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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 p 1 =1 and rest of p=0? In general p i s 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 dp j =-dp k

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Entropy - 3 In microcanonical ensemble all p i s are the same By differentiating the above equation second time with respect to p j Thus S has a maximum in equilibrium for isolated system. Also, since p i =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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