Guided reading · At your own pace
From wandering to a molecular number
Choose what to measure, explain a growing spread, then ask what the observations can actually identify.
7 stops. Start at the beginning or choose any stop. Every question and explanation is available below without submitting an answer.
This path strings together pages of the edition; it is not a reviewed account of how the paper came about. A laboratory works out what a claim implies, and what it shows is a calculation, not an observation.
With JavaScript, the guide travels with you above the paper or experiment. Without it, return to this outline using your browser's Back command; the full route remains readable here.
Stop 1 · read
Start with a visible speck
Begin with the no-algebra entrance. Compare signed displacement with the size of a displacement.
Consider: Can the average signed displacement vanish while the particles keep spreading?
Read the explanation without answering
Left and right displacements can cancel in the average. Squared displacements do not cancel. A small mean is not evidence that nothing moved.
Stop 2 · experiment
Build the spread from steps
Inspect the independent-step model and its variance. Distinguish this teaching walk from a model of every molecular collision.
Consider: Which assumption makes the cross terms disappear when the displacement sum is squared?
Read the explanation without answering
Independent, zero-mean increments give zero mean cross terms. The variance then adds across steps. Correlated increments need a different calculation.
Stop 3 · experiment
Read a probability distribution
Compare the distribution at two times. Inspect a selected interval rather than treating the height of the density as its probability.
Consider: Does four times the observation time mean four times the typical displacement?
Read the explanation without answering
In free diffusion the one-coordinate mean square grows with time, so its square root grows with the square root of time. Interval probability is area under the density, not its height.
Stop 4 · experiment
Find diffusion without following every collision
Follow the balance between directional drag and random spreading, keeping its equilibrium assumptions visible.
Consider: Why can an auxiliary force help determine diffusion even though it drops out of the final relation?
Read the explanation without answering
The force connects drift and osmotic equilibrium. Matching the two descriptions relates diffusion to mobility and temperature. The zero-force case is a limit of that argument, not a division of zero by zero.
Stop 5 · experiment
Ask what a recording identifies
Inspect the diffusion estimate and its assumptions before interpreting a molecular number. Keep radius, viscosity and temperature explicit.
Consider: Is a diffusion estimate alone enough to determine a molecular number?
Read the explanation without answering
The molecular-number inference also needs the physical inputs in the diffusion relation. A statistical interval does not account for every possible calibration error or a wrong physical model.
Stop 6 · experiment
Separate the particle from its measurement
Compare the particle, camera and estimate. Look for assumptions broken by blur, localization error or observation timing.
Consider: Could changing the camera change the estimate without changing the liquid?
Read the explanation without answering
A recording is a measurement model as well as a particle model. Observational effects can alter inferred diffusion even when the underlying motion is unchanged.
Stop 7 · read
Return to the whole argument
Read the paper's two routes to diffusion and its closing observable prediction. Use the facsimile and the edition's stated availability of each reading face.
Consider: How would you explain the chain from visible displacements to molecules, naming what must be known independently?
Read the explanation without answering
A complete account separates the displacement statistic, the diffusion model, the relation to mobility, the calibrated inputs, and the uncertainty of the inference. Finishing this route is not a test of mastery.
Carry the argument beyond this route
You have reached the outline's final stop, not a certification of understanding. Return to any question, try a changed assumption, or explain which step would fail if that assumption changed.