Two Einstein solids A and B are sealed in a rigid insulating box and weakly coupled

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 Microstates, multiplicity and the statistical entropy

Problem

Two Einstein solids A and B are sealed in a rigid insulating box and weakly coupled, so that they may exchange whole units of energy hνh\nu but the total number of units is fixed. Solid A has NA=3N_A = 3 oscillators, solid B has NB=5N_B = 5 oscillators, and the total energy is q=4q = 4 units. Each oscillator holds a whole number of units above its lowest level, and by the postulate of equal a priori probabilities every accessible microstate of the pair is equally likely. The multiplicity of an Einstein solid is Ω(N,q)=(q+N−1q)\Omega(N,q) = \binom{q+N-1}{q}. ((a)) Predict which value of qAq_A, the number of units in solid A, is most probable. ((b)) Find the probability of that macrostate. ((c)) Find the mean number of units in solid A, ⟨qA⟩\langle q_A\rangle, and compare it with the value q NA/(NA+NB)q\,N_A/(N_A+N_B) that an equal share per oscillator would give.

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