Boltzmann distributionlaw
Imagine a crowd of people on an escalator in the subway: there are always fewer people at the top because it’s harder to get there. In the world of atoms and molecules, a similar principle applies: the greater the energy of a state, the fewer molecules “climb” to that “energy floor.” The Boltzmann distribution is a mathematical rule that says the chance of finding a particle in a state with a given energy drops exponentially as the energy increases. It’s like having a mountain, and with height, the number of tourists would decrease according to a strict exponential law, determined by the steepness of the slope (the temperature).
How it works
Imagine the Earth's atmosphere: air molecules are distributed by height according to Boltzmann in the gravitational field. At sea level there are more of them, at the summit of Everest — three times fewer, and at the orbit of the ISS — almost none. In electronics, charge carriers in semiconductors are distributed by energy according to this law: when you heat a diode, the number of electrons with enough energy to overcome the barrier increases, and the current grows. Also, lasers work on the basis of this law: you need to create a population inversion, i.e., a situation opposite to the Boltzmann one, to make atoms emit light synchronously.
💡 It is precisely because of the Boltzmann distribution that water evaporates at room temperature: the fastest molecules at the surface, having energy above average, can overcome the attraction and fly away. Moreover, this law explains why it’s hard to breathe at high altitudes: the density of air (the number of molecules at a given height in the gravitational field) drops exponentially with height — exactly according to Boltzmann, where the role of energy is played by gravitational potential energy.
P_i = \frac{1}{Z} e^{-\frac{E_i}{k_B T}}
P_i is the probability of the system being in a microstate with energy E_i; Z is the partition function; k_B is Boltzmann's constant (k_B ≈ 1.38×10⁻²³ J/K); T is the absolute temperature.
Z = \sum_j e^{-\frac{E_j}{k_B T}}
Z is the normalization factor, sum over all possible microstates j with energies E_j.