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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

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**Problem 7.2 From definitions of canonical ensemble averages calculate**

and and show that

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**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

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**Formula for Number of Particles**

Number of particle by definition Since

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**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

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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

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**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

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**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

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