Week 11: A reservoir that counts states, and Fock space

Why systems go downhill: a reservoir that counts states (Datta Ch. 15, Fig. 15.2)

A small system shares energy quanta ε with a reservoir of N_R oscillators (an "Einstein solid"). Every move simply passes one quantum between two randomly chosen members: no force pushes the system down. Yet the system ends up mostly in its low states, because giving a quantum to the reservoir multiplies the reservoir's number of states W (Eq. 15.12). The histogram converges to the Boltzmann law p_n ∝ e^−E_n/kT with kT fixed by dS/dE of the reservoir (Eq. 15.13).

Reservoir oscillators N_R Quanta per reservoir oscillator Q/N_R Small system

Reservoir temperature kT/ε = 1/ln(1 + N_R/Q)–

Measured p(0)/p(1) · theory e^ε/kT–

Mean system energy ⟨n⟩ · theory–

Moves made–

Fock space: one law of equilibrium, three famous answers (Sec. 15.5)

Interaction energy U₀/kT for two levels Number of degenerate levels g (strong interaction)