Ch3. The Canonical Ensemble
• Microcanonical ensemble: describes isolated systems of
known energy. The system does not exchange energy
with any external system so that (N,V,E) are fixed.
• Macrostate is determined by (N, V, E)
• Distinct microstate number W(N,V,E) or G(N,V,E;D) so
that entropy S=k lnW(N,V,E) or S=k lnG(N,V,E;D)
• Microcanonical ensemble
  q , p   const
0
G N ,V , E , D    /  0
If E-D/2 H(q,p) E+D/2
otherwise
 – allowed volume in phase space;
0 –volume of one microstate (~h3)
1
The Canonical Ensemble
• Describes a system of known temperature, rather than
known energy. The energy is variable due toi exchange
of energy with external system at common T.
• Macrostate is determined by (N, V, T). T is common
between the system and reservoir.
exchange E
system
Fixed T at
equilibrium
Heat reservoir
The probability Pr


Pr is the probability that a system, at any time t,
is found to be in one of the states with energy
value Er.
The energy E can be any value from 0 to infinity. Pr is
the probability that E=Er.
The entropy of the system is given by
S   k ln Pr   k  Pr ln Pr
r
with
P
r
1
r
E 
E
r
r
Pr
3
Pr in microcanonical
ensemble
Er  E  E
 fixed E
Pr  1 / W  N , V , E 
P
r
r

1
W
Each mocrostate is equally accessible
1
r
S   k  Pr ln Pr   k 
r
r
 1 
ln    k ln W
W W
1
4
3.1 Equilibrium between a
system and a heat reservoir




Consider a system A immersed in a very large heat reservoir A’.
They are thermal equilibrium with common T at time t.
System A: in a state of Er
Reservoir A’: in a microstate of Er’
Composite system A(0) (=A+A’): Conservation of energy
Er+Er’=E(0) = const
A’
(Er’;T)
W’(Er’) – number of reservoir’s states
with given energy value of Er’
Pr – Probability that system A is
found to have E=Er.
Pr ~ W’(Er’) = W’(E(0) -Er)=e-bEr
A
(Er;T)
Remark

Normalization
Pr 
e

 E r /( kT )
e
 E r /( kT )
• When a system is in thermal
equilibrium with external
reservoir, its energy Er is
exchanged with reservoir.
r
with

Pr  1
r
b=kT
• The system energy Er can be
any values from 0 to infinity.
• The probability that the
system has an energy E=Er
is Pr given by …
6
3.2 A system in the canonical
ensemble

An ensemble: N identical system (i=1,2, …N)
sharing a total energy E=NU.




U =E/N is the average energy per system in the
ensemble.
The number of different ways of E distributes
among N members according to the mode {nr}
(distribution set) W{nr}~
The average number of systems having energy Er
is <nr> ~
The most probable distribution set {nr*} – to
make W{n*r} maximum
Canonical distribution
For a single system, the probability that
the system has energy E=Er is Pr

Pr 
nr
e



 bEr
e
 bEr
r
with
P
r
1
r
b=kT
8
3.3 Physical significance of the various
statistical quantities in the canonical
ensemble

The canonical distribution-
the probability that the
system has energy E=Er
Pr 
nr

e

 bEr
e
e

 bEr
 bEr
Q N (V , T )
b=kT
r

Partition function of the system with (N,V,T)
Q N (V , T ) 
e
 bEr

e
r

 E r / kT
r
The average energy of the system with (N,V,T)
U  Er 

r
Ere
 bEr


b
ln Q N (V , T ) 
Physical significance in the canonical
ensemble

Helmholtz free energy
A  U  TS   kT ln Q N (V , T )

Entropy
S ( N , V , T )  k ln Pr   k b




Specific heat
Gibbs free energy
Pressure
Chemical potential

b
ln Q N (V , T )  k ln Q N (V , T )
Example – single quantum oscillator

The state of a single oscillator
 n    ( n  1 / 2 ),




n  0 ,1, 2 ,3,....
Partition function (N=1)
Average energy of one oscillator
The average number of quantum
The entropy