Brownian Motion RMS Calculator

Example data table

Formula used

For isotropic diffusion in d dimensions, the mean square displacement is ⟨r²⟩ = 2 d D t, where D is the diffusion coefficient and t is time. The root-mean-square displacement is RMS = √(⟨r²⟩) = √(2 d D t).

If you choose the fluid/particle model, the calculator estimates D using the Stokes–Einstein relation for a spherical particle: D = k_BT / (6π η r), where η is dynamic viscosity and r is particle radius.

How to use this calculator

1. Select a model mode: enter D directly or compute it.
2. Choose the diffusion dimension d (1D, 2D, or 3D).
3. Enter time and its unit, then select the output length unit.
4. Click Compute RMS to display results above the form.
5. Use Download CSV or Download PDF to export values.

Professional notes on Brownian RMS displacement

1) What the RMS metric represents

Brownian motion produces a distribution of displacements, not a single path. The root-mean-square (RMS) displacement summarizes that spread. In isotropic diffusion, the expected mean square displacement grows linearly with time, so RMS grows with the square root of time. This is why doubling time increases RMS by about 41%, not 100%.

2) Link between MSD and RMS

The calculator first evaluates the mean square displacement (MSD) using ⟨r²⟩ = 2 d D t. RMS is then √(⟨r²⟩). For example, with D = 4×10⁻¹³ m²/s, t = 10 s, and d = 3, MSD becomes 2.4×10⁻¹¹ m² and RMS is about 4.90 µm.

3) Why dimensionality matters

Dimensionality reflects constraints: 1D along a channel, 2D tracking on a surface, or 3D in bulk fluid. Since MSD scales with d, RMS scales with √d. At fixed D and t, 3D RMS is about √(3/2) ≈ 1.225 times the 2D RMS.

4) Diffusion coefficient as an input

If you already have D from tracking or literature, direct mode is the most transparent route. For micron-scale particles in water, effective values commonly fall near 10⁻¹³–10⁻¹² m²/s, while small molecules can reach 10⁻¹⁰–10⁻⁹ m²/s. Always match the measurement conditions and temperature.

5) Estimating D with Stokes–Einstein

When viscosity and particle size dominate, the Stokes–Einstein relation gives D = kB T / (6π η r). At T = 298 K with water-like viscosity η ≈ 1 mPa·s, a r = 500 nm sphere yields D on the order of 10⁻¹³ m²/s, consistent with typical colloidal tracking experiments.

6) Unit handling and scaling checks

This tool converts time to seconds internally and reports RMS in your chosen length unit (nm to m). A quick sanity check is the scaling law: if you multiply time by 100, RMS should multiply by 10. If your results break this pattern, re-check time units and whether the intended D is in m²/s.

7) Measurement and model caveats

Real data can deviate from ideal diffusion. Drift adds a systematic component; confinement reduces the apparent dimension; and viscoelastic media can create subdiffusion where MSD grows slower than linear time. For optical microscopy, localization error can inflate short-time MSD, so consider using times where MSD is comfortably above the squared position uncertainty.

8) How to interpret outputs in practice

Use RMS to plan observation windows, pixel sizes, or simulation step sizes. For example, if RMS over 1 s is ~2 µm in 2D, a field-of-view of 100 µm supports many seconds of tracking without frequent boundary losses. Export the CSV/PDF to document assumptions, parameters, and computed MSD/RMS for reports.

FAQs

1) What is the difference between MSD and RMS?

MSD is the expected value of squared displacement, ⟨r²⟩. RMS is the square root of MSD, giving a typical displacement scale in the same units as position.

2) Which dimension should I choose for my experiment?

Use 1D for motion along a line or narrow channel, 2D for surface or planar tracking, and 3D for bulk diffusion. Choose the dimension that matches physical constraints and your measurement method.

3) Can I use this for very small molecules?

Yes, if you supply an appropriate diffusion coefficient. Small molecules often have D around 10⁻¹⁰–10⁻⁹ m²/s in water, leading to much larger RMS over the same time than colloids.

4) When is Stokes–Einstein a good approximation?

It works best for spherical particles in Newtonian fluids at low Reynolds number, with well-defined hydrodynamic radius. It can fail for non-spherical shapes, strong slip, or complex/viscoelastic media.

5) Why does RMS scale with the square root of time?

Diffusion accumulates many random steps with zero mean. Variances add linearly in time, so MSD grows ∝ t. Taking the square root makes RMS grow ∝ √t.

6) How should I handle temperature in the fluid model?

Enter temperature in kelvin or Celsius as selected. Higher temperature increases D linearly, while viscosity often decreases with temperature, which can further increase D and therefore RMS.

7) My data shows non-linear MSD versus time. What then?

That suggests anomalous diffusion or drift. Consider fitting MSD to ⟨r²⟩ ∝ t^α, subtracting drift, or using a model matched to confinement or viscoelasticity. The tool assumes normal diffusion with α=1.