PHY.2:5.2 - Construct a mechanism whose short-time consequence differs from diffusion

A spreading population of moving particles is approximately diffusive over long observations. The target question is whether that approximation can describe the first short interval after a localized release. Consider a proposed mechanism in which motion persists for a while before its direction changes.

Build a simple source in thought: independently driven shuttles on a line move at speed u, reversing direction at random times. The waiting times are independent exponentials with reversal rate lambda. A drive maintains the speed, and a controller supplies the reversals. Treat reversal duration as negligible relative to the intervals being considered. This source combines sustained physical motion with stochastic switching. Using it for the particles proposes a relation between their direction persistence and their observed spreading.

The motion already supplies a useful qualitative consequence. Starting at x=0, a shuttle travels path length u*t in elapsed time t; reversals can only reduce its distance from the start. Thus abs(x(t)) <= u*t. Reaching a point at distance d requires at least time d/u. An ideal diffusion law with positive diffusivity instead assigns positive probability beyond every finite distance at every positive time. These are different predictions even when their long-time spreading agrees. An arrival before d/u would contradict the proposed bounded-speed target mechanism under its stated speed and preparation. Whether an earliest-arrival observation would discriminate the mechanisms also depends on its resolution and on the probability predicted for such arrivals. The bound is already a source result; the target correspondence remains a hypothesis.

For a small interval dt, reversal has probability lambda*dt to first order. If p_plus and p_minus are the position densities of right-moving and left-moving shuttles, transport and exchange between the two states give:

partial_t p_plus  = -u*partial_x p_plus  - lambda*p_plus + lambda*p_minus
partial_t p_minus = +u*partial_x p_minus + lambda*p_plus - lambda*p_minus.

The total density is p=p_plus+p_minus and the flux is j=u*(p_plus-p_minus). Adding and subtracting the equations yields:

partial_t p = -partial_x j
partial_t j = -u^2*partial_x p - 2*lambda*j.

Direction is the additional state that lets the source retain motion between changes. Omitting it too early would erase the short-time effect under investigation.

Release all shuttles at x=0 with equal probabilities of the two directions, on an unbounded line. The mean position stays zero. Multiplying the equations by x and x^2 and integrating, with vanishing boundary terms, gives for the mean-square displacement M(t):

M''(t) + 2*lambda*M'(t) = 2*u^2
M(0) = M'(0) = 0
M(t) = (u^2/lambda)*[t - (1-exp(-2*lambda*t))/(2*lambda)].

At short times M(t) is approximately u^2*t^2: particles mostly retain their direction. At long times its leading growth is (u^2/lambda)*t, corresponding to diffusion coefficient D=u^2/(2*lambda). The source thus produces a long-time diffusion law while giving a different short-time consequence. If u and lambda both have numerical value 1 in the chosen units, M(1) is approximately 0.568 square length units, while the leading diffusion expression gives 1.

The first result is a conditional explanation and a candidate change of observation interval. Long-time diffusive behavior alone leaves the proposed persistence mechanism unresolved. A useful return is to compare direction correlation or early spreading when either could distinguish it from a competing mechanism. An existing measurement can suffice. A physical shuttle apparatus adds no value if the derivation already supplies the consequence the work needs.

Now suppose target turns have an appreciable duration, or their occurrence depends on how long the present run has lasted. The source’s instantaneous, memoryless reversal rule no longer supplies that target behavior. Construct the missing turn state or waiting-time dependence, recover its physical interpretation, and derive its effect on the requested interval. The previous model remains a limiting construction where those effects are negligible. For a material shuttle implementation, finite acceleration also limits how short an interval it can reproduce.