Response to the following problem:
Given N indistinguishable, quasi-independent particles capable of existing in energy levels ?1, ?2,· .. , with degeneracies gb g2, ... , respectively; in any given macro state in which there are N, particles in energy level ?1, N2 particles in energy level ?2, ... , assume the thermodynamic probability to be given by the Bose-Einstein expression,
ΩBE = (g1 + N1)!(g2 +N2)!.../g1!N1!g2!N2!
Using Stirling's approximation and the method of Lagrangian multipliers, render
In ΩBE a maximum,subject to the equations of constraint ∑Ni = N = const.and ∑Ni?i = U = const., and show that
Ni = gi/λe-β?i - 1