Heat flow in quantile coordinates — interactive experiment

Numerical experiment · deterministic quantile trajectories

Follow equal amounts of mass through the heat flow

A mixture of three Gaussians diffuses. Each colored marker keeps the same cumulative mass rank s, so its position is q(s, t). These are deterministic transport paths, not Brownian sample paths.

Tracked quantiles 40 Ranks s = (i + ½)/n; changing n only changes the display. Initial cluster separation a 2.00 Move from overlapping bumps to three distinct clusters. Heat time t 0.00 Each component variance increases by 2t.

Paused at t = 0.00

Density and equal-mass markers

InitialAt time t

Markers are not equally spaced in space: they crowd where the density is high. Color follows cumulative mass rank, from blue through teal to orange.

The quantile function q(s, t)

Horizontal axis: mass rank s. Vertical axis: position x. The curve stays increasing; the gray curve is q(s, 0).

The paths t ↦ q(s, t)

Horizontal axis: heat time. Vertical axis: position x. Paths cannot cross, but they need not move outward at every instant.

Total mass: 1
Mean (conserved): —
Variance: —
Variance increase: 0.000

Numerics: exact Gaussian convolution followed by numerical inversion of the mixture CDF. No time-discretized JKO scheme is used here. Curves show central quantiles only: the endpoint quantiles s = 0 and s = 1 are infinite.