Brownian

Brownian refers to the random motion of small particles suspended in a fluid, first observed by the botanist Robert Brown in 1827 while examining pollen grains in water. He noted that the particles moved erratically and without a discernible pattern. The phenomenon is now understood as the result of countless collisions with molecules of the surrounding fluid, transmitting momentum and energy to the suspended bodies.

Theoretical explanations emerged from the kinetic theory of heat. Albert Einstein, in 1905, provided a quantitative description linking Brownian motion to molecular agitation and diffusion, predicting how far particles would wander over time and establishing a measurable connection to molecular properties. Jean Perrin conducted experiments that confirmed these predictions, providing strong evidence for the molecular nature of matter and the reality of atomic-scale motion.

In mathematics, Brownian motion describes a continuous-time stochastic process with specific probabilistic properties. A standard Brownian motion, or Wiener process, W_t, starts at zero, has independent and stationary increments with Gaussian distribution, and almost surely exhibits continuous paths. The variance grows linearly with time, Var(W_t) = t. Brownian motion serves as a canonical model of random diffusion and is foundational to Ito calculus and many stochastic differential equations, often written in differential form as dX_t = μ dt + σ dW_t.

Applications extend across disciplines. In physics and chemistry, Brownian motion models diffusion and transport phenomena. In finance, geometric Brownian motion is used to model stock price dynamics. In numerical analysis and physics, Brownian paths underpin simulations of diffusion processes and methods for solving certain partial differential equations. The concept illustrates how microscopic randomness can give rise to macroscopic diffusion behavior.