Paint with Brownian motion. Each mode is a different stochastic differential equation.
The simplest stochastic process: position evolves under pure white-noise forcing,
$$dX_t = \sigma\, dW_t,$$
where \(W_t\) is a standard Wiener process. The variance of position grows linearly
with time: \(\mathrm{Var}(X_t) = \sigma^2 t\). In the continuum limit, the density
of many such particles satisfies the heat equation
\(\partial_t \rho = \tfrac{\sigma^2}{2}\Delta \rho\) -- the microscopic side of the
diffusion term in the
Reaction-Diffusion toy.
Click or drag to spawn glowing plumes.
Add a drift toward the minimum of a potential \(V\):
$$dX_t = -\nabla V(X_t)\, dt + \sigma\, dW_t.$$
Particles thermalize in basins of \(V\) and occasionally hop between them
(thermal activation, Kramers' escape law). In the long-time limit the density
relaxes to the Boltzmann distribution
\(\rho_\infty \propto \exp(-2V/\sigma^2)\). This is the stochastic counterpart
to the deterministic gradient flow \(\dot X = -\nabla V(X)\) that underlies
Justin's dissertation work -- see
/p/research/.
Left-click to drop attractive wells. Right-click (or shift-click) for repulsive barriers.
Mean-reverting noise:
$$dX_t = -\theta(X_t - \mu)\, dt + \sigma\, dW_t.$$
Each particle is tethered to its nearest attractor \(\mu\) by a linear restoring
force of strength \(\theta\). The stationary distribution is Gaussian with
variance \(\sigma^2 / (2\theta)\), so cranking \(\theta\) tightens the cloud
and cranking \(\sigma\) loosens it.
Click to add attractors. Particles pool around the nearest one.
Replace the Gaussian increment with a heavy-tailed jump law,
\(\mathbb{P}(|\Delta X| > r) \sim r^{-\alpha}\) with \(\alpha \in (0, 2)\).
Variance is infinite for \(\alpha < 2\); the walk is dominated by rare,
very long flights. Observed in animal foraging, turbulent transport, and
photon scattering in disordered media.
Click to spawn walkers. Watch for the sudden long jumps.
A toy active-matter model. Position drifts along a heading \(\hat n\) with
fixed speed \(v\); the heading itself diffuses with rotational noise \(\sigma_r\):
$$dX_t = v\,\hat n_t\, dt + \sigma_t\, dW_t,
\qquad d\theta_t = \sigma_r\, dW^{(\theta)}_t.$$
Small \(\sigma_r\) gives long persistent runs (vortices emerge); large \(\sigma_r\)
recovers Brownian motion. The continuum limit is run-and-tumble dynamics.
Click to spawn particles. Right-click to flip nearby headings.
Walkers do unbiased random walks until they touch a growing cluster, at which
point they stick. The result has fractal dimension
\(d_f \approx 1.71\) in 2D (Witten and Sander, 1981). DLA structures show up
in mineral dendrites, lightning, viscous fingering, and electrodeposition.
$$d_f \approx 1.71$$
Click anywhere to drop extra nucleation seeds. Multiple clusters grow and eventually fuse. Use Pause to admire the structure.
The deterministic limit of these particle dynamics is the reaction-diffusion / gradient-flow PDE
explored in the
Reaction-Diffusion toy. The
energy-stable schemes in Justin's dissertation discretize that continuum side; this toy renders
the microscopic side directly. See
/p/research/.