where N i is the number of particles in state i and N is the total number of particles in the system. We may use the Boltzmann distribution to find this probability that is, as we have seen, equal to the fraction of particles that are in state i. So the equation that gives the fraction of particles in state i as a function of the energy of that state is N i N = exp ⁡ ⁡ ( − − ε ε i k T ) ∑ ∑ j = 1 M exp ⁡ ⁡ ( − − ε ε j k T )