Grand Canonical Ensemble and Criteria for Equilibrium

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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

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

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

Formula for Number of Particles
Number of particle by definition Since

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

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

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

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