Presets
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Fluid
Water at 20 C is 1.0 mPa s.
Beads
Time and view
Choose the time step to see the ballistic or the diffusive regime.
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The science
Why a bead jiggles
A micrometre bead in water is hit about 1020 times a second by solvent molecules. Each kick is tiny, but the sum does not cancel exactly: the bead does a random walk (Robert Brown, 1827). Einstein (1905) showed that the mean square displacement grows linearly in time and that its slope gives the diffusion coefficient:
⟨Δr2⟩ = 2 d D t (d = 1, 2 or 3 dimensions), D = kBT / (6 π η a)The second formula is the Stokes-Einstein relation: thermal energy divided by the Stokes drag coefficient γ = 6πηa. Perrin used it to measure Avogadro's number from the motion of beads.
The model in this page: Langevin dynamics
Each bead obeys m dv/dt = −γv + √(2γkBT) ξ(t). The page uses the exact update of the Ornstein-Uhlenbeck process for velocity and position (correlated Gaussian noise), so the result does not depend on the step size:
v(t+Δt) = v e−Δt/τ + noise, τ = m/γ, ⟨Δx2⟩ = 2 (kBT/m) τ2 (t/τ − 1 + e−t/τ)For t << τ (some 10 ns for a 1 µm bead) the motion is ballistic, ⟨Δx2⟩ = (kBT/m) t2, and the bead has a Maxwell-Boltzmann velocity with (1/2) m ⟨vx2⟩ = (1/2) kBT (equipartition). For t >> τ it is diffusive. Choose the "Ballistic regime" preset to see the crossover on the log-log graph.
Honest simplifications
- The fluid is a continuum with constant viscosity (Stokes drag, no hydrodynamic memory, no added mass or Basset force); those matter at sub-microsecond times for real beads, where the true velocity correlation decays as a power law t-3/2.
- 2D motion in a periodic box is shown; the bead mass assumes density 1050 kg/m3 (polystyrene). No walls and no bead-bead interactions.
- The thermal speed (a few mm/s for a 1 µm bead) is not observable in a real microscope at normal frame rates because the direction changes in nanoseconds; that is exactly why the measured tracks look like a random walk.
What to try
- Run Water, 1 um bead and press Collect 2000 steps now: the measured D agrees with Stokes-Einstein within a few percent.
- Halve the bead radius: D doubles (the white dot moves up the green curve in graph 4).
- Run Viscous glycerol mix: the same bead diffuses 20 times slower.
- Run Hot water 90 C: D rises through both T and the lower viscosity.
- Run Ballistic regime: the MSD slope on the log-log graph is 2, not 1.