Watch the path before averaging it
Drop a speck of dust into water and magnify it. Molecules strike it from all sides, and its position jitters. Brownian motion is the name for this irregular movement. A self-propelled particle jitters too, but it also pushes along a body direction, like a tiny boat whose rudder slowly wanders.
At first the boat follows an almost straight path. Wait longer and the path bends, loops and crosses itself. A single photograph hides the difference: both a passive grain and an active swimmer may sit at the same place. A short movie reveals whether the next step tends to continue the previous one.
A particle becomes active when it continually converts supplied energy into motion. Heating a passive suspension can also make grains spread faster, but it does not give each grain a long-lived body direction. The direction is the extra piece of information we need to keep.
Memory has a time and a distance
Imagine marking the heading of every swimmer at the start. Later, compare each new heading with its own initial one. When the swimmers have largely forgotten where they faced, one persistence time has passed. The persistence length is the distance a freely swimming particle covers during that time.
A persistence length of ten body lengths means a swimmer can reach the far side of a small crowd before it turns much. The same swimmer in a much larger crowd may turn repeatedly before escaping. Compare the particle size with this length to judge how long a collision can persist.
Worked example
A swimmer travels six micrometres before it turns
Take an ideal two-dimensional swimmer that moves 2 micrometres each second and remembers its heading for 3 seconds. Its persistence length is 6 micrometres. Observe it for a tenth of a second: most tracks look nearly straight. Observe it for a minute: each track has changed direction many times.
Now double the swim speed and keep the turning process the same. The persistence length doubles. Over long times, the active contribution to the spreading rate becomes four times as large, because every nearly straight segment is twice as long and its squared displacement is four times as large. Doubling the persistence time instead gives twice the active spreading rate.
Why the average path can stay at the start
Release many swimmers with headings spread equally in every direction. Their average position can stay at the release point even though each swimmer travels a long way. The useful measurement is the average squared distance from the starting position: it counts how widely the group has spread without letting leftward and rightward motion cancel.
For a free swimmer, the propulsion contribution to this measure first grows like elapsed time squared, then roughly in proportion to elapsed time. Independent positional Brownian motion also contributes a term linear in time. The early active part records directional memory; the late active part records many differently directed runs. Crowding, walls and forces will alter this simple reference curve. We first need to recognise the free curve so that those changes have a meaning.