# 布朗运动模拟 - Brownian Motion Simulation

展示粒子随机运动和扩散现象的交互式模拟

> 标准页面: https://elysiatools.com/zh/visualizations/brownian-motion

- **分类:** Physics

## 概述

Interactive simulation of Brownian motion demonstrating random particle movement and Einstein's diffusion theory. Features the fundamental equations: Mean Square Displacement <x²> = 2Dt, Einstein Relation: D = k_BT/(6πηr), Stokes' Law: F = 6πηrv, and Random Step: dx = √(2Ddt)·ξ where ξ~N(0,1). Real-time visualization includes: (1) Main canvas showing large particle's random trajectory with color-graded trail (blue→red indicating time progression), background small particles simulating thermal molecular motion, coordinate axes with origin marker, and displacement vector showing distance from starting point; (2) Distribution histogram displaying final displacement statistics from 100-trial runs comparing experimental data against theoretical Rayleigh distribution; (3) Real-time statistics: elapsed time, current displacement, mean square displacement (MSD), and diffusion coefficient D. Adjustable parameters: temperature T (100-500 K), particle radius r (0.1-5 μm), fluid viscosity η (0.1-10 mPa·s), time step dt (0.001-0.1 s), trail length (10-500 steps), and background particle count (0-200). Control modes: Start/Pause for single simulation, Reset to return to origin, and 'Run 100 Trials' for statistical analysis generating displacement distribution histogram. Display options: show/hide particle trail, background particles, coordinate grid, and color gradient trail. Educational content covers historical discovery by Robert Brown (1827), Einstein's 1905 theoretical explanation proving atomic existence, diffusion coefficient calculation using Boltzmann constant (k_B = 1.38×10⁻²³ J/K), Stokes' drag force balancing thermal fluctuations, and applications in statistical mechanics, polymer science, finance (stock models), biology (cell membranes), and nanotechnology. The simulation accurately demonstrates how higher temperature, smaller particles, or lower viscosity increase diffusion rate, and validates the theoretical relationship MSD = 2Dt over time. Multi-language support (zh, en, fr, de, es, ru, pt).

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