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 identical Einstein solids, each of N oscillators, are sealed in a rigid insulated box and weakly coupled, exchanging energy only in whole units hν. The total number of units q=qA+qB is fixed, and both solids are in the many-units regime, where lnΩ(N,q)=N[ln(q/N)+1].
((a)) Show that (d2lnΩtot/dqA2) evaluated at the even split qA=q/2 equals −8N/q2. Taking the peak to be Gaussian, Ωtot∝exp[−(qA−q/2)2/(2σqA2)], show that the relative width σqA/(q/2) equals 1/2N, and evaluate it for N=5.0×1021.
((b)) Show that, within the Gaussian approximation of (a), a split qA=(q/2)(1+ε) is less probable than the even split by the factor e−Nε2, and give the value of the exponent Nε2 for ε=1.0×10−5 and the same N.
((c)) For N atoms of a gas in a volume V the volume-dependent part of the entropy is NkBlnV if the atoms are treated as labelled, and NkB[ln(V/N)+1] if they are identical (the labelled count divided by N!). Show that only the second doubles when V and N are doubled together, and evaluate the spurious excess given by the first for N=7.5×1022.
((d)) Four two-state spins are known to have exactly two spins up, and the six microstates of that macrostate are equally likely. Use the Gibbs formula S=−kB∑iPilnPi to find S, showing that it agrees with kBlnΩ. Then find S for a distribution over the same six microstates in which one has probability 0.50 and each of the other five has probability 0.10.