It describes the behaviour of a collection of a particle N(E) with gives energy through N(E)α exp(-E/KT)
where K is Boltzmann constant. It can be explained as a collection of particles in random motion and colliding with each other, we require finding out the concentration of charges particles in the energy range E to (E+dE). Consider the process in which two electrons with energies E_{1} and E_{2} interact and then move off in different directions, with energies E_{3 }and E_{4}. Let the probability of an electron having an energy E be P (E), where P (E) is the fraction of electron with an energy E. The probability of this event is then P (E1), P (E_{2}). The probability of reverse process in which electrons with energies E_{3} and E_{4} interact is P (E_{3}) P (E_{4). }In the thermal equilibrium, that is, the system is in equilibrium, the forward process must be just as likely as the reverse process, so.
P (E_{1)} P (E_{2) =}P (E_{3)} P (E_{4)}
Furthermore, the energy in this collision must be conserved, so we also need
E_{1}+E_{2= }E_{3+}E_{4}
Therefore, we need to find the P (E) that let satisfies both equation and equation
P (E) =A exp (_E/KT)
K= Boltzmann constant, T= Temperature, A=constant
Equation is the Boltzmann probability function is shown. The probability of finding a particle at any energy E states therefore exponentially with energy.
Suppose that we have N_{1} at an energy level E_{1} and N_{2} particles at a higher energy E_{2}. As the temperature increases N_{2}/N_{1} also increases. Therefore increasing the temperature postulates the higher energy levels.