The restless path of a tracer

Can random motion have a predictable law?

Can random motion have a predictable law?

  1. Run the tracer and inspect the trail from two camera angles.
  2. Double radius while holding temperature fixed.
  3. Compare the theoretical RMS displacement at equal time.

What to measure

Units and parameter definitions are shown beside every laboratory control and live reading.

The idea underneath

A suspended tracer receives many uneven microscopic impulses. Its individual path is irregular, but the mean squared displacement of an ensemble follows a simple law. The laboratory integrates independent Gaussian increments with variance 2DΔt in each direction. This is an overdamped diffusion model, not a resolved simulation of individual solvent impacts. The effective coefficient increases with temperature and decreases with tracer radius, following the qualitative Stokes–Einstein scaling for fixed viscosity. Compare the ensemble spread, not the endpoint of a single random run.

⟨r²⟩=6Dt · D∝T/a

Units and parameter definitions are shown beside every laboratory control and live reading.

Build the argument

  1. Specify the ensemble, particle mass, temperature and boundary conditions.
  2. Average over many particles before comparing microscopic and macroscopic quantities.
  3. For independent isotropic diffusion, each coordinate has variance 2Dt, giving mean squared displacement 6Dt in three dimensions.

Worked example 1

D=0.06 m²/s over 10 s gives RMS=√3.6=1.897 m.

Worked example 2

Four times the time gives twice the RMS distance.

Applications

Microscopic statistics connect to pressure, diffusion and transport. The visible particles represent an ensemble; their rendered size and playback speed are illustrative.

Common misconception

Temperature is a distribution of molecular kinetic energies, not the speed of every molecule. A stochastic trajectory cannot predict a particular future collision.

A further investigation

RMS displacement grows with time as…

Check your understanding

RMS displacement grows with time as…

  • √t
  • t²
  • A constant
Explanation

Mean squared displacement is linear in time; its square root is the RMS distance.

Model limits

Gaussian overdamped diffusion at fixed viscosity; no hydrodynamic interactions. Distances are model coordinates.

MIT OpenCourseWare · Random walks and diffusion