In an ideal gas, the Boltzmann distribution predicts a distribution of particle energies $E_i$ proportional to $ge^{-E_i/k_bT}$.
But, doesn't entropy dictate that the system will always progress towards a state of maximum disorder? In other words the system evolves towards a macro-state which contains the maximum possible number of indistinguishable micro-states. This happens when all particles have the same energy, which seems to contradict the Boltzmann distribution.
I'm pretty sure I've misinterpreted entropy here, but I'd be please if someone could explain how!
Best Answer
In any system in equilibrium, the entropy of such system is the maximum given a set of constrains. If you think of a microcanonical ensemble, the total energy is fixed while in an canonical ensemble of particles the temperature is the one being held constant.
This distribution probability you mention, is for a canonical situation. Given that the temperature is being held fixed, the many different microstates available for such macrostate are given by that exponential function which depends of the energy of the particles and the temperature.