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7.10 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: P, exp(-BE,) where B = 1/(k T), and Z = exp(-B E,) is the partition function. (a) Demonstrate that the entropy can be written S = -k P, In P,. (b) Demonstrate that the mean Helmholtz free energy is related to the partition function according to Z = exp(-BF). %3D

7.10 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: P, exp(-BE,) where B = 1/(k T), and Z = exp(-B E,) is the partition function. (a) Demonstrate that the entropy can be written S = -k P, In P,. (b) Demonstrate that the mean Helmholtz free energy is related to the partition function according to Z = exp(-BF). %3D

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7.10 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: exp(-BE,) P, = where B = 1/(k T), and Z = Eexp(-B E,) is the partition function. (a) Demonstrate that the entropy can be written S=ΣΡ In P, (b) Demonstrate that the mean Helmholtz free energy is related to the partition function according to Z = exp(-8F).
Transcribed Image Text:7.10 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: exp(-BE,) P, = where B = 1/(k T), and Z = Eexp(-B E,) is the partition function. (a) Demonstrate that the entropy can be written S=ΣΡ In P, (b) Demonstrate that the mean Helmholtz free energy is related to the partition function according to Z = exp(-8F).
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