Using the partition function find the entropy

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ber70
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Question is on attachment file.
 
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  • Typeset problem statement: using Z = sum over j of e^(-beta E_j), show that S = -k_B sum over j of P_j ln(P_j), where P_j = e^(-beta E_j) / Z is the probability of state j with energy E_j.
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I solved it.
 
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  • Handwritten statistical mechanics derivation: S = (E - F)/T, sum of P_j = 1, beta = 1/(k_B T), then E = -d/d(beta) ln Z, expanded through sums over E_j and P_j, ending in S = -k_B sum P_j ln P_j.
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