Brownian Motion Learning Path

Definition and physical setting of Brownian motion

Brownian motion is the random motion of particles suspended in a medium such as a liquid or a gas. In the traditional mathematical description, it is modeled by the Wiener process (often called “Brownian motion” itself). Physically, the motion appears as irregular fluctuations of a particle’s position within a small region, followed by relocation to another region; this repeats over time. This behavior is characteristic of a fluid at thermal equilibrium at a given temperature. In equilibrium there is no preferred direction of flow: the fluid’s overall linear and angular momenta remain zero over time. The particle’s kinetic energy is part of the fluid’s internal energy, consistent with the equipartition theorem, where translational, rotational, and vibrational molecular motions collectively contribute to the caloric component of internal energy. The phenomenon is named after Robert Brown, who observed pollen grains in water undergoing continuous jittery motion under a microscope. The underlying physical picture is that the particle is bombarded by molecules whose directions and impacts change rapidly, producing the seemingly random trajectory; because the full many-body dynamics cannot be solved directly, probabilistic models are used to describe the motion.

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Wiener process as mathematical Brownian motion

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