Subsections

# Equivalence of Ensembles

## Canonical and Grand Canonical Ensembles Revisited

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!

## Method of Maximal Term Or Did Your Teachers Cheat You?

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!

Next: Homework Problems Up: Ensembles and Thermodynamic Potentials Previous: General Case: How to

© 1997 Boris Veytsman and Michael Kotelyanskii
Tue Sep 9 22:39:08 EDT 1997