A system in contact with a reservoir at temperature has discrete microstates of energies

PHYSICS · Multi-part · Fermi University Physics · Thermal, Statistical & Complex Systems · Statistical foundations and the equilibrium ensembles · Statistical foundations and the equilibrium ensembles

Thermal Physics, Ch The Boltzmann factor, the canonical ensemble and the partition function

Problem

A system in contact with a reservoir at temperature TT has discrete microstates ss of energies EsE_s, which depend on the volume VV but not on TT. Its probabilities are P(s)=e−βEs/ZP(s) = e^{-\beta E_s}/Z with β=1/kBT\beta = 1/k_BT and Z=∑se−βEsZ = \sum_s e^{-\beta E_s}. ((a)) Starting from U=∑sEsP(s)U = \sum_s E_sP(s), show that U=−(∂ln⁡Z/∂β)VU = -(\partial \ln Z/\partial\beta)_V, and hence that U=kBT2(∂ln⁡Z/∂T)VU = k_BT^{2}(\partial \ln Z/\partial T)_V. Say why VV is the variable held fixed. ((b)) Apply the result to a centre with three non-degenerate states of energies 00, ε\varepsilon and 3ε3\varepsilon: obtain UU as a function of βε\beta\varepsilon, and evaluate it as a multiple of ε\varepsilon at the temperature where kBT=εk_BT = \varepsilon. Check the number by computing ∑sEsP(s)\sum_s E_sP(s) directly. ((c)) What does UU tend to as T→∞T \to \infty?

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