Fractional Laplacian · diffusion that can jump over walls

Fractional Diffusion Maze

Spatial fractional derivatives help us model processes that are both diffusive and nonlocal, with particles that can occasionally "jump" or "teleport", spattering through space via Lévy flights. It captures many phenomena more accurately than classical diffusion models. These include the migration patterns of sharks and other large predators[1], patterns in disease spread[2], sediment transport in riverbeds[3], and the exploratory strategies of various microorganisms[4]. Here we build a simple model that illustrates the difference between fractional and standard diffusion. The left window shows ordinary diffusion: Brownian walkers creep along corridors and can never reach a sealed chamber. The right window shows fractional diffusion: Lévy walkers can teleport, taking heavy-tailed jumps that sail over walls, so ink appears in places no corridor can reach.

  1. D.W. Sims et al. — Scaling laws of marine predator search behaviour, Nature 451 (2008). doi
  2. D. Brockmann, L. Hufnagel, T. Geisel — The scaling laws of human travel, Nature 439 (2006). doi
  3. R. Schumer, M.M. Meerschaert, B. Baeumer — Fractional advection-dispersion equations for modeling transport at the Earth surface, J. Geophys. Res. 114 (2009). doi
  4. G. Ariel et al. — Swarming bacteria migrate by Lévy walk, Nature Communications 6 (2015). doi
Brownian motion α = 2 · reflecting walls
In sealed-off chambers
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In the probe tile
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Lévy flights α = 1.00 · jumps pass over walls
In sealed-off chambers
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In the probe tile
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T = 0.32corridor times
Probability per tile

Hover a tile to read both values. Click to act with the tool chosen below.

Jump lengths have tail ∝ r−2−α. Smaller α means longer, more frequent leaps. α → 2 recovers the left panel.

Walls

Maze
Click on the maze to
Color range

Lévy tails are algebraic, so with a wide range the whole maze lights up almost at once. A narrow range shows where the mass actually concentrates.

Grid

Arrival at the probe tile

BrownianLévy

What is being solved

  • Space. Each corridor tile is a 4×4 (or 6×6) block of cells. The jump rate between cells i, j is C|k|−2−α in grid units, plus a nearest-neighbour correction −Cζ(α/2)β(α/2) that makes the lattice symbol |ξ|α + O(|ξ|4). Without it the operator is off by up to 25% at α = 1.7. As α → 2 the weights become the 5-point Laplacian exactly.
  • Time. There is no time stepping. The solver builds a Krylov basis for u(t) = e−tAu0 (Lanczos, full reorthogonalisation, the constant null vector removed exactly) and keeps going until the a-posteriori residual bound ‖u − um‖₂ ≤ ‖u0‖ βm∫₀t|emTe−τTme1|dτ is below 10−12 for every time on the slider. The bound only needs the operator to be symmetric and positive semidefinite, which holds in all three wall modes. Scrubbing the slider evaluates that one solution; nothing accumulates.
  • Time units. Both panels use the same T = t/τα, where τα = wα is the time each process needs to spread about one corridor width w (diffusivity 1).

Solver checks

BrownianLévy
Unknowns––
Krylov dim.––
Error bound––
|mass − 1| now––
Solve time––

Offline checks of this exact code: the Krylov solution matches a dense eigendecomposition to 4×10−14 for α from 0.3 to 2; the operator's symbol matches |ξ|α to 2×10−6 at ξ = 0.05; and refining the grid from 4 to 8 cells per corridor changes probe values by under 0.4%. The absorbing and line-of-sight operators were rebuilt independently in Python (exact segment–square visibility tests, brute-force lattice sums for the killing rate); the visibility graphs are identical and the solutions agree to the reference's own accuracy. Switch the grid to 6 cells to see how much each picture moves.