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Boltzmann distribution
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Probability falls exponentially with energy.
P ∝ e^{−E/kT}At temperature T, the probability of a state with energy E is proportional to e^{−E/kT}. Lower-energy states are more populated; the ratio of populations of two levels is e^{−ΔE/kT}.
Excited state with ΔE = 10 kT is 4.5×10⁻⁵ as populated.
Includes degeneracy factors.
In more detail
The Boltzmann distribution gives the probability that a system is in a state of energy E at temperature T, proportional to exp(−E/kT). Higher-energy states are less probable.
Real-world uses
- Predicting populations of energy levels in atoms and molecules.
- Explaining how lasers invert populations to amplify light.
- Scientists use the Boltzmann distribution to model the populations of energy levels in molecules and atoms at a given temperature.
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Frequently asked questions
Why does temperature matter?
At higher temperature, higher-energy states become more populated.
Is it the same as the Maxwell distribution?
Related, since the Maxwell distribution follows from the Boltzmann distribution for kinetic energy.
Why do high-energy states have fewer particles?
At a finite temperature, the probability falls exponentially with energy, so higher states are less populated.
Related topics in Kinetic Theory & Statistical Physics
Compressibility factor · Liquefaction of gases · Microstates and macrostates · Partition function · Canonical ensemble · Fermi–Dirac statistics