Ludwig Eduard Boltzmann is responsible for a lot of equations and in various fields, different equations are termed the Boltzmann equation.
In the field of stellar structure, an equation giving the count of atoms at different possible atomic excitation levels is given the name, or sometimes Boltzmann relation (which is also ambiguous). One form:
Ni+1 gi+1 ———— = ———— e-(Ei+1-Ei)/kT Ni gi
This equation is related to the Boltzmann distribution (frequency of particles at various states, not the same as the Maxwell-Boltzmann distribution, which is the distribution of (say) the kinetic energy of particles in (say) a gas) incorporating a Boltzmann factor (e-(E1-E2)/kT) giving the relative probability of two energy levels, and relating the ratio of the two terms, Nx/gx (the number of atoms in specific quantum state) to it.