Model random motion and connect data to theory. Choose dimensions, units, and calculation pathway easily. Get diffusion, displacement, and time estimates in seconds now.
⟨r²⟩ = 2 n D t so D = ⟨r²⟩ / (2 n t).D = kB T / (6 π η r), for spherical particles in a viscous fluid.rrms = √(⟨r²⟩) = √(2 n D t).t = rrms² / (2 n D).n is the number of spatial dimensions, η is dynamic viscosity, and kB is Boltzmann’s constant.
| Scenario | Inputs | Key output | Interpretation |
|---|---|---|---|
| 2D trajectory MSD | MSD = 0.50 (µm²), t = 2 s, n = 2 | D ≈ 0.0625 µm²/s | Slow diffusion typical of crowded media. |
| Stokes–Einstein estimate | T = 298 K, η = 1 mPa·s, r = 0.5 µm | D ≈ 0.44 µm²/s | Order-of-magnitude check for spheres in water. |
| Predict displacement | D = 0.25 µm²/s, t = 10 s, n = 2 | RMS ≈ 2.236 µm | Expected spread grows like √t. |
Brownian diffusion describes how random thermal kicks spread particles over time. The central metric is the diffusion coefficient D (m²/s). In microscopy, you often see D reported in µm²/s for convenience. The calculator links measured displacements to D, and it can also predict how far a particle is expected to wander after a chosen time interval.
For ideal diffusion, the mean squared displacement (MSD) grows linearly: ⟨r²⟩ = 2nDt. Here n is the number of spatial dimensions. Many tracking experiments are effectively 2D because the camera observes motion in the image plane. Choosing 1D, 2D, or 3D directly changes the inferred D for the same MSD and time.
Diffusion spans a wide range. Small molecules in water can reach about 10⁻⁹ m²/s, while micron-scale beads are often around 10⁻¹³ to 10⁻¹² m²/s depending on size and fluid conditions. Inside crowded gels or cytoplasm, effective diffusion may drop further due to obstacles and binding events.
When you know temperature T, viscosity η, and particle radius r, the Stokes–Einstein relation estimates diffusion: D = kBT/(6π η r). As a quick check, a 0.5 µm radius sphere at 298 K in a 1 mPa·s fluid gives D on the order of 0.4 µm²/s, matching common lab-scale expectations.
Because D ∝ 1/(ηr), doubling viscosity halves diffusion, and doubling radius halves diffusion. This is why temperature control and accurate size estimates matter in quantitative work. Even modest viscosity shifts (for example, adding glycerol or polymers) can change diffusion by multiples, not just a few percent.
RMS displacement is the square-root form of MSD: rrms = √(2nDt). It is useful for planning imaging windows. If you know D and choose a time step, RMS estimates the expected spread. The √t scaling means displacement grows slowly, so longer times are needed to see big changes.
Rearranging the same model gives t = rrms²/(2nD). This helps you infer how long diffusion would take to reach a target spread, or whether an observed displacement is unusually fast. Large deviations from the estimate can indicate drift, active transport, or non-diffusive dynamics.
Use consistent units and confirm what your “MSD” represents. MSD should be averaged over many steps or trajectories for stable estimates. The calculator converts inputs internally to SI and reports multiple D unit scales. If your results differ by orders of magnitude, re-check dimension choice, time units, and whether your length input is already squared.
If motion is measured in an image plane, 2D is usually appropriate. Choose 3D only when your displacement data represents full spatial motion, such as 3D tracking or volumetric imaging.
MSD is the average of squared displacement, ⟨r²⟩, over a time lag t. Enter it in (length)² units, then provide the matching time lag used to compute that MSD.
In ⟨r²⟩ = 2nDt, the same MSD is shared across more dimensions as n increases. Therefore D must be smaller for larger n when MSD and time are fixed.
Yes. Select °C and the calculator converts to kelvin internally using T(K) = T(°C) + 273.15. Diffusion depends on absolute temperature, not on Celsius directly.
Use mPa·s or cP for many liquids; they are numerically identical. Use Pa·s for strict SI inputs. Water near room temperature is roughly 1 mPa·s.
Random steps add in a way that MSD grows linearly with time. Taking the square root gives RMS, so RMS grows as √t. This is a key signature of normal diffusion.
Non-linearity can reflect drift, confinement, active transport, or measurement noise. Try shorter lag times, remove drift, or fit MSD versus time with an appropriate model rather than a single slope.
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.