Mean Squared Displacement Calculator

Trajectory definition: MSD(τ) = ⟨|r(t+τ) − r(t)|²⟩.

Normal diffusion model: MSD = 2 d D t.

Anomalous diffusion model: MSD = 2 d D_α t^α.

1. Select a mode: trajectory data or diffusion model.
2. Pick spatial dimension d, then choose input units.
3. For trajectories, paste rows: t, x, y, z.
4. For models, enter D, α (if needed), and times.
5. Compute, then export results using CSV or PDF.

Example 2D trajectory input (time in seconds, position in µm):

1) Why MSD is a standard mobility metric

Mean squared displacement (MSD) summarizes how far a particle or tracer moves over a chosen time lag. In microscopy, single‑particle tracking, molecular dynamics, granular flows, and turbulence studies, MSD converts complex paths into a comparable curve. The slope and curvature of MSD versus lag reveal transport regimes, confinement, and drift without requiring a full dynamical model.

2) What the curve actually measures

The trajectory definition uses squared step lengths: MSD(τ) = ⟨|r(t+τ) − r(t)|²⟩. Averaging across all starting times increases statistical power, especially for noisy tracks. For 2D imaging, MSD is commonly reported in m² (or µm²) while τ is in seconds. This calculator outputs SI units so results are consistent across experiments.

3) Normal diffusion benchmark and typical scales

For ideal Brownian motion, MSD grows linearly: MSD = 2 d D t. As a practical reference, small molecules in water at room temperature often have D around 10⁻⁹ m²/s, while micron‑scale beads can be near 10⁻¹³ to 10⁻¹² m²/s, depending on viscosity and size. In 2D, a D of 10⁻¹² m²/s implies an MSD of about 4×10⁻¹² m² at t = 1 s.

4) Detecting anomalous transport with α

Many systems deviate from linear growth. A power‑law form MSD ≈ A τ^α captures this behavior. Subdiffusion (α < 1) can indicate crowding, viscoelastic media, or trapping. Superdiffusion (α > 1) can reflect active transport, intermittency, or Lévy‑like steps. The built‑in log‑log fit estimates α and reports R² to gauge how well the power law describes your lags.

5) Choosing lag range and sample size

MSD at large lags uses fewer point pairs, which increases uncertainty. For N samples, lag k uses roughly N−k pairs. A common practice is to interpret only the first 10–25% of lags when tracks are short. This tool displays the number of pairs per lag, helping you judge where the curve becomes statistics‑limited.

6) Units, conversions, and reporting

Tracking software often exports positions in pixels, nm, or µm, and times in ms. Conversions are critical because squared units amplify scale errors. Here you select input units, while results are computed in meters and seconds. When reporting, state the dimension d, calibration, time step, and whether drift removal was performed upstream.

7) Interpreting fits into diffusivity estimates

For normal diffusion, D can be estimated from the linear coefficient: D = MSD/(2 d t). For power‑law behavior, the fit gives A, and an effective generalized coefficient can be approximated as D_α = A/(2 d). Compare α across conditions rather than relying on a single D value when transport is non‑Brownian.

8) Export workflow for documentation

After computation, download the table as CSV for spreadsheets or plotting software, or export a PDF summary for lab notes and reports. Keeping the lag table, pair counts, fitted α, and units together makes it easier to reproduce figures and defend analysis choices in publications or audits.

1) What is the difference between MSD and RMS displacement?

MSD is the average of squared displacements. RMS displacement is the square root of MSD, giving a distance scale. RMS is easier to interpret visually, while MSD is linear for Brownian motion.

2) Why does MSD become noisy at large τ?

Large lags use fewer point pairs (N−k). Fewer averages increase variance and sensitivity to outliers. Interpreting early lags or using longer trajectories usually improves stability.

3) Can I use uneven time steps?

This tool infers a representative sampling interval from your time column. Strongly uneven sampling can bias τ values. If possible, resample or export data at a constant frame interval before calculating MSD.

4) How do I handle drift or flow?

Systematic drift adds a quadratic component and can mimic superdiffusion. Remove drift by subtracting a mean velocity or referencing a stationary marker, then recompute MSD for the corrected trajectory.

5) What does α < 1 usually indicate?

Subdiffusion often reflects trapping, crowding, viscoelastic response, or confinement. It can also appear from localization noise or motion blur, so validate with controls and compare across lag ranges.

6) What coordinate columns should I paste?

Provide rows as time followed by coordinates: t,x for 1D; t,x,y for 2D; or t,x,y,z for 3D. Separators may be commas, tabs, or spaces, and an optional header line is allowed.

7) What is a reasonable max lag setting?

A practical starting point is 20–50 lags for long tracks, or about 10–25% of your total samples for shorter tracks. Use the “Pairs” column to avoid lags dominated by low counts.