1. Grand Canonical Ensemble and Criteria for Equilibrium
Lecture 7
Grand Canonical Ensemble and Criteria for Equilibrium
Problem 7.2
Grand Canonical Ensemble
Entropy and equilibrium

2. Problem 7.2 From definitions of canonical ensemble averages calculate
and
and show that

3. Probabilities in Grand Canonical Ensemble
Number of particle can vary
Where Ξ is the grand canonical partition function
Which can be also written as
Where Q(N) is canonical partition function for system with N particles

4. Formula for Number of Particles
Number of particle by definition
Since

5. Entropy - 1 Consider Quantity S’ in terms of state probabilities
Where k is a constant.
What is S’ when p1=1 and rest of p=0?
In general pis are many and very small and thus S’ is large and positive

6. Entropy - 2
Consider differential of S’ in terms of state probabilities
But since therefore
Consider changing two states, j and k probabilities a bit - to conserve total probability dpj=-dpk

7. Entropy - 3 In microcanonical ensemble all pis are the same
By differentiating the above equation second time with respect to pj
Thus S’ has a maximum in equilibrium for isolated system. Also, since pi =1/Ω, for microcanonical ensemble

8. Other thermodynamic functions and equilibrium
In microcanonical we showed that
reaches maximum in equilibrium. Following similar procedures one can show that
has minimum in equilibrium for canonical ensemble, and
has maximum for grand canonical ensemble