2/6/2014phy 770 spring 2014 -- lectures 7 & 81 phy 770 -- statistical mechanics 11 am-12:15 pm...

PHY 770 Spring 2014 -- Lectures 7 & 8 12/6/2014

PHY 770 -- Statistical Mechanics11 AM-12:15 PM & 12:30-1:45 PM TR Olin 107

Instructor: Natalie Holzwarth (Olin 300)Course Webpage: http://www.wfu.edu/~natalie/s14phy770

Lecture 7 & 8 -- Appendix A & Chapter 2Introduction to Probability and Its Role in Statistical Physics

1. Probability distribution functions2. Central limit theorem3. Liouville theorem and its quantum equivalent4. Relationship between entropy and notions from

probability theory

http://www.wfu.edu/~natalie/s14phy770


Some ideas from probability theory

xPx

x

X

X : outcome ofy Probabilit

:X of valuePossible

: variableRandom

-- Notation

N

iiX

iX

i

xP

xP

Nixx

1

1

0

,....2,1 case; Discrete

functiony probabilit of Properties


Some ideas from probability theory -- continued

212

1

1

:diviation Standard

:ueMoment val

: valueAverage

XXσ

xPxX

xPxX

X

N

iiX

ni

n

N

iiXi

dxxPxX

dxxP

xP

xx

Xnn

X

X

1

0

: where variablecontinuous aFor



dkkexP

n

XikdxxPeek

Xikx

X

n

nn

Xikxikx

X

2

1

:ansformFourier tr inverse theusing that,Note

!

:function sticcharacteri theansfroms;Fourier tr of useClever

1

323

3

2221

1

0

23

!exp

:Cummulants

lim1

:ipsrelationsh usefulOther

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ikk

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iX

nn

n

X

n

nX

n

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xxxP

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12

1

1

2

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:Example

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!24

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8

1

4

1

0....

! :From

42

53

1

XX

XXX

n

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n

nn

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X


Some ideas from probability theory – continued

Example: Consider a random walk in one dimension for which the walker at each step is equally likely to take a step with displacement anywhere in the interval d-a≤x≤d+a (a<d).Each step is independent of the others. After N steps, the displacement of the walker is S=X1+X2+….XN

What is the average <S> and standard deviation sS?

ka

kaedxe

adxxPeek

adxadaxP

ikdad

ad

ikxX

ikxikxX

X

sin

2

1 :step single aFor

otherwise0

for 2

1


Some ideas from probability theory – continued

ka

kaekk

NSka

kaeek

NikdN

XS

ikdikxX

sin

:steps) ( for function sticCharacteri

sin

:step single afor function sticCharacteri

aN

SS

dNNaSNdS

kdNNakiNdka

kaek

k

S

Nikd

S

3

3

1

...2

1

6

11

sin

: of powersin Expansion

22

2222

2222


Some ideas from probability theory – continuedTypical probability functions

Binomial distribution Gaussian distribution Poisson distribution

Binomial distribution

Consider a process with 2 outcomes:0 with probability p1 with probability q=1-p

For N “trials” of the process, n0 denotes the number outcomes 0 and n1 denotes the number of outcomes 1, with N=n0+n1.

11

!!

!

111

nNnN qp

nNn

NnP


Binomial distribution continued:

Npq

NpqNpn

Npn

qpqpnNn

NnP

qpnNn

NnP

N

NN

n

nNnN

nN

nNnN

221

1

0 1101

111

: thatshowCan

1!!

!

: thatNote

!!

!

1

11

1

11


Example: Dice throws

On average, how many times must a die be thrown until “4” appears?

1

1

1

1

1

: throwsof #Mean

:nth throwon 4 gettingfirst ofy Probabilit

throwoneon 4 gettingnot ofy probabilitLet

)61( throwoneon 4 getting ofy probabilitLet

2

01

1

1

pq

p-qdq

dp

qdq

dpnpqm

pqP

q

pp

n

n

n

n

nn


Gaussian distribution

Consider the binomial distribution in the limit of large N and large pN:

2

22

2exp

2

1

2exp

2! :ionapproximat Stirling

!!

!

nn

Npq

nnnPnP

e

nnn

Npn

qpnNn

NnP

NN

n

nNnN


Poisson distribution

Consider the binomial distribution in the limit of large N and pN =a<< N:

1!

: thatNote

!

!!

!

0

aa

n

an

an

N

nNnN

een

ea

n

eanP

aNpn

qpnNn

NnP


Poisson distribution example

Consider a monolayer thin sheet of gold foil as a target for neutron scattering. Assume that the probability that in any given pulse of the beam the probability that the beam will scatter from the gold nuclei is given by the Poisson distribution with a=2. Determine the probability that n=0 and that n=2.

090.0!4

24

180.0!3

23

271.02

22

271.01

21

135.020

!

24

23

22

21

20

eP

eP

eP

eP

eP

n

eanP

Poisson

Poisson

Poisson

Poisson

Poisson

an

Poisson


Central limit theoremConsider N independent stochastic variables Xi, i=1,2,..N. What is the distribution of their sum YN=(X1+…XN)/N

2

2

2

22

22

2

22

1/)...(

1

2exp

22exp

2

1

2exp...

2

11

/

........

:for function sticCharacteri

1

21

XX

X-ikyY

X

N

N

X

NX

NXXNxxxik

NY

N

NyN

N

kedkyP

N

k

N

k

Nk

xPxPedxdxk

Y

N

N


N

y

NyP

XXY /2

exp/2

12

2

2

Central limit theoremConsider N independent stochastic variables Xi, i=1,2,..N. What is the distribution of their sum YN=(X1+…XN)/N

Distribution function for Y is a Gaussian distribution centered at <x> and with variance NX /


Justification of statistical treatment of macroscopic systemsClassical mechanics argument ant the Liouville theorem

Liouville’s theorem: Imagine a collection of particles obeying the Canonical equations of motion in phase space.

in timeconstant is 0

: thatshows theormsLiouville'

,,

:space phasein particles of on"distributi" thedenote Let

3131

Ddt

dD

tppqqDD

D

NN


Proof of Liouville’e theorem:

t

D

v

v

Dt

Dv

:equation Continuity

N21N21

,,,,,

:gradient ldimensiona 6 a have also We

,,,,,

: vectorldimensiona 6 theis velocity thecase, in this :Note

2121

ppprrr

ppprrrv

N

N

NN


N

j j

j

j

jN

jj

jj

j

N

jj

jj

j

p

p

q

qDp

p

Dq

q

D

Dpp

Dqq

Dt

D

3

1

3

1

3

1

v

022

jjjjj

j

j

j

qp

H

pq

H

p

p

q

q


03

1

3

1

dt

dDp

p

Dq

q

D

t

D

pp

Dq

q

D

t

D

N

jj

jj

j

N

jj

jj

j

N

j j

j

j

jN

jj

jj

j p

p

q

qDp

p

Dq

q

D

t

D 3

1

3

1

0


Complexity and entropy

Microscopic definition of entropy

nEknENS NB ,ln),,( N

In this case, we have N particles having a total energy E and a macroscopic parameter n.

denotes the multiplicity of microscopic states having the same parameters. Each of these states are assumed to equally likely to occur.

nEN ,N


Example: Suppose you have N spin-1/2 particles. How many microscopic states does the system have?

For N=10: ↑↓ ↑↑↓ ↓↓↑↓↑ total = 210=1024For N=100 total= 2100=1030

Now, consider N spin-1/2 particles with n ↑.

!!

!

nNn

NnN N


Spin-1/2 system continued

N

n

NN

N

n

nnNN

nNn

N

banNn

Nba

ba

0

0

211!!

!

!!

!

and fixedfor :ondistributi binomial theRecall

nNn

NNNN nNn

N

nNn

Nnn

2

1

2

1

!!

!

!!

!

2

1

2

1

systerm? for this states cmicroscopi ofFraction

NF



2/ and 2/ where

2exp

2

1

For

2

1

2

1

!!

!

!!

!

2

1

2

1

systerm? for this states cmicroscopi ofFraction

2

2

NNn

nnn

N

nNn

N

nNn

Nnn

N

NN

N

nNn

NNNN

F

NF



!!

!lnln),(

!!

!

:system for thisEntropy

nNn

NknknNS

nNn

Nn

BNB

N

N

N

nNn

N

BNB

NN

nNn

NknknNS

NNNN

eNNN

lnln),(

ln!ln

2! :ionapproximat Stirling

N



!!

!lnln),(

nNn

NknknNS BNB N

2lnln),(

lnln),(

BnNn

N

B

nNn

N

BNB

NknNn

NknNS

nNn

NknknNS

N


Relationship between probability function and entropy

BN

NN

NN knNSnnPn /),(exp11

systermfor states cmicroscopi ofFraction

NN

NF


Spin-1/2 system continued – effects of Magnetic field

H=0 H>0

↓ mH

↑ -mH

H

ENn

HnNHHnNHnE

22 : thatNote

2


Spin-1/2 system continued – effects of Magnetic field -- continued

H

EN

H

ENk

H

EN

H

ENkNNkHENS

HE

B

BB

22ln

22

22ln

22ln),,(

:ionapproximat Stirling

);, (fixed case for thisentropy eApproximat


Spin-1/2 system continued – effects of Magnetic field -- continuedBig leap:

Assume the microscopic entropy function IS the same as the macroscopic entropy found in classical thermodynamics

TE

S

NH

1

,

H

EN

H

ENk

H

EN

H

ENkNNkHENS

B

BB

22ln

22

22ln

22ln),,(

H

EN

H

EN

H

k

TE

S B

NH

2

2ln

2

1

,

2/6/2014phy 770 spring 2014 -- lectures 7 & 81 phy 770 -- statistical mechanics 11 am-12:15 pm...

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