Brownian Motion Calculation with Code Implementation

Brownian motion is a fundamental physics model describing the irregular movement of microscopic particles suspended in a fluid medium. Numerical simulations of Brownian motion provide crucial insights into the characteristic behaviors of stochastic processes.

The core algorithm implementation employs random number generators to simulate particle displacement. In two-dimensional space, each computational time step generates normally distributed random displacements in both x and y directions. The particle trajectory emerges through cumulative summation of these stochastic displacements, typically implemented using array operations to store position history.

Three critical parameters govern the simulation accuracy: the diffusion coefficient (D) determines displacement magnitude through the relation Δx = √(2D*dt)*randn(), the time step (dt) controls temporal resolution, and the total step count defines trajectory length. Optimal parameter selection balances computational efficiency with physical realism, often requiring dimensional analysis for proper scaling.

This simulation methodology extends beyond physics to financial mathematics for stock price modeling and ecology for biological migration patterns. The framework can be adapted to simulate diverse random walk processes by modifying distribution types (Gaussian, Lévy, etc.) and implementing boundary conditions or drift terms through additional code modules.