Brownian Motion

A Brownian Motion is a random continuous motion of a particle group suspended in a fluid resulting from their collision with the fluid's molecules.

• AKA: Pedesis.
• Context:
  • It can (typically) describe the erratic movement of particles, such as pollen grains, in a liquid or gas.
  • It can (often) serve as a fundamental concept in statistical mechanics and stochastic processes.
  • It can provide the basis for various algorithms in computational simulations, such as Monte Carlo methods.
  • ...
• Example(s):
  • As observed in pollen grains suspended in water, showing erratic movements under a microscope, first recorded by Robert Brown.
  • As observed in financial model that uses Brownian motion to simulate stock price fluctuations over time.
  • As observed in a particle simulation that uses Brownian motion principles to model the diffusion of pollutants in the atmosphere.
  • ...
• Counter-Example(s):
  • Deterministic motion, which follows predictable and non-random paths, unlike the random nature of Brownian motion.
  • Circular motion, where particles move in a defined circular path, unlike the unpredictable trajectory of particles in Brownian motion.
• See: Diffusion, Stochastic Processes, Wiener Process, Transport Phenomena, Robert Brown (Botanist), Atomic Theory, Avogadro's Number.

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References

2024

• (Wikipedia, 2024) ⇒ https://en.wikipedia.org/wiki/Brownian_motion Retrieved:2024-9-13.
  • Brownian motion is the random motion of particles suspended in a medium (a liquid or a gas).

    This motion pattern typically consists of random fluctuations in a particle's position inside a fluid sub-domain, followed by a relocation to another sub-domain. Each relocation is followed by more fluctuations within the new closed volume. This pattern describes a fluid at thermal equilibrium, defined by a given temperature. Within such a fluid, there exists no preferential direction of flow (as in transport phenomena). More specifically, the fluid's overall linear and angular momenta remain null over time. The kinetic energies of the molecular Brownian motions, together with those of molecular rotations and vibrations, sum up to the caloric component of a fluid's internal energy (the equipartition theorem).This motion is named after the botanist Robert Brown, who first described the phenomenon in 1827, while looking through a microscope at pollen of the plant Clarkia pulchella immersed in water. In 1900, the French mathematician Louis Bachelier modeled the stochastic process now called Brownian motion in his doctoral thesis, The Theory of Speculation (Théorie de la spéculation), prepared under the supervision of Henri Poincaré. Then, in 1905, theoretical physicist Albert Einstein published a paper where he modeled the motion of the pollen particles as being moved by individual water molecules, making one of his first major scientific contributions.

    The direction of the force of atomic bombardment is constantly changing, and at different times the particle is hit more on one side than another, leading to the seemingly random nature of the motion. This explanation of Brownian motion served as convincing evidence that atoms and molecules exist and was further verified experimentally by Jean Perrin in 1908. Perrin was awarded the Nobel Prize in Physics in 1926 "for his work on the discontinuous structure of matter". The many-body interactions that yield the Brownian pattern cannot be solved by a model accounting for every involved molecule. Consequently, only probabilistic models applied to molecular populations can be employed to describe it. Two such models of the statistical mechanics, due to Einstein and Smoluchowski, are presented below. Another, pure probabilistic class of models is the class of the stochastic process models. There exist sequences of both simpler and more complicated stochastic processes which converge (in the limit) to Brownian motion (see random walk and Donsker's theorem).

2014

• (Wikipedia, 2014) ⇒ http://en.wikipedia.org/wiki/Brownian_motion Retrieved:2014-7-26.
  • Brownian motion or pedesis (from "leaping" ) is the random motion of particles suspended in a fluid (a liquid or a gas) resulting from their collision with the quick atoms or molecules in the gas or liquid. The term "Brownian motion" can also refer to the mathematical model used to describe such random movements, which is often called a particle theory. This transport phenomenon is named after the botanist Robert Brown. In 1827, while looking through a microscope at particles found in pollen grains in water, he noted that the particles moved through the water but was not able to determine the mechanisms that caused this motion. Atoms and molecules had long been theorized as the constituents of matter, and many decades later, Albert Einstein published a paper in 1905 that explained in precise detail how the motion that Brown had observed was a result of the pollen being moved by individual water molecules. This explanation of Brownian motion served as definitive confirmation that atoms and molecules actually exist, and was further verified experimentally by Jean Perrin in 1908. Perrin was awarded the Nobel Prize in Physics in 1926 "for his work on the discontinuous structure of matter" (Einstein had received the award five years earlier "for his services to theoretical physics" with specific citation of different research). The direction of the force of atomic bombardment is constantly changing, and at different times the particle is hit more on one side than another, leading to the seemingly random nature of the motion. The mathematical model of Brownian motion has numerous real-world applications. For instance, Stock market fluctuations are often cited, although Benoit Mandelbrot rejected its applicability to stock price movements in part because these are discontinuous.

    Brownian motion is among the simplest of the continuous-time stochastic (or probabilistic) processes, and it is a limit of both simpler and more complicated stochastic processes (see random walk and Donsker's theorem). This universality is closely related to the universality of the normal distribution. In both cases, it is often mathematical convenience rather than the accuracy of the models that motivates their use.

1905

• (Einstein, 1905b) ⇒ Albert Einstein. (1905). “Über die von der molekularkinetischen Theorie der Wärme geforderte Bewegung von in ruhenden Flüssigkeiten suspendierten Teilchen.” In: Annalen der Physik, 17(8). doi:10.1002/andp.19053220806
  • “On the Motion - Required by Molecular Kinetic Theory of Heat – of Small Particles Suspended in a Stationary Liquid”

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