# Brownian Motion & Random Walk

Explore random processes from physics to finance, Einstein's diffusion theory, and geometric Brownian motion

> Canonical page: https://elysiatools.com/en/visualizations/brownian-motion-random-walk

- **Category:** Math

## Overview

Comprehensive interactive visualization exploring Brownian motion and random walks from physics to finance, bridging statistical mechanics and mathematical finance. Features multiple simulation modes: Physics mode (standard Brownian motion with Einstein diffusion theory, particle trajectory visualization, ensemble average calculation), Finance mode (geometric Brownian motion for stock price simulation, Black-Scholes option pricing, volatility and drift analysis), and Math mode (simple random walk, Central Limit Theorem demonstration, Wiener process properties). Real-time particle tracking with 1-100 particles, adjustable parameters: diffusion coefficient D (0.1-10), drift rate μ (-2 to 2), time step dt (0.001-0.1), and volatility σ for finance. 2D/3D view switching with interactive rotation for 3D trajectories. Statistical analysis including mean square displacement ⟨x²⟩ with linear regression to verify Einstein relation ⟨x²⟩ = 2Dt, position distribution histogram with theoretical Gaussian overlay, variance σ² calculation, and ensemble average plots. Financial instruments: stock price paths using geometric Brownian motion dS = μS dt + σS dW_t with analytic solution S_t = S_0 exp((μ - σ²/2)t + σW_t), European call/put option pricing via Black-Scholes formula C = S·N(d₁) - K·e^(-rT)·N(d₂), realized volatility tracking, maximum drawdown calculation, and risk-neutral valuation demonstration. Initial conditions: single particle at origin, multiple particles, grid layout, or random distribution. Preset scenarios: Einstein's diffusion verification (1905), stock price simulation, pollen grain observation (Brown 1827), option pricing, and CLT demonstration. Mathematical theory coverage: simple random walk S_{n+1} = S_n + ξ_n, continuum limit to Brownian motion, Fokker-Planck diffusion equation, Itô calculus with (dW_t)² = dt, scaling law x ~ √t, and Wiener process properties (W_0=0, independent increments, continuous paths). Historical context timeline: Robert Brown (1827), Louis Bachelier (1900), Albert Einstein (1905), Jean Perrin (1908, Nobel 1926), Norbert Wiener (1923), Black-Scholes-Merton (1973, Nobel 1997). Educational experiments: verify Einstein relation, observe Gaussian emergence from discrete steps, drift effect analysis, stock market scenarios comparison (bull/bear markets), and Monte Carlo option pricing. Keyboard shortcuts: Space (start/pause), S (step), R (reset), P (pause). Multi-language support (zh, en, es, fr, de, ru, pt).

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