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).
History
L. Boltzmann began his research by extending Maxwell's (James Clerk Maxwell) distribution for an ideal gas to systems with an external field. In 1871, he formulated the general principle: if a system can be in discrete states, then its equilibrium distribution is determined only by energy and temperature. In 1877, he linked entropy to the logarithm of the number of microstates (the famous Boltzmann formula), which led to a deep understanding of the statistical nature of the second law of thermodynamics. In the 20th century, quantum mechanics expanded the application of the distribution, and Erwin Schrödinger in his lectures 'What is Life?' used it to explain order and disorder in biological systems.
How it works
The Boltzmann distribution works for classical and quantum systems in thermal equilibrium, when quantum exchange effects can be neglected (i.e., for 'distinguishable' particles in the classical limit). It describes the populations of energy levels of atoms and molecules in spectroscopy, impurity distributions in semiconductors, thermal ionization, diffusion. Limits of applicability: the distribution fails when Fermi–Dirac statistics (for fermions, e.g., electrons in metals at low temperatures) or Bose–Einstein statistics (for bosons, like photons) are important, where the Pauli principle or Bose condensation must be accounted for. It is also valid only in equilibrium; in non-equilibrium systems, populations may be different.
Fun fact
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.
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