Thermal activation

Thermal activation is barrier crossing enabled by thermal fluctuations. A system can remain for a long time near a local minimum of an energy landscape, then receive a fluctuation large enough to cross an energy barrier. Its most probable state and its waiting time for escape are therefore different questions.

In a simple high-barrier regime, the escape rate is approximated by $r(T)=r_0\exp [-\Delta E/(k_{B}T)]$, an Arrhenius equation. The barrier $\Delta E$ controls the strong temperature dependence, while the prefactor $r_0$ depends on dynamics near the initial state and transition region. Plotting $\log r$ against $1/T$ tests the exponential model: a constant prefactor gives slope $-\Delta E/k_B$. Interpreting a slope still requires care if several escape paths compete or the prefactor varies with temperature. A large barrier can delay approach to thermal equilibrium even when the final equilibrium state is known.

For gapped excitations with $C(T)\simeq A\exp [-\Delta /(k_{B}T)]$ and approximately constant prefactor, two heat capacity measurements at $T_2>T_1$ give the displayed activation-energy estimate. A more general prefactor $A{T}^p$ changes its numerator to $\ln [C(T_2)/C(T_1)]-p\ln (T_2/T_1)$. The two-point estimate measures an effective exponential scale; it does not independently determine the prefactor or establish a detailed excitation density of states.