Brownian Motion: A Comprehensive Guide
Introduction
Brownian motion, also known as Wiener process or Wiener path, is the irregular movement of tiny particles suspended in a liquid or gas, resulting from their collision with fast-moving atoms or molecules.
Historical Background
In 1827, Scottish botanist Robert Brown observed the erratic motion of pollen grains in water under a microscope. The phenomenon was later named Brownian motion in his honor.
Mechanism
Brownian motion is caused by the incessant bombardment of particles by surrounding molecules. In a fluid, these collisions cause the particle to move in random directions, resulting in the observed zigzag trajectory.
Mathematical Description
The mathematical model of Brownian motion, known as the Wiener process, is a continuous-time stochastic process. The position of the particle is represented by a random variable X(t) that follows a normal distribution with mean and variance proportional to time.
Applications
Brownian motion finds applications in various fields such as physics, chemistry, biology, and finance:
- Diffusion and coagulation processes
- Modeling stock market fluctuations
- Understanding protein interactions
Conclusion
Brownian motion is a fundamental physical phenomenon that has been extensively studied and applied. From its origins in microscopic observations to its far-reaching implications in diverse scientific disciplines, Brownian motion continues to captivate scientists and inspire new discoveries.
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