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Lecture 5
Canonical Ensemble
Difficult to control total energy
E
; fixed temperature is more practical.
On the technical side—microcanonical ensemble is difficult to use in practice because of difficulty in
evaluating the volume of phase space or number of states accessible to the system.
Total system can be expressed by a microcanomical ensemble with total Energy
t
E
()
tt
E
“volume” of the phase space accessible to the total system for Energy between
and
t
t
t
E
E
E
The probability that the total system is an element of volume
t
d
of the appropriate phase space is
d
C d
if the energy between
and
t
t
t
E
E
E
=0 otherwise
What is the probability that the subsystem is in
s
d
without specifying the condition of the reservoir?
volume of the phase space of then reservoir
(so that total energy is in
about
).
s
s
r
d
C d
EE
Our task is to evaluate
r
 if we know the subsystem is in
s
d
, how much phase space is accessible to
the reservoir?
(I will discuss a spinsystem example to elaborate)
Entropy of Reservoir
/
r
Sk
r
r
r
S
k n
e
Now
r
t
s
E
E
E
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We may take
st
EE
Expand:
()
(
)
(
)
(
)
rt
r
r
r
t
s
r
t
s
t
SE
S E
S E
E
S E
E
E
Then
(
)
(
)
1
exp
exp
r
t
r
t
rs
t
S E
S E
E
k
k
E
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 Spring '10
 Chakabarti
 mechanics, Energy, Statistical Mechanics

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