From persistent motion to a justified diffusion calculation

1. Keep the preparation with the microscopic law. In PHY.8:5.3, particles start at the origin on an unbounded line, with either direction equally likely. Each moves at speed v>0 and reverses direction at independent Poisson events of rate alpha>0. These are supplied physical premises; a fitted position histogram would not establish the reversal mechanism or its time scale.

2. Turn those premises into the collective evolution. PHY.6 balances transport and transitions between the two directions. With n the total position-probability density and j its current, the resulting relations are

partial_t n = -partial_x j,

partial_t j = -v²*partial_x n - 2*alpha*j.

The ideal point preparation leaves probability atoms at x=±v*t for particles that have not yet reversed; these density equations are understood in the distributional sense. PHY.8 connects their solution to the observable needed here: mean-square displacement at a specified time. The current carries directional persistence even though the mean position stays zero.

3. Decide whether the faster response can be omitted. PHY.5 compares the observation time with the current’s relaxation time 1/(2*alpha) and checks the spatial variation. When the reduction is justified, MMP.9 supplies the mathematical reduction: replace the relaxed current by j approximately -D*partial_x n, with D=v²/(2*alpha), to obtain a diffusion equation. The question still decides whether its consequence is accurate enough.

For v=2 cm/s and alpha=1/s, PHY.8’s full account gives

E[x(t)²] = (v²/alpha)*(t-(1-exp(-2*alpha*t))/(2*alpha)),

while diffusion gives 2*D*t. At 20 s these are about 78 and 80 cm². The relative overestimate is about 2.56%, so a 3% allowance for this observable permits the reduced result under the supplied premises. This is a comparison with the microscopic model, not an experimental validation of it.

4. Return through the consequence that changed. Ask instead for the spread at 0.1 s. The full result is about 0.03746 cm²; diffusion gives 0.4 cm². Restore the current response for this early question. A finer computation of the same diffusion equation cannot recover what its reduction removed. A question about arrival at a boundary would need the corresponding boundary conditions and its own comparison; agreement on mean-square displacement does not supply that answer.

A simulation used to obtain these timed predictions must implement the stated reversals and their physical time. An artificial sampler that reproduces a position distribution supplies a different result. If the conditional prediction already serves the work, no new measurement is mandatory; if the physical mechanism or preparation is the consequential uncertainty, that becomes the next physical question.