Explore random motion using diffusion theory, probabilities, and simulations visually instantly here. Choose units, enter parameters, and compare predictions with measured tracks easily now.
| Temperature (K) | Viscosity (mPa*s) | Radius (nm) | Estimated D (m^2/s) | Time (s) | 3D RMS (um) |
|---|---|---|---|---|---|
| 298.15 | 1.00 | 500 | ~4.39e-13 | 1 | ~0.051 |
| 310.00 | 0.70 | 500 | ~6.49e-13 | 2 | ~0.088 |
Brownian motion is the irregular movement of microscopic particles driven by thermal collisions with surrounding molecules. In liquids and gases, the effect becomes measurable when particle sizes are in the nanometer to micrometer range. This calculator links observable displacements to diffusion physics using consistent SI conversions and clear outputs.
The diffusion coefficient D sets the scale of random spreading. Typical values vary widely: small dye molecules in water can reach about 10−9 m2/s, while a 500 nm bead may be near 10−13–10−12 m2/s. When you enter D, the calculator predicts mean-squared displacement and RMS distance for selected times.
For ideal diffusion, the mean-squared displacement follows ⟨r2⟩ = 2·d·D·t. Dimension d matters because each axis contributes equally. In 3D, the expected spread grows faster than in 1D for the same D and t. The calculator reports both ⟨r2⟩ and the RMS displacement √⟨r2⟩.
When D is unknown, it can be estimated from temperature T, viscosity η, and particle radius r using D = kBT / (6π η r). Increasing temperature raises D, while higher viscosity or larger radius lowers D. This mode is useful for quick planning, sensitivity checks, and order-of-magnitude validation against experiments.
If you measure D from tracking or dynamic light scattering, the inverse relation r = kBT / (6π η D) estimates an effective hydrodynamic radius. The estimate reflects surface coatings and solvent interactions, not just geometric size. The calculator returns r in meters and nanometers for convenient reporting.
In one dimension, displacements follow a Gaussian distribution with width set by 2Dt. The calculator can compute the probability density p(x,t) for a chosen displacement x, which helps interpret histogram bins from tracking data. Large |x| values become exponentially unlikely as exp(−x2/(4Dt)).
The simulation mode generates a stepwise trajectory using per-axis increments with standard deviation σ = √(2DΔt). Over many steps, the simulated path fluctuates, but the predicted MSD remains a reliable benchmark. The table shows sample points along the path and a final |r| value for comparison with RMS predictions.
Start by choosing viscosity and temperature consistent with your setup, then compute D from Stokes–Einstein or enter measured D directly. Use MSD mode to plan observation times that yield resolvable displacements, and use simulation to communicate expectations visually. Export CSV or PDF outputs to document parameters, assumptions, and results.
Begin with MSD mode if you already know D, or Stokes–Einstein mode if you know temperature, viscosity, and radius. These provide a quick baseline before running simulations.
Each spatial axis contributes 2Dt to the variance. In d dimensions, variances add, giving ⟨r2⟩ = 2·d·D·t. Higher dimension increases expected spreading.
Near room temperature, water is about 1 mPa*s. Warmer water is less viscous, increasing D. If you need high accuracy, use a temperature-specific viscosity value from a reliable source.
The inverse Stokes–Einstein estimate gives a hydrodynamic radius, influenced by surface coatings, solvent layers, and interactions. It can be larger than the geometric radius measured from images.
Use enough steps so total time matches your experiment and dt is small compared with the timescale you care about. Hundreds to thousands of steps are typical for quick intuition checks.
It is the likelihood per meter of observing displacement x at time t for ideal diffusion. It helps compare theory to displacement histograms, especially when bin widths are known.
The formulas assume normal diffusion in a uniform medium. In confinement, viscoelastic fluids, or crowded environments, diffusion can become anomalous and effective D may depend on time or scale.
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.