Can I derive the Boltzmann distribution by an invariance argument?

Like Andreas, I find a maximum entropy argument to be intellectually appealing. However, he says the solution can be found by Lagrange multipliers and I don't know the justification for using Lagrange multipliers. That is, in the space of all probability distributions on the particles, how do you know the maximum entropy solution is really accessible to variational methods?

For a derivation not using Lagrange multipliers, see the bottom of page 9 through page 11 at http://www.math.uconn.edu/~kconrad/blurbs/analysis/entropypost.pdf.

For me the clearest derivation of the Boltzmann distribution is by maximizing the entropy $\sum n_i \ln(n_i)$ unter the constraint of constant total energy $\sum n_i E_i = \text{const.}$ and constant total particle number $\sum n_i = \text{const.}$. The Lagrange multiplicator for the first constraint gives $\beta$. You can immediately see that a shift of the energies does not change the distribution.

I sketched this here and in section IV of my ancient preprint Effective statisical physics of Anosov systems.