Simulations

Interactive explorations of stochastic processes and phase transitions. Below are three fundamental systems in statistical physics, accompanied by their rigorous mathematical descriptions.

1. The Wiener Process (Brownian Motion)

The Wiener process $W_t$ is the scaling limit of a random walk. It is the fundamental object in stochastic calculus and is characterized by:

Starts at 0: $W_0 = 0$ almost surely.
Independent Increments: $W_t - W_s \perp \mathcal{F}_s$ for $t > s$ .
Gaussian Increments: $W_t - W_s \sim \mathcal{N}(0, t-s)$ .
Continuity: Paths $t \mapsto W_t$ are continuous almost surely.

Theorem (Lévy's Construction).

The Wiener process can be constructed as a random Fourier series (Karhunen-Loève expansion):

W_t = \frac{t}{\sqrt{\pi}} 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. This series converges uniformly on $[0, 1]$ almost surely, proving existence.

Fractal Properties: While continuous, $W_t$ is nowhere differentiable. It is a fractal with Hausdorff dimension transition $1.5$ (for the graph) and $2$ (for the path in $\ge 2$ dimensions). The process is Self-Similar: $\frac{1}{\sqrt{c}} W_{ct} \stackrel{d}{=} W_t$ . This scale invariance explains why it emerges universally in diverse multi-scale systems.

Fig 1.1: 2D Brownian Motion (Wiener Process)

2. The Ising Model & Universality

The Ising Model is the “hydrogen atom” of statistical mechanics—the simplest system exhibiting a phase transition. Consider a lattice $\Lambda$ of spins $\sigma_i \in \{-1, +1\}$ . The Hamiltonian is:

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

The probability of a configuration is given by the Gibbs measure $\mu(\sigma) = \frac{1}{Z} e^{-\beta H(\sigma)}$ .

Phase Transition: In $d \ge 2$ , there exists a critical temperature $T_c$ (Curie point).

For $T > T_c$ : The system is disordered (paramagnetic). Correlations decay exponentially.
For $T < T_c$ : The system is ordered (ferromagnetic). Spontaneous magnetization occurs ( $m \ne 0$ ).

Theorem (Peierls Argument (1936)).

For sufficiently low temperatures in 2D, the ordered state is stable. Proof Sketch: A domain wall (boundary between + and - spins) of length $L$ costs energy $2JL$ . The number of such contours grows as $3^L$ . The probability is bounded by $\sum (3 e^{-2\beta J})^L$ . For large $\beta$ (low $T$ ), this sum is small, implying that global order is practically strictly enforced.

Critical Exponents: Near $T_c \approx 2.269$ (Onsager’s solution), observables follow power laws: $m \sim (T_c - T)^\beta$ , $\chi \sim |T - T_c|^{-\gamma}$ . These exponents depend only on dimension, not lattice details—a phenomenon called Universality.

Temperature: 2.27Critical

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

3. The Galton Board & The Heat Equation

The Galton Board demonstrates the Central Limit Theorem physically. Balls bounce left or right with $p=0.5$ . The final position is the sum of independent Bernoulli trials: $X = \sum_{i=1}^N B_i$ .

Theorem (De Moivre-Laplace Theorem).

As $N \to \infty$ , the probability mass function of the Binomial distribution converges to the Gaussian PDF:

\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)

Connection to PDEs: Let $u(x, t)$ be the density of particles. The discrete random walk converges to the diffusion equation (Heat Equation):

\frac{\partial u}{\partial t} = D \Delta u

The Gaussian distribution is simply the Green’s function (fundamental solution) of the heat equation. It represents the entropy-maximizing spreading of mass over time.

Fig 1.3: Galton Board (Central Limit Theorem)