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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 *E*_{A}=*E*_{i}?

- All states in the
*microcanonical ensemble*with the same energy*E*are equally probable. If_{0}*A*_{i}is fixed, only*B*can change(1) - Total energy:
- Probability for
*A*_{i}

**Naïve approach:**- Let us expand
*W*_{B}: **Why it does not work:**- Taylor expansion works
*only*if the next term is smaller than the previous. We need(2) *But**S*_{B}and*E*are extensive variables. The derivative is_{0}*intensive*and does*not*blow up. We are*not*guaranteed that (2) works!^{} **A better way:**- Let us expand
*S*_{B}(*E*) instead: Thermodynamic limit () Large term*S*_{B}(*E*) is canceled by denominator in (1). We are left with the second term._{0} **Answer:**- For the canonical ensemble
(3) *Gibbs formula*.

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

Particular case:

© 1997

Thu Sep 4 21:28:23 EDT 1997