Random Walk MSD Calculator

Calculator

Model mode Compare both modelsStep model onlyDiffusion model only

Choose whether to compare or focus on one model.

Dimensions (d) 123

1D, 2D, or 3D motion.

Number of steps (N)

Total increments in the walk.

Step length (ℓ)

Positive distance per step.

Step length unit mcmmmµmnmftin

Used for reporting RMS and MSD units.

Time step (Δt)

Needed for time-based calculations.

Time unit smsµsnsminhr

Converted to seconds internally.

Diffusion coefficient (D)

Used by diffusion model MSD = 2 d D t.

D unit m²/scm²/smm²/sµm²/s

Choose typical lab units.

Table points

Rows between 0 and N for the table.

Calculate Reset

Formula used

• Random-walk step model: for independent isotropic steps of fixed length ℓ, the mean squared displacement is MSD = N · ℓ².
• Diffusion model: for normal diffusion in d dimensions, MSD(t) = 2 · d · D · t, with t = N · Δt.
• RMS displacement: the root-mean-square distance is r_rms = √MSD.
• Effective diffusion from steps: if Δt is given, D_eff = MSD / (2 · d · t).

How to use this calculator

1. Select a mode to compare models or use one model.
2. Set the dimension d and the total steps N.
3. Enter the step length ℓ and choose its unit.
4. For time-based results, provide Δt and (optionally) D.
5. Click Calculate to view results above the form.
6. Use Download CSV or Download PDF in the results panel.

Example data table

Random walk MSD: professional notes

1) Why MSD matters in random motion

Mean squared displacement (MSD) converts a noisy trajectory into a single, stable measure of spreading. In experiments, MSD helps summarize particle wandering, molecular transport, or sampling error. In simulations, it validates whether a model behaves as expected across steps and time.

2) Step model interpretation

The step model assumes each move has fixed length ℓ and a random direction. Independence is the key assumption: no memory, no drift, and no preferred direction. Under these conditions, the expected value grows linearly with steps: MSD = N·ℓ².

3) Diffusion model interpretation

The diffusion form MSD(t) = 2·d·D·t is widely used for Brownian-like motion. Here D is the diffusion coefficient and d is the spatial dimension. For 2D motion, MSD increases as 4Dt; for 3D motion, it increases as 6Dt.

4) Linking steps to diffusion

When you provide a time step Δt, the calculator can estimate an effective diffusion value from the step model. This is helpful for mapping discrete walks to continuous transport. If the two models disagree strongly, it usually signals inconsistent ℓ, Δt, or D inputs.

5) Dimension choice and practical data

Pick d = 1 for constrained channels, d = 2 for planar motion, and d = 3 for bulk media. Using the wrong dimension changes the slope of MSD versus time and can distort fitted D values. Always match d to the geometry of your measurement or simulation domain.

6) Unit discipline for clean reporting

MSD has squared length units, so converting length units also squares the scaling factor. This tool converts inputs to SI internally, then reports MSD and RMS using your chosen length unit. That makes plots, tables, and exported CSVs consistent for lab notebooks and reports.

7) Using the table for slopes and checks

The generated table samples k from 0 to N, allowing quick slope inspection. For diffusion, the slope of MSD versus time should be 2·d·D. For the step model, MSD versus step index should be a straight line with slope ℓ².

8) Common caveats in real systems

Deviations from linear growth can indicate drift, confinement, correlations, or anomalous diffusion. Short-time MSD may be biased by localization noise, camera blur, or finite sampling. Use this calculator as a baseline before introducing more complex models.

FAQs

1) What does MSD represent physically?

MSD is the expected squared distance from the start. It summarizes how fast trajectories spread, even when individual paths look noisy or irregular.

2) When should I use the step model?

Use it for discrete walks with fixed step length and random direction, especially when you track motion per step rather than per unit time.

3) When should I use the diffusion model?

Use it when motion is well-described by normal diffusion, and you have a diffusion coefficient and time scale. It is standard for Brownian-like transport.

4) Why does dimension change the result?

Diffusion spreads independently along each axis. More dimensions add more degrees of freedom, increasing MSD slope: 2·d·D·t.

5) What is RMS displacement in the results?

RMS displacement is √MSD. It is a typical distance scale that’s easier to interpret than a squared quantity, especially in reporting.

6) How can I estimate D from my MSD data?

Fit a straight line to MSD versus time in the linear regime. The slope equals 2·d·D, so D = slope/(2·d).

7) Why might my MSD curve be non-linear?

Non-linearity can come from drift, confinement, step correlations, obstacles, or measurement noise. It may also indicate anomalous diffusion or changing conditions.