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Subsections
Compare the partition function and grand partition function:
we see that
| |
(2) |
We can express one partition function through other!
Two connections between canonical and grand canonical ensembles:
- 1.
- A system in canonical ensemble can be divided into parts.
Each part can be considered in grand canonical ensemble with
!
- 2.
- A system in grand canonical ensemble can be considered as
a set of systems, each of them in canonical ensemble with
These two ways reflect a deep connection between the function and its
Legendre Transform.
Through this method we can connect any two ensembles!
In our calculations we often substituted for
N. But in canonical ensembles , in grand canonical
ensemble . Did we cheat?
- Theorem:
- suppose we have a sum
| |
(3) |
and
Then at
- Proof:
- For any N
Take log:
At we have , and the
theorem is proven.
- Consequence:
- In the sum (2) all terms are . In the thermodynamic limit
with such N that
or
Your teachers did not cheat you--at least in the thermodynamic
limit!
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Up: Ensembles and Thermodynamic Potentials
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© 1997
Boris Veytsman
and Michael Kotelyanskii
Tue Sep 9 22:39:08 EDT 1997