Model barrier crossing in noisy physical systems. Enter potential parameters, friction, and thermal energy quickly. Get rate constants, then export tables for reports today.
The escape rate is commonly written as a prefactor times a Boltzmann factor: k = A · exp(-ΔE / (kBT)), where ΔE is the barrier height and kB is Boltzmann’s constant.
Notes: ω0 is the well curvature scale, ωb is the barrier curvature magnitude, and γ is a damping rate. This tool reports all internal values in SI units.
| Model | ΔE (eV) | T (K) | ω0 (rad/s) | ωb (rad/s) | γ (1/s) | k (1/s) |
|---|---|---|---|---|---|---|
| Kramers | 0.50 | 300 | 1.0×1012 | 5.0×1011 | 1.0×1013 | ~ order of 10? (depends on inputs) |
| High damping | 0.70 | 350 | 8.0×1011 | 4.0×1011 | 2.0×1013 | Lower than Kramers prefactor case |
| Transition-state | 0.40 | 300 | 1.0×1012 | — | — | Upper-bound style estimate |
For a reproducible example, enter the first row values and calculate. The exact output depends on the exponential sensitivity to ΔE and T.
Kramers’ rate quantifies how often a system escapes a metastable state due to thermal noise. It is widely used for chemical reactions, magnetic switching, Josephson junctions, nucleation, and biomolecular folding. The key feature is exponential sensitivity: modest changes in barrier height or temperature can change the escape rate by orders of magnitude.
Imagine a particle trapped in a potential well separated from another region by an energy barrier ΔE. Random thermal fluctuations supply energy, while friction dissipates it. The escape process depends on both the “attempt” dynamics near the well bottom and how the system moves near the saddle (barrier top).
The dominant term is the Boltzmann factor exp(−ΔE/kBT). At room temperature, kBT ≈ 0.0259 eV (≈ 4.14×10−21 J), so a 0.50 eV barrier gives ΔE/kBT ≈ 19.3 and exp(−ΔE/kBT) ≈ 4.1×10−9. This “data point” explains why barrier heights are often quoted in eV for nanoscale and molecular systems.
The prefactor A captures local curvatures: ω0 characterizes the well (attempt scale) and ωb characterizes the barrier top. In many condensed‑matter and molecular models, ω values fall between 109 and 1013 rad/s. This calculator lets you enter ω in rad/s or Hz (internally converted to rad/s).
Friction changes how energy diffuses across the barrier. In the moderate‑to‑high damping Kramers form, A includes √((γ/2)2+ωb2)−γ/2, which decreases as γ grows. In the high‑damping limit (γ ≫ ωb), the approximation A ≈ (ω0/2π)(ωb2/γ) becomes useful and highlights the 1/γ scaling.
Always keep units consistent: ω0 and ωb must share the same unit choice, and γ must be a rate (1/s). If you enter frequencies in Hz, the calculator multiplies by 2π to obtain rad/s. Results include ΔE in SI joules, the dimensionless ratio ΔE/(kBT), and the computed prefactor.
The escape rate k is reported in 1/s. The mean escape time is τ = 1/k, a convenient metric for reliability and lifetime estimates. For example, if k = 10−3 s−1, then τ ≈ 1000 s. If k = 103 s−1, then τ ≈ 1 ms.
Because rates span many decades, it is common to run multiple “what‑if” cases: varying ΔE, T, or γ to see sensitivity. After computing, export to CSV for spreadsheets or to PDF for reports. This workflow is helpful for documenting assumptions, sharing results, and maintaining reproducibility across parameter sweeps.
ΔE is the energy difference between the stable minimum and the saddle point. Larger ΔE dramatically reduces the escape probability through the exponential factor exp(−ΔE/kBT).
Use it when you want a friction‑free reference or an upper‑bound style estimate. It does not correct the prefactor for damping, so it can overpredict rates in strongly dissipative systems.
ω0 measures curvature near the well minimum (attempt dynamics). ωb measures curvature magnitude near the barrier top (saddle). Both affect the prefactor, not the exponential barrier term.
If γ is much larger than ωb (rule of thumb: γ/ωb ≥ 10), the high‑damping approximation is often reasonable. Otherwise, prefer the moderate‑to‑high damping Kramers expression.
The exponential term is very sensitive. Small increases in ΔE or small decreases in T can reduce k by orders of magnitude. Double‑check units, especially Hz vs rad/s, and energy units.
Yes. The calculator converts kJ/mol and kcal/mol to joules per particle using Avogadro’s constant, then uses SI units internally for ΔE/(kBT) and the rate calculation.
Exports include model choice, inputs, converted SI quantities, ΔE/(kBT), exp(−ΔE/kBT), the prefactor, the escape rate k, and the mean escape time τ. Use them to archive computed scenarios.
Important Note: All the Calculators listed in this site are for educational purpose only and we do not guarentee the accuracy of results. Please do consult with other sources as well.