Professor Nano
5 minutes
Softmax as the Maximum-Entropy Distribution at Fixed Mean Energy
Maximising \( H(p) \) subject to \( \sum_i p_i = 1 \) and \( \sum_i p_i E_i = \bar{E} \) with Lagrange multipliers gives \( p_i \propto e^{-\beta E_i} \), the Boltzmann form, where \( \beta = 1/T \) is the multiplier.