Possible to derive Boltzmann distribution using W, not lnW?

[Darren73]

Hi all, in following the many available derivations of the Boltzmann distribution I was trying to do it by maximizing W, which is N choose n₁,n₂,...n_t., instead of lnW, because it should give the same answer (since W is monotonically increasing with lnW, am I wrong?).

So given the two constraint equations of constant particle number and energy, $$g=\sum_{i}n_{i}=N, h=\sum_{i}n_{i}\epsilon_{i}=E$$And the Stirling approximation of W, $$W=N^{N}n_{1}^{-n_{1}}n_{2}^{-n_{2}}...n_{t}^{-n_{t}}$$

And maximizing W with the above constraints (using Lagrange multipliers) gives the following t equations, $$\frac{\partial W}{\partial n_{i}}-\alpha\frac{\partial g}{\partial n_{i}}-\beta\frac{\partial h}{\partial n_{i}}=0$$

Which gives,$$\frac{\partial W}{\partial n_{i}}-\alpha-\beta\epsilon_{i}=0$$ $$\frac{\partial W}{\partial n_{i}}=C_{i}n_{i}^{n_{i}}\left(\ln n_{i}+1\right)$$ $$C_{i}n_{i}^{n_{i}}\left(\ln n_{i}+1\right)=\alpha+\beta\epsilon_{i}$$

Where C_i is some constant of the other n_j's and N. Proceeding from this point has proven fruitless for me to isolate n_i and apply the constraints. Does anyone know if this can be done? Or do you have to use lnW? It would seem odd to me if this cannot be done by maximizing W directly. And they should give the same distribution, namely $n_{i}=N\exp -\beta \epsilon_{i}$, correct?

 

[Dr. Courtney]

Given this long without a reply, perhaps a search at scholar.google.com would yield some insight.