Twelve two-state units are fixed in a row; each shows either "up" or "down" and has…
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
Twelve two-state units are fixed in a row; each shows either "up" or "down" and has no energy, so the row is isolated and every arrangement is equally likely.
((a)) Give the probability of the single arrangement ↑↑↑↑↑↑↑↑↑↑↑↑, and the probability of the single arrangement ↑↓↑↓↑↓↑↓↑↓↑↓.
((b)) How many times more likely is the macrostate "six up" than the macrostate "twelve up"?
((c)) Two Einstein solids, each with 5 oscillators, are weakly coupled in an insulated box and share q=4 energy units, the multiplicity of each solid being Ω(N,q)=(qq+N−1). A student argues: "every microstate of the isolated pair is equally likely, so every value of qA from 0 to 4 is equally likely, each with probability 1/5=0.20." Compute the probabilities of the splits qA=2 and qA=0 and say where the argument goes wrong.