2.4 The Boltzmann factor and thermal activation

The Gaussian of the Maxwell–Boltzmann distribution is one appearance of a deeper rule. The probability of finding a system in a state of energy $E$ , in equilibrium with a heat bath at temperature $T$ , is weighted by the Boltzmann factor

\boxed{\;P(E) \;\propto\; e^{-E/k_B T}.\;}

It is the universal cost of energy: a state that costs an energy $E$ to occupy is suppressed by $e^{-E/k_B T}$ relative to the ground state. Whether $E$ is the kinetic energy of a molecule, the potential energy of an atom raised against gravity, or the barrier of a chemical reaction, the same exponential governs how often the energy is paid.

▶ The two-level population ratio Derivation

Take a system with just two states, a ground state at energy $0$ and an excited state at energy $\Delta E$ . The Boltzmann factors are $e^{0} = 1$ and $e^{-\Delta E/k_B T}$ , so the equilibrium population ratio is

\frac{n_\text{excited}}{n_\text{ground}} \;=\; e^{-\Delta E/k_B T}.

The control variable is the dimensionless ratio $\Delta E/k_B T$ , not $\Delta E$ alone: at $\Delta E \ll k_B T$ the two states are equally populated; at $\Delta E \gg k_B T$ the excited state is empty; the crossover is at $\Delta E \sim k_B T$ . ✓

ΔE1.00

k_BT1.00

ΔE / k_BT1.00

e^(−ΔE/k_BT)3.68e-1

Energy gap ΔE = 1.00 Temperature k_BT = 1.00

The Boltzmann factor governs every thermally activated process. When ΔE/k_BT = 1 the excited state holds e⁻¹ ≈ 37% of the ground-state population; at ΔE/k_BT = 10 the ratio is 4×10⁻⁵ — effectively zero. The dimensionless ratio ΔE/k_BT, not ΔE alone, decides which states matter.

Slide the gap $\Delta E$ and the temperature. When $\Delta E \sim k_B T$ the two states populate comparably; when $\Delta E \gg k_B T$ the excited state empties. What matters is always the ratio $\Delta E/k_B T$ .

Three faces of the same exponential

The barometric formula. A molecule of mass $m$ at height $z$ has gravitational energy $mgz$ , so the number density falls as $n(z) = n_0\, e^{-mgz/k_B T}$ . The atmosphere thins exponentially, with a scale height $k_B T/mg$ that is the height at which the Boltzmann cost reaches $k_B T$ .
The two-state paramagnet. A magnetic moment in a field $B$ has energy $\mp\mu B$ for the two orientations; the population difference $\tanh(\mu B/k_B T)$ is the magnetisation, and it saturates exactly when $\mu B \gg k_B T$ .
Thermal activation. A process that must surmount an energy barrier $E_a$ proceeds at a rate $k = A\, e^{-E_a/k_B T}$ — the Arrhenius law — because the fraction of attempts carrying enough energy to clear the barrier is the Boltzmann factor.

The exponential is what makes thermal physics so sharply temperature-sensitive. Near $\Delta E/k_B T \approx 5$ , a change in the gap of just $k_B T$ — a few percent of the gap — changes the population by a factor of $e \approx 2.7$ . Small shifts in energy become large shifts in probability, which is why barriers and gaps measured in a handful of $k_B T$ dominate the behaviour of thermal systems.