Translation Rate Estimator Calculator

Formula used

• Kinematic: \( v = \Delta x / \Delta t \) where \(\Delta x\) is displacement magnitude.
• Diffusive: Einstein relation \( D = k_B T / \gamma \). Mean-squared displacement \(\langle r^2\rangle = 2 n D t\). Effective speed \(v_\mathrm{eff} = \sqrt{\langle r^2\rangle}/t\).
• Activated: Arrhenius hopping \( k = A\,\exp(-E_a/(k_B T)) \). With step length \(a\), effective speed \(v \approx a\,k\).

How to use this calculator

1. Select a model matching your physical regime.
2. Enter a time interval and choose the unit.
3. Provide model inputs (displacement, diffusion parameters, or hopping parameters).
4. Optionally add ± uncertainties and enable Monte Carlo estimation.
5. Click Estimate. Results appear above the form.
6. Use CSV or PDF export to save the computed outputs.

Example data table

Translation rate estimator: practical interpretation

1) What “translation rate” means here

The calculator converts common physical inputs into a translation speed (m/s). It supports three regimes: kinematic motion from measured displacement, diffusive motion from thermal fluctuations, and activated hopping with Arrhenius-controlled steps. Outputs are in SI, with intermediate fields like D, γ, and hop rate k. Pick the regime that matches your data and assumptions.

2) Kinematic regime: measured motion

Kinematic estimation uses v = Δx/Δt. If a tracer moves Δx = 1 mm in Δt = 0.5 s, the speed is 2×10⁻³ m/s. Use this mode for tracked trajectories, stage motion, or drift measurements where a single displacement over a time window is meaningful.

3) Diffusive regime: Brownian transport

Diffusion has no single velocity, so the tool reports an effective speed using RMS displacement. With ⟨r²⟩ = 2nDt, v_eff = √(⟨r²⟩)/t = √(2nD/t). For D ≈ 1×10⁻¹² m²/s, t = 1 s, n = 3, v_eff is on the order of 10⁻⁵ m/s.

4) Stokes–Einstein inputs with practical ranges

If you enter viscosity and radius, drag is γ = 6πηr and D = k_BT/γ. Many aqueous systems sit near η ≈ 0.5–5 cP and T ≈ 280–330 K. Particle radii commonly span 50–500 nm in colloids and nanoparticle tracking, making D and v_eff strongly size dependent.

5) Using drag γ directly

When friction is measured (trap calibration, microrheology, or coarse-grained models), input γ directly. Because D ∝ 1/γ, a 20% change in γ produces roughly a 20% change in D and a ~10% change in v_eff (through the square root).

6) Activated hopping: Arrhenius-controlled steps

Activated motion uses k = A exp(−E_a/(k_BT)) and v ≈ a k. Many systems use A ≈ 10⁸–10¹³ s⁻¹ and E_a ≈ 0.05–0.8 eV. A 0.1 eV increase in E_a near 300 K can cut the rate by ~50×.

7) Comparing regimes on the same timescale

Kinematic and activated modes can look steady over a chosen interval, but diffusion depends on t. Because v_eff ∝ 1/√t, short windows inflate it. Use consistent Δt when comparing samples.

8) Uncertainty, reporting, and exports

Turn on uncertainties to propagate measurement error with Monte Carlo sampling (mean and 95% interval). Report the chosen model and key inputs (n, T, η and r or γ, and for hopping: A, E_a, a). Use CSV/PDF exports to archive parameters alongside derived speeds for lab notes or simulation logs.

FAQs

1) Which model should I choose?

Use kinematic for tracked displacements, diffusive for Brownian spreading, and activated for thermally assisted step motion with a barrier and attempt rate.

2) Why does diffusion return an “effective” speed?

Diffusion has no single velocity. The calculator converts the RMS displacement √⟨r²⟩ over time t into v_eff = √⟨r²⟩ / t to summarize typical transport.

3) What does the spatial dimension n change?

The mean-squared displacement scales as 2nDt. Larger n increases the expected spread for the same D and t, raising the reported v_eff.

4) When should I enter drag γ instead of viscosity and radius?

Enter γ when friction is measured or modeled directly, such as trap calibration, microrheology, or coarse-grained simulations where an effective drag is known.

5) How sensitive is activated hopping to E_a?

Very sensitive. Because k ∝ exp(−E_a/(k_BT)), small changes in E_a can shift the rate by orders of magnitude, especially near room temperature.

6) What uncertainty values should I enter?

Use one-sigma measurement uncertainty in the same units as each input. If you only know bounds, choose uniform mode and enter half-width uncertainty as the ± value.

7) Are the exports identical to the on-screen results?

Yes. The CSV and PDF exports include the same computed fields shown in the results table, helping you store parameters and outputs consistently for reports.