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 E1 and E2 interact and then move off in different directions, with energies E3 and E4. 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 (E2). The probability of reverse process in which electrons with energies E3 and E4 interact is P (E3) P (E4). 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 (E1) P (E2) =P (E3) P (E4)
Furthermore, the energy in this collision must be conserved, so we also need
E1+E2= E3+E4
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 N1 at an energy level E1 and N2 particles at a higher energy E2. As the temperature increases N2/N1 also increases. Therefore increasing the temperature postulates the higher energy levels.