Subsections

# Canonical Ensemble

## Gibbs formula

Let us take a part of Microcanonical Ensemble M. This part is described by canonical ensemble , if the size of the rest (thermal bath ) tends to infinity.

What is the probability for A to be in the microscopical state i with energy EA=Ei?

• All states in the microcanonical ensemble with the same energy E0 are equally probable. If Ai is fixed, only B can change    (1)
• Total energy:

• Probability for Ai

We want to use thermodynamic limit: , . We hope that the answer does not depend on the nature of the thermal bath.
Naïve approach:
Let us expand WB:

Why it does not work:
Taylor expansion works only if the next term is smaller than the previous.

We need
 (2)
But SB and E0 are extensive variables. The derivative is intensive and does not blow up. We are not guaranteed that (2) works!

A better way:

Thermodynamic limit ()

Large term SB(E0) is canceled by denominator in (1). We are left with the second term.

For the canonical ensemble
 (3)
This is Gibbs formula .
In microcanonical ensemble all states are equal. In canonical ensemble states with lower energy are more equal than others!

Dependence on the thermal bath: only through T. All thermal baths with the same T are equivalent.

## Averaging

Particular case:

Next: Statistical Mechanics and Thermodynamics Up: Probabilities of Macroscopic States. Previous: Entropy and Probabilities

© 1997 Boris Veytsman and Michael Kotelyanskii
Thu Sep 4 21:28:23 EDT 1997