Download PPTX

PHY 770 -- Statistical Mechanics
11 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 2
Introduction to Probability and Its Role in Statistical Physics
1.
2.
3.
4.
2/6/2014
Probability distribution functions
Central limit theorem
Liouville theorem and its quantum equivalent
Relationship between entropy and notions from
probability theory
PHY 770 Spring 2014 -- Lectures 7 & 8
1
2/6/2014
PHY 770 Spring 2014 -- Lectures 7 & 8
2
Some ideas from probability theory
Notation - Random variable :
X
Possible value of X :
x
Probability of outcome x : PX  x 
Properties of probability function
Discrete case; x  xi
PX  xi   0
i  1,2,....N
N
 P x   1
i 1
2/6/2014
X
i
PHY 770 Spring 2014 -- Lectures 7 & 8
3
Some ideas from probability theory -- continued
N
Average value :
X   xi PX  xi 
i 1
N
Moment value :
X
n
  xi PX  xi 
n
i 1
Standard diviation : σ X 
X
2
 X
1 2
For a continuous variable x where    x   :
PX  x   0

 P x dx  1
X


Xn 
n
x
 PX x dx

2/6/2014
PHY 770 Spring 2014 -- Lectures 7 & 8
4
Some ideas from probability theory -- continued
Clever use of Fourier transfroms; the characteristic function :



n 1
 X k   eikx   eikx PX x dx  
ik n
Xn
n!
Note that, using the inverse Fourier transform :
1
PX  x  
2

ikx
e
  X k dk

Other useful relationships :
1
d n X k 
X  n lim
i k 0 dk n
Cummulants : Cn  X 
n
  ik n

 X k   exp 
Cn  X 
 n 1 n!

C1  X   X
C3  X   X
2/6/2014
C2  X   X 2  X
3
3 X
2
X 2 X
2
3
PHY 770 Spring 2014 -- Lectures 7 & 8
5
Some ideas from probability theory -- continued
Example :
2

1 x2
PX  x   

0

 X k   e ikx 
for x  1
otherwise

ikx
e
 PX x dx 

2

1
ikx
2
e
1

x
dx 

1
2
J1 k 
k
Using the assending series expansion :
2
1 k2 1 k4
 X k   J1 k   1 

k
4 2! 8 4!
From :  X k   e ikx 



n 1
ikx
e
 PX x dx  
ik n
Xn
n!
X  X 3  X 5  ....  0
X
2/6/2014
2
1

4
X
4
1

8
PHY 770 Spring 2014 -- Lectures 7 & 8
6
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?
1

PX  x    2a

0
for d  a  x  d  a
otherwise
For a single step :  X k   e ikx
2/6/2014
1
ikx
ikd sin ka 
  e ikx PX  x dx 
e
dx

e
2a d a
ka


PHY 770 Spring 2014 -- Lectures 7 & 8
d a
7
Some ideas from probability theory – continued
Characteristic function for a single step :
sin ka 
 X k   eikx  eikd
ka
Characteristic function for S ( N steps) :


S k    X k N   eikd
sin ka  

ka 
N
Expansion in powers of k :
sin ka  
1 2 2 2

1
2


S k    eikd

1

iNd
k

Na

N d k  ...


ka 
2

6

1
 S  Nd
S 2  Na 2  N 2 d 2
3
N
2
sS  S2  S 
a
3
N
2/6/2014
PHY 770 Spring 2014 -- Lectures 7 & 8
8
Some ideas from probability theory – continued
Typical probability functions
 Binomial distribution
 Gaussian distribution
 Poisson distribution
 Binomial distribution
Consider a process with 2 outcomes:
0
with probability p
1 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.
PN n1  
2/6/2014
N!
p n1 q  N  n1 
n1! N  n1 !
PHY 770 Spring 2014 -- Lectures 7 & 8
9
Binomial distribution continued:
PN n1  
N!
p n1 q  N  n1 
n1! N  n1 !
Note that :
N
N
 PN n1   
N!
N
p n1 q  N  n1   p  q   1
n1  0 n1! N  n1 !
n1  0
Can show that :
n1  Np
n1
2
  Np   Npq
2
s N  Npq
2/6/2014
PHY 770 Spring 2014 -- Lectures 7 & 8
10
Example: Dice throws
On average, how many times must a die be thrown
until “4” appears?
Let p  probability of getting 4 on one throw ( p  1 )
6
Let q  probability of not getting 4 on one throw
Probability of first getting 4 on nth throw : Pn  pq n 1

d  n
Mean # of throws : m   npq  p  q
dq n 0
n 1
d 1
1
1
p
p

2
dq 1-q
1  q  p
2/6/2014
n 1
PHY 770 Spring 2014 -- Lectures 7 & 8
11
Gaussian distribution
Consider the binomial distribution in the limit of large N
and large pN:
N!
PN n  
p n q  N n 
n! N  n !
n  Np
n
Stirling approximation : n! 2n  
e
n
 n  n 2 
 n  n
1

PN n   PN  n exp 
exp 
2



2
Npq
2
s
2

s



2/6/2014
PHY 770 Spring 2014 -- Lectures 7 & 8

2




12
Poisson distribution
Consider the binomial distribution in the limit of large N
and pN =a<< N:
PN n  
N!
p n q  N n 
n! N  n !
n  Np  a
a n e a
PN n  
n!
a n e a a a
Note that : 
e e  1
n!
n 0

2/6/2014
PHY 770 Spring 2014 -- Lectures 7 & 8
13
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.
a nea
PPoisson n  
n!
PPoisson 0   2 0 e  2  0.135
21 e  2
PPoisson 1 
 0.271
1
22 e 2
PPoisson 2  
 0.271
2
23 e  2
PPoisson 3 
 0.180
3!
24 e 2
PPoisson 4  
 0.090
4!
2/6/2014
PHY 770 Spring 2014 -- Lectures 7 & 8
14
Central limit theorem
Consider N independent stochastic variables Xi, i=1,2,..N.
What is the distribution of their sum
YN=(X1+…XN)/N
Characteristic function for YN :


Y k    dx1....  dx N eik ( x  x ... x
1
2
N
)/ N
PX 1  x1 ....PX N  x N 


  X k / N 
N
 1ks
 1 
 2 N
2
1
PY  y  
2
2/6/2014

2
X
2

 ... 

N
 k 2s X2
 exp 
N 
 2N
2 2

sX
k
-iky


exp
e
dk

 2N


 

PHY 770 Spring 2014 -- Lectures 7 & 8
N
2s X2



 Ny 2
exp 
2
s
2
X




15
Central limit theorem
Consider N independent stochastic variables Xi, i=1,2,..N.
What is the distribution of their sum
YN=(X1+…XN)/N
PY  y  

y2 

exp 
2
2
2s X / N
 2s X / N 
1
Distribution function for Y is a Gaussian distribution
centered at <x> and with variance s X / N
2/6/2014
PHY 770 Spring 2014 -- Lectures 7 & 8
16
Justification of statistical treatment of macroscopic systems
Classical mechanics argument ant the Liouville theorem
Liouville’s theorem:
Imagine a collection of particles obeying the
Canonical equations of motion in phase space.
Let D denote the " distribution" of particles in phase space :
D  Dq1  q3 N , p1  p3 N , t 
Liouville's theorm shows that :
dD
0
 D is constant in time
dt
2/6/2014
PHY 770 Spring 2014 -- Lectures 7 & 8
17
Proof of Liouville’e theorem:
v
D
t
v
Continuity equation :
D
   vD 
t
Note : in this case, the velocity is the 6 N dimensional vector :
v  r1 , r2 ,  rN , p 1 , p 2 ,  p N 
We also have a 6 N dimensional gradient :
   r1 ,  r2 ,   rN ,  p1 ,  p 2 ,   p N

2/6/2014

PHY 770 Spring 2014 -- Lectures 7 & 8
18
D
   vD 
t
3N 



q j D    p j D 
  
p j
j 1 

 q j
3N 
3 N  q
 j p j 
D
D 
  
q j 
p j   D  


p j 
j 1 
j 1 
 q j
 q j p j 
 2H
2H


 
q j p j q j p j  p j q j
q j
2/6/2014
p j

0


PHY 770 Spring 2014 -- Lectures 7 & 8
19
0
3N 
3 N  q
 j p j 
D
D
D 
  
q j 
p j   D  


t
p j 
j 1 
j 1 
 q j
 q j p j 
3N 
D
D
D 
  
q j 
p j 
t
p j 
j 1 
 q j
D 3 N  D
D  dD


q j 
p j  
0
t j 1  q j
p j  dt
2/6/2014
PHY 770 Spring 2014 -- Lectures 7 & 8
20
Complexity and entropy
Microscopic definition of entropy
S ( N , E , n)  k B lnNN E , n 
In this case, we have N particles having a total energy E and a
macroscopic parameter n.
NN E , n  denotes the multiplicity of microscopic states
having the same parameters. Each of these states are
assumed to equally likely to occur.
2/6/2014
PHY 770 Spring 2014 -- Lectures 7 & 8
21
Example:
Suppose you have N spin-1/2 particles. How many
microscopic states does the system have?
For N=10: ↑↓ ↑↑↓ ↓↓↑↓↑
For N=100
total = 210=1024
total= 2100=1030
Now, consider N spin-1/2 particles with n ↑.
N!
NN n  
n! N  n !
2/6/2014
PHY 770 Spring 2014 -- Lectures 7 & 8
22
Spin-1/2 system continued
Recall the binomial distribution : for fixed a and b
a  b 
N
N
N!

a N nb n
n  0 n! N  n !
N
N!
N

 1  1  2 N
n  0 n! N  n !
Fraction of microscopic states for this systerm?
n
1
1
N!
N!  1   1 
FN n   N NN n   N

   
2
2 n! N  n ! n! N  n !  2   2 
2/6/2014
PHY 770 Spring 2014 -- Lectures 7 & 8
N n
23
Spin-1/2 system continued
Fraction of microscopic states for this systerm?
n
1
1
N!
N!  1   1 
FN n   N NN n   N

   
2
2 n! N  n ! n! N  n !  2   2 
For N  
N n
2




n

n
1

FN n  
exp 
2


2
s
2s N
N


where n  N / 2 and s N  N / 2
2/6/2014
PHY 770 Spring 2014 -- Lectures 7 & 8
24
Spin-1/2 system continued
Entropy for this system :
N!
NN n  
n! N  n !

N! 

S ( N , n)  k B ln NN n   k B ln
 n! N  n ! 
Stirling approximation :
N ! 2N N N e  N
ln  N !  N ln  N   N


NN

S ( N , n)  k B ln NN n   k B ln n
N n 
 n N  n  
2/6/2014
PHY 770 Spring 2014 -- Lectures 7 & 8
25
Spin-1/2 system continued

N! 

S ( N , n)  k B ln NN n   k B ln
 n! N  n ! 


NN

S ( N , n)  k B ln NN n   k B ln n
N n 
 n N  n  
N


N
  Nk ln 2 
S ( N , n )  k B ln
B
 n n N  n N  n 


2/6/2014
PHY 770 Spring 2014 -- Lectures 7 & 8
26
Relationship between probability function and entropy
Fraction of microscopic states for systerm
1
1
FN n   PN n  
NN n  
expS ( N , n) / k B 
NN
NN
2/6/2014
PHY 770 Spring 2014 -- Lectures 7 & 8
27
Spin-1/2 system continued – effects of Magnetic field
↓ mH
↑ -mH
H=0
H>0
E   m n H  m N  n H  mNH  2 m n H
E
N
Note that : n  
2 2 mH
2/6/2014
PHY 770 Spring 2014 -- Lectures 7 & 8
28
Spin-1/2 system continued – effects of Magnetic field -- continued
Approximate entropy for this case (fixed E , H );
Stirling approximation :
N
E  N
E 
 ln 

S ( N , E , H )  k B N ln N  k B  



2
2
m
H
2
2
m
H

 

N
E  N
E 
 ln 

 k B  
  2 2 mH 
2
2
m
H

 

2/6/2014
PHY 770 Spring 2014 -- Lectures 7 & 8
29
Spin-1/2 system continued – effects of Magnetic field -- continued
Big leap:
Assume the microscopic entropy function IS the same as
the macroscopic entropy found in classical
thermodynamics
S
1
 

 
 E  H , N T
N
E  N
E 
 ln 

S ( N , E , H )  k B N ln N  k B  



 2 2 mH   2 2 mH 
N
E  N
E 
 ln 

 k B  



2
2
m
H
2
2
m
H

 

 S 


 E  H , N
2/6/2014

E
N
1
kB
2 mH
 
ln
E
T 2 mH
N
2 mH

PHY 770 Spring 2014 -- Lectures 7 & 8






30