of the outcome of throwing both dice together?

what is the chance, rolling two fair dice, that we shall roll an eleven, a seven?

Given two fair dicewhat is the average sum and product of what turns up

Now we roll one hundred dice all at the same time. What is the average of the sum of thedice-values. How can one define a root mean square fluctuation in this value? How big isit?


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Probability Distributions

8

So far we have talked about discrete outcomes. A die may take on one of six possiblevalues. But measured things are often continuous. For example, in one dimension, the

probability that a quantum particle will be found between x and x+dx is given in terms ofthe wave function, . In this context, the squared wave function appears as|(x)|2dx

(x)a probability density. In general, we shall use the notation for a probability density, sayingthat is the probability for finding a particle between x and x+dx. The generalproperties of such probability densities are simple. They are positive. Since the totalprobability of some x must be equal to one they satisfy the normalization condition

(x) dx

+

(x) dx = 1

For example, in classical statistical mechanics, the probability density for finding aparticle with x-component of momentum equal to p is

2

m1/2

exp[

p2

/(2m)]

This is called a Gaussian probability distribution, i.e. one that is based on exp(-x2). Suchdistributions are very important in theoretical physics.


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Statistical Mechanics: Hamiltonian Description (reprise)

9

The first idea in statistical mechanics is that the relative probability of a configurationlabeled by , is given in terms of the Hamiltonian H or equivalently the energy E() in

that the relative probability of such a configuration is exp(-H/T)=exp[-E()/T] where Tis

the temperature measured in energy units. (This is usually written kTwhere k is theBoltzmann constant.) The general formulation of statistical mechanics is then that

= eE()/T / Z

Z= e

E()/T

with

For example a single atomic spin in a magnetic field has a Hamiltonian which can be writtenas H=-B, with B thez component of the magnetic field and being thez component ofthe spin. In this model, this component takes on the values =1. (Most often, one putssubscripts z on the spin and magnetic field.)

The relative probabilities for positive and negative values of the spin are respectively eh and

e-hwith h=B/(kT) so that the partition function is

Z= eh + e-h = 2 cosh h

and the probabilities are

(=1)= eh/zand (=-1)= e-h/Zso that we can calculate

= (=1)(1)+(=-1)(-1)=(eh + e-h )/z=tanh h


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Perimeter Institute Lecture Notes on Statistical Physics part 2: Basics of Statistical Physics Version 1.7 9/13/09 LeoKadanof10

Averages from Derivatives (reprise)

10

z= exp(h) = 2coshh

homework: Read Chapters 1 and 2 in textbook.

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

d(lnz) / dh= exp(h)

/ z=< >= tanhh

note how the second derivative gives the mean squared fluctuations.

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 but not Z itself. The derivatives described above arerespectively called the magnetization, M= and the magneticsusceptibility, , = dM/dH. The analogous first derivative with respect to isminusthe energy. The next derivative with respect to is proportional tothe specific heat, or heat capacity, another traditional thermodynamicquantity.

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

/ z=< ( < >)2 >

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


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One and Many

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Imagine a material with many atoms, each with its own spin. The systemhas a Hamiltonian which is a sum of the Hamiltonia of the different atoms

H= h

=1

N

and a probability distribution

= exp(H) / Z= (1/ Z) exp(h)=1

N

which is a product of pieces which belong to the different atoms. Thedifferent pieces are then statistically independentof one another. Notethat the partition function is

Z=

=1

N

exp(h) = (2coshh

=1

)N = zN

so that the entire probability is a product of N pieces connected with the N atoms

The appearance of a product structure depends only upon having a Hamiltonianwhich is a sum of terms referring to individual parts of the system

Hamiltonian is sum stat mech probability is product statistical independence

{} = [exp(h

) / z]

ii.4


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Structural invariance

12

Note how the very same structure which applies to one atom exp(-H)/Z

carries over equally to many atoms.

This structural invariance is characteristic of the mathematical basisof physical theories. Newtons gravitational theory seemed naturalbecause the same law which applied to one apple equally applies toan entire planet composed of apples.

This same thing works for electromagnetism.

A wave function is the same sort of thing for one electron or many.The structure of space and time has a similar invariance property.Remember that a journey of a thousand miles starts with but a singlestep. The similarity between a single step and a longer distance is akind of structural invariance. This invariance of space is called a scaleinvariance. It is quite important in all theories of space and time.


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Intensive and Extensive

13

Imagine a situation in which we have many independent identical subsystems, callediid by the mathematicians for independent and identically distributed. Then

Z( )=z()N as in equation ii.4. In this kind of system, the free energy, the entropy,and the average energy will be precisely N times that for the single subsystem.

Quantities that are a sum or integral over the entire system are called extensive.One which define local properties or are ratios of extensive quantities are calledintensive. The former are almost always linear in N, the latter are almost alwaysindependent ofN.

The mean square fluctuation in energy is extensive. Why?? The root mean square

fluctuation in energy, []1/2, is not extensive. Why?? In the limiting case inwhich the number of subsystems (particles) is very large, which is the relative size ofthe average energy and the root mean square fluctuation in the energy?

I have just told you the reason why classical thermodynamics does not consider

fluctuations in things. Again, what is this reason?

How big is N, say for the particles in this room?

In terms of real MKS numbers, roughly what is the average kinetic energy of theseparticles and what is the size of its fluctuation?


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Gaussian Statistical Variables

14

A Gaussian random variable, X, is one whichhas a probability distribution which is theexponential of a quadratic in X.

The sum of two statistically independent Gaussian

variables is also Gaussian. How does the variance addup?

A Gaussian variable is an extreme example of astructurally stable quantity.

Central Limit Theorem: A sum of a large number ofindividually quite small random variables need not besmall, but that sum is, to a good approximation, aGaussian variable, given only that the variance of theindividual variables is bounded.

Carl Friedrich Gauss (1777 1855)

A Gaussian distribution has a lot of structurally invariant properties.

(x)= [/(2)]1/2 exp[(x)2/2]

1/ is the variance of this distribution.


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Gaussian integrals and Gaussian probability distributions

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I= dx exp(ax2 / 2+bx+ c)

Gaussian integrals are of the form

with a, b, and cbeing real numbers, complex numbers, or matrices.They are very, very useful in all branches of theoretical physics.

There is a canonical form for Gaussian probability distributions, namely

(X= x) = ( 2)1/2 exp[(x < X>)2 / 2]

For Gaussian probability distributions, there is a very important result:

We define the probability that the random variableXwill take on the value

betweenxandx+dxas (X=x)dxormoresimplyas(x)dx

Notice how the that appears in the numerator of the probability distribution reappears in thedenominator of the average.

produced by completeing the square. Here 1/is the variance and is the average of therandom variable, X.

prove this< exp(iqX) >= exp(iq< X>)exp[q

2 / (2)] ii.5


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Gaussian Distributions

16

According to Ludwig Boltzmann (1844 1906) andJames Clerk Maxwell (1831-1879) theprobability distribution for a particle in a weakly interacting gas as is given by

Here, the potential holds the particles in a box of volume , so that U is zero inside a boxof this volume and infinite outside of it. As usual, we go after thermodynamic properties bycalculating the partition function,

(p,r) = (1/ z)exp(H)H= [px

2

+ py2

+ pz2

] / 2m+U(r)

z= [ dpexp(p2 / (2m ))]3 = (2m/ )3/2

In the usual way, we find that the average energy is 3/(2) = (3/2)kT. The classical result is theaverage energy contains a term 1/2kT for each quadratic degree of freedom. Thus a harmonicoscillator has =kT.

Hint for theorists: Calculations of Z (or of its quantum equivalent, the vacuum energy) areimportant. Once you can get this quantity , you are prepared to find out most other thingsabout the system.

ii.6


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The usual way

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Z= d3p d3r exp(-[p2/(2m)+U(r)])=(2m/)3/2

Let = [p2

/(2m)+U(r)]

ln Z/ =-(1/Z) d3p d3r exp(-)= -=3/(2)=(3/2)kT

(The usual way)2

2 ln Z /2= -/= ???

Gaussian Averages

Let one particle be confined to a box of volume . Let U(r) be zero inside the boxand + infinity outside. Then, in three dimensions

Let N particles be confined to a box of volume . What is the RMS fluctuation in thepressure?


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Rapidly Varying Gaussian random variable

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Later on we shall make use of a time-dependent gaussian random variable, (t). Inits usual use, (t) is a very rapidly varying quantity, with a time-integral whichbehaves like a Gaussian random variable. Specifically, it is defined to have twoproperties:

< (t)>=0

X(t) = du (u) is a Gaussian random variable with variance |s-t|.

Here defines the strength of the oscillating random variable.

st


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Approximate Gaussian Integrals

19

It is often necessary to calculate integrals like

in the limit as M goes to infinity. Then the exponential varies over a wide range and theintegral appears very difficult. But, in the end its easy. The main contribution will come atthe maximum value of f in the interval [a,b]. Assume there is a unique maximum and thesecond derivative exists there. For definiteness say that the maximum occurs atx=0, witha


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The usual way: in generalLets start from a Hamiltonian on a Lattice

W{}=-H{} = hr

r

r + Krs

This Hamiltonian defines what is called the Ising model. The first sumis a sum over all lattice sites, r. The second sum is a sum over

nearest neighbors. The field, hr, which depends upon r, multiplies aspin variable which is different, of course, different on each site. Thenotation {} says that these things depend upon many spin variables.You can assume that the spin variables take on the value +1 or -1 ifyou want, but the argument is very general and the result does notdepend upon what might be. Start from Z =Tr exp W{}where Tr means a summation over all possible values of all the spinvariables. It is a repeat of the argument that we have given before to

say that

= ln Z/ hr

The partial derivative means that we hold all other hs constant. The second derivative isgiven by

- = ]> =2

ln Z/h

s h

r

=

/h

r

Calculation of Averages and Fluctuations: Reprise


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To see this we go slowly

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/ hr = Tr[s exp(W{})]/Tr exp(W{}) /hr

= Tr[s r exp(W{})]/Z

-Tr[s exp(W{})] Tr[r exp(W{})]/Z2

=< s r > - < s > < r >

This derivative can act in two places, one here and one here. In bothcases the derivative bring down a factor ofr . Since the denominator

is called Z, we have that this expression is equal to

which then simplifies to

This expression is called the spin-spin correlation function. It expresseshow fully the fluctuations in the values of the spins in the two places arecorrelated. In general this kind of correlation falls off with distance, buthow it falls off tells us a lot about what is going on.

=


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Sums and Averages in Classical Mechanics

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The probability distribution for a single particle in a weakly interacting gas as is given by

Here, the potential holds the particles in a box of volume , so that U is zero inside a boxof this volume and infinite outside of it.The partition function, is

(p,r) = (1/ z)exp(H)

H= [px2

+ py2

+ pz2

] / 2m+U(r)

z= [ dpexp(p2 / (2m ))]3 = (2m/ )3/2

The average of any function ofp and r is given by

= dp dr(p,r) g(p,r)

Since there are N particles in the system N dp dr(p,r) is the number of particles which haveposition and momentum within dp drabout the phase space pointp,r. The quantity

N (p,r)=f(p,r) is called the distribution function. The totalamount of the quantity represented byg(p,r) is given in terms of the distribution function as

total amount ofg = dp drf(p,r) g(p,r)

Example: We calculated the average energy < p2/(2m) >=3 k T/2= dp dr(p,r)p2/(2m)The total energy in the system is dp drf(p,r)p2/(2m)= 3N k T/2.


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More sums and averages

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The normalization condition for the probability is dp dr(p,r) ==1

The normalization for the distribution function is dp dr f(p,r) =N

The pressure, P, is defined as the total momentum transfer to a wall per unit of area and unit oftime. Call these dA and dt. Since a low density gas is the same all over, the number hitting isthe number within the distance px/m dt of the area, for px >0, and hence the number within the

volume px/m dt dA which is dp f(p,r) px/m dt dA with the integral covering all ps with thecondition that px >0. If a particle hits the wall and bounces back it transfers momentum 2px.

Therefore the total momentum transferred is dp f(p,r) px/m dt dA 2px once again with the

condition that px >0. An integral over all momenta would give a result twice as large. In theend we get that the pressure is

P= dp f(p,r) px2/mwhich is then NkT as we knew it would be.

The partition function is the sum over all variables of exp(-H). For large N, it can beinterpreted as W exp(-) , where W is the number of configuration which enter.Boltzmann got W in terms of the entropy as ln W=S/k. We put together previous results andfindW exp(-3N/2)= zN = N (2 m kT)3N/2 so that S/k= N [ln +3 ( ln (2 e m kT))/2 ]


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Homework

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Work out the value of the pressure for a classical relativistic gas with H=|p|c.

Do this both by using kinetic theory, as just above, and also by differentiatingthe partition function.

Statistical Mechanics started with Boltzmanns calculation off(p,r,t) for a gas of N particles in

a non-equilibrium situation. He calculated a result for the entropy as a logarithm of the

number of configurations, or in terms of an integral roughly of the form dp f(p,r) ln f(p,r).From our knowledge of equilibrium statistical mechanics how might we guess that this integralwas related to the entropy?

The length of a random walk ofN steps is of the form of a sum ofN iid variables jeach of which takes on the values 0 and 1 with equal probability. What is the

probability that a walker who takes 1000 steps will end up at a point at least 30 stepsaway from her starting point? (A relative accuracy of two decimal digits will suffice.)

The Hamiltonian for N particles in a fluid is H=p2 /(2m) +

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Part 2 Basics of Statistical Physics1