Simulations

The Wiener Process

The Wiener process $W_t$ is the scaling limit of a random walk, characterized by four properties:

$W_0 = 0$ a.s.
Independent increments: $W_t - W_s \perp \mathcal{F}_s$ for $t > s$ .
Gaussian increments: $W_t - W_s \sim \mathcal{N}(0, t-s)$ .
Continuous paths a.s.

Theorem (Lévy's Construction).

The Wiener process can be constructed via the Karhunen-Loève expansion:

W_t = t \cdot Z_0 + \sum_{n=1}^\infty \frac{\sqrt{2}}{\pi n} Z_n \sin(n \pi t)

where $Z_n \sim \mathcal{N}(0, 1)$ are i.i.d. The series converges uniformly on $[0, 1]$ a.s.

Despite being continuous, $W_t$ is nowhere differentiable. The graph has Hausdorff dimension $3/2$ ; the path in $\ge 2$ dimensions fills space with dimension $2$ . The self-similarity $\frac{1}{\sqrt{c}} W_{ct} \stackrel{d}{=} W_t$ is behind its appearance across scales in physics and finance.

Fig 1.1: 2D Brownian Motion (Wiener Process)

The Ising Model

Spins $\sigma_i \in \{-1, +1\}$ on a lattice $\Lambda$ , with Hamiltonian

H(\sigma) = -J \sum_{\langle i, j \rangle} \sigma_i \sigma_j

and Gibbs measure $\mu(\sigma) = \frac{1}{Z} e^{-\beta H(\sigma)}$ .

In $d \ge 2$ there is a critical temperature $T_c$ . Above $T_c$ correlations decay exponentially (paramagnetic phase). Below $T_c$ the system magnetizes spontaneously.

Theorem (Peierls Argument (1936)).

For sufficiently low temperatures in 2D, the ordered state is stable. A domain wall of length $L$ costs energy $2JL$ , while the number of such contours grows as $3^L$ . The probability of a contour is bounded by $(3 e^{-2\beta J})^L$ , which is summable for large $\beta$ .

Near $T_c \approx 2.269$ (Onsager), observables follow power laws: $m \sim (T_c - T)^\beta$ , $\chi \sim |T - T_c|^{-\gamma}$ . The exponents depend on dimension but not on lattice details (universality).

Temperature: 2.27Critical

Fig 1.2: 2D Ising Model (Draw to Magnetize)

The Galton Board

Each ball bounces left or right with $p=1/2$ at every peg. The final position is $X = \sum_{i=1}^N B_i$ , a sum of independent Bernoulli trials.

Theorem (De Moivre-Laplace Theorem).

As $N \to \infty$ :

\binom{N}{k} p^k (1-p)^{N-k} \approx \frac{1}{\sqrt{2\pi N p(1-p)}} \exp \left( - \frac{(k - Np)^2}{2 N p(1-p)} \right)

In the continuum limit, the particle density $u(x, t)$ satisfies the heat equation $\partial_t u = D \Delta u$ . The Gaussian is its Green’s function — the density that maximizes entropy subject to a variance constraint.

Fig 1.3: Galton Board (Central Limit Theorem)

Geodesics on Curved Manifolds

On a Riemannian manifold $(M, g)$ , a geodesic $\gamma(t)$ satisfies the equation

\frac{d^2 x^k}{dt^2} + \Gamma^k_{ij} \frac{dx^i}{dt}\frac{dx^j}{dt} = 0

where $\Gamma^k_{ij}$ are the Christoffel symbols of the Levi-Civita connection. On surfaces of constant positive curvature ( $S^2$ ), geodesics are great circles. On the torus $T^2$ , Gaussian curvature $K = \cos\theta / (R + r\cos\theta)r$ changes sign, positive on the outer equator, negative on the inner, zero at top and bottom. On a saddle (hyperbolic paraboloid), $K < 0$ everywhere and geodesics diverge.

Vertices are colored by Gaussian curvature, warm tones for $K > 0$ and cool tones for $K < 0$ .

K > 0 outer, K < 0 inner, K = 0 top/bottom0 geodesics

Fig G.1: Geodesic Explorer (Double-Click Surface to Add Geodesics)

Weather-Driven Flow

A particle system with drift and diffusion set by live weather from Chapel Hill, NC (Open-Meteo API). Wind velocity $(v_x, v_y)$ determines the mean drift, temperature $T$ scales the diffusion coefficient $\sigma^2 \propto (T - T_0)^+$ , precipitation adds a gravitational component, and cloud cover modulates particle density.

Fig L.1: Weather-Driven Particle Field (Live)

Ornstein-Uhlenbeck on Live Forecast

The 24-hour temperature forecast $\mu(t)$ for Chapel Hill serves as the mean function for a time-inhomogeneous Ornstein-Uhlenbeck process:

dX_t = \theta(\mu(t) - X_t) \, dt + \sigma \, dW_t

The stationary variance $\text{Var}(X_t) = \frac{\sigma^2}{2\theta}(1 - e^{-2\theta t})$ gives the $2\sigma$ confidence band. The forecast curve is deterministic, the sample paths are stochastic. Real data from Open-Meteo.

Fig L.2: Ornstein-Uhlenbeck Paths on Live Forecast (Live)

Bayesian Visitor Profile

A Dirichlet prior $\text{Dir}(\alpha_1, \ldots, \alpha_6)$ over six interest categories, updated from scroll position and visible content. Each observation increments the corresponding $\alpha_i$ and the posterior mean $\mathbb{E}[\pi_i] = \alpha_i / \sum_j \alpha_j$ concentrates as evidence accumulates, with credible intervals narrowing at $O(1/\sqrt{n})$ . State persists in localStorage.

Fig L.3: Bayesian Visitor Profile (Live)