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2.
Elements of Ensemble Theory
1. Phase Space of a Classical System
2. Liouville’s Theorem & Its Consequences
3. The Microcanonical Ensemble
4. Examples
5. Quantum States & the Phase Space
At equilibrium:
Each macrostate represents a huge number of microstates.

Observed values ~ time averaged over microstates.
Ensemble theory:
Ensemble average = Time average
2.1.
Phase Space of a Classical System
Hamiltonian formulism of mechanics:
Each dynamical state of a system of N particles is specified by 3N
position coordinates { qi }, and 3N momentum coordinates { pi },
i.e., by a point { qi , pi } in the 6N –D phase space of the system.
Time evolution of the system is described by a path in phase space satisfying
qi 
H
 pi
pi  
H
 qi
For a conservative system, its phase space
trajectory is restricted to the hypersurface
For a system in thermal equilibrium with a
heat reservoir, its phase space trajectory is
restricted to the “hypershell”
i  1,2,
,3N
H  qi , pi   E
H  qi , pi    E  E , E  E

Ensemble of a system:
Set of identical copies of a system that includes every distinct state that satisfies
the given boundary conditions and gives rise to the given macrostate.
In the thermodynamical limit ( N, V   , N / V = finite ) ,
the phase points of the ensemble form a continuum.
The ensemble average of a dynamical function f is given by
f
d


3N
q  d 3 N p f  q, p    q, p, t 
d
3N
q d
An ensemble is stationary if
3N
p   q, p, t 

0
t

 = density function
 f  is t – indep.
Ensemble of a system in equilibrium must be stationary.
Liouville’s Theorem & Its Consequences
2.2.
d   qi , pi , t   
 
  H   H 
 


 
qi 
pi  
 


dt
t

q

p

t

q

p

p

q
i 
i 
i
i
i
i
i
i 


d  

  , H
dt
t

  f H  f H 
f
,
H



 

 pi  qi 
i   qi  pi
where the Poisson bracket is defined as :
# of phase points are conserved
Liouville’s theorem

d
0
dt
The current density of phase points is a 6N-D vector
j v
v   qi , pi 
j v

v   qi , pi 
   qi   pi 
 
 

q

pi   
  j  

 
i
 pi 
 pi 
i   qi
i   qi
where
  qi  pi
i   q   p
i
 i
  qi  pi 
i   q   p     , H
i 
 i

  H
 H 

  
0
 i   qi  pi  pi  qi 
Hence, the Liouville eq. is just the equation of continuity of the phase points :
d  

  j  0
dt
t

0
For a system in equilibrium
t

  H  H 

 , H    
0

q

p

p

q
i 
i
i
i
i 

  H  H 

 , H    
0

q

p

p

q
i 
i
i
i
i 
Solution 1:
 q, p   
 const
 0
  q, p   
otherwise
  region of acceptible microstates in phase space.
Principle of equal a priori probability :
Solution 2:

1


d 3 N q d 3 N p f  q, p  Microcanonical
ensemble
  q, p     H  q, p  
 , H   
i
Chap 3:
f 
d   H H H H 


0
d H   qi  pi  pi  qi 

H 

kT


  q, p   exp  
Canonical
ensemble
2.3.
 const
  q, p   
 0
The Microcanonical Ensemble
1.   Hypersurface
 q, p   
H  q, p   E
H  q, p   E 
2.   Hypershell
otherwise
 f   ensemble average of f
= time average of  f 
for systems in eqm.
=  time average of f 
2 av are indep
= long time average of f over 1 ensemble member
= f measured
Let 0  fundamental volume of one microstate.
S  k ln   k ln

0
Then   
0
1

2
2.4.
Examples
   d 3N q d 3N   q, p    d 3 N q d 3 N p
Microcanonical

Classical ideal gas:
 d
 d
3N
3N
q V N
p
3N
pr2

E

2m
2
r 1

Surface area of an n-D sphere (App.C) is
R  2mE


d 3N
d 3N p  
2 mE 

yr2
 m
d 3N y
r 1
2  n /2
Sn  R  
R n 1
  n / 2
1 
1 


R  2m  E     2m  E   
2 
2 


2  3 N /2
 3 N 1 /2 
p
2
mE



  3N / 2 

3N
E 

2m 

E

2m
N  2 mE 
 V
  3N / 2 
3 N /2


E
 2 mE 
 V N
  3N / 2 
3 N /2
Sec 1.4 :

E
3N     V 

  
2  E   h3 
N
 2 mE 
 3N / 2  !
3 N /2

Volume of state per degree of freedom = h
0 


 h3N
Single Free Particle Confined to V
1  P  
1  p  
P2
E
2m
p2

2m
1
1 4 3
3
3
d
q
d
p

V
P
3 
3

p

P
h
h
3
d 1
dp


p 
1
2
V
4

p
p  g  p  p
3
h
1  E  
1    
V 4
3/2
2
mE


h3 3
d 1
2 V
  3  2m 3/2  1/2   a   
d
h
Simple Harmonic Oscillator
p2 1 2
H
 kq
2m 2

q  A cos  t   
k
m

p  m  A sin  t   
E

q2
p2

1
2
2
A  m  A
1
m  2 A2
2

q2
p2

1
2
2E / m 
2mE
2E
m 2
2mE 
1  E  

1    
d 1


 

d


h
Ellipse
E

 n    n  
2
1


2.5.
Quantum States & the Phase Space
Phase space volume of 1 quantum state may
be estimated from the uncertainty principle:
Among the first to suggest 0 


 h DoF are Tetrode, Sackur, & Bose.
p
Bose (black-body radiation) :
q p 
h
c
for photons
2
4

h

h

3




   d q  d p  V 4

      h V c 3 
 c   c 
3
3
2
4 2
Cf. Rayleigh’s number of modes V

3
c

0  h 3
Caution: Above formulae consider only a single polarization component.