Equilibrium distribution functions are determined for fermions (e.g. electrons) and bosons (e.g. photons). The Saha–Boltzmann equation, the Maxwellian distribution, and relativistic Maxwell–Juttner distribution are derived. The relativistic equation of state for a distribution where particle velocities approach the speed of light is examined.

Max Planck was an expert on classical thermodynamics and turned his attention to the understanding of black-body radiation in 1895. He worked out the form the radiation formula had to have from the emission and absorption of radiation of a dipole in thermal equilibrium with the radiation it emits. The formula related the mean energy density of radiation to the average energy of the oscillator. Using new precision measurements of the black-body spectrum, Planck derived the primitive form of the black-body spectrum by a combination of theory and the empirical results of experiment, but without any physical interpretation of the significance of the formula.

The story of the discovery of quantisation and quanta begins with the numerous problems facing physicists at the end of the nineteenth century. A number of experimental results did not fit naturally into the scheme of classical physics. Among the most important of these was the spectrum of black-body radiation, which was being determined much more precisely experimentally in the last decade of the nineteenth century and which had to be explained theoretically. Important clues came from the pioneering studies of the origin of the Stefan-Boltzmann law and Wien's displacement law, the latter involving the clever use of dimensional methods.