Consider a system in thermal equilibrium with a heat bath held at absolute temperature T. The...

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Consider a system in thermal equilibrium with a heat bath held at absolute temperature T. The probability of observing the system in some state r of energy E, is is given by the canonical probability distribution: where ß = 1/(k T), and Pr = exp(-ß Er) Z z = Σ exp(-βE,) r is the partition function. (a) Demonstrate that the entropy can be written S = -kΣP, In P₁. (b) Demonstrate that the mean Helmholtz free energy is related to the partition function according to Z = exp(-BF). Consider a system in thermal equilibrium with a heat bath held at absolute temperature T. The probability of observing the system in some state r of energy E, is is given by the canonical probability distribution: where ß = 1/(k T), and Pr = exp(-ß Er) Z z = Σ exp(-βE,) r is the partition function. (a) Demonstrate that the entropy can be written S = -kΣP, In P₁. (b) Demonstrate that the mean Helmholtz free energy is related to the partition function according to Z = exp(-BF).