A Very Simple Derivation of the Boltzmann Distribution

Consider a system at equilibrium in which particles can occupy either to two energy states, S_{1} or S_{2}. The flow rate of particles from S_{1} to S_{2} will be proportional to the number of particles, n_{1<}, in S_{1} and fraction of the particles in S_{1} with sufficient energy to get over the barrier. This fraction will be a function f(a) of the height of the energy barrier, a, that the particles must surmount in order to reach state S_{2},and is equal to n_{1}f(a).

Similarly, the flow from S_{2} to S_{1} will be n_{2}f(b). At equilibrium these flows are equal, and furthermore, if the height of the barrier is increased by an amount c, the n_{1} and n_{2} will remain unchanged, and the new rates of flow will remain equal, but at different values from before. That is, that is, the two rates are n_{1}f(a) = n_{2}f(b) and n_{1}f(a+c) = n_{2}f(b+c). Dividing one by the other gives f(a)/f(a+c) = f(b)/f(b+c). The only function f satisfying this relationship is f(x) = exp(dx) where d is any constant. Thus, the fraction of particles possessing sufficient energy to cross the barrier is an exponential function of the height of the barrier and n_{1}/n_{2} = exp(d(a-b)).