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Brownian motion

There are two meanings of the term Brownian motion: the physical phenomenon that minute particles immersed in a fluid will experience a random movement, and one of the mathematical models used to describe it.

The mathematical model can also be used to describe many phenomena not resembling (other than mathematically) the random movement of minute particles. An often quoted example is stock market fluctuations, and another important example is the evolution of physical characteristics in the fossil record.

Brownian motion is the simplest stochastic process on a continuous domain, and it is a limit of both simpler (see random walk) and more complicated stochastic processes. This universality is closely related to the universality of the normal distribution. In both cases, it is often mathematical convenience rather than actual accuracy as models that dictates their use. All three quoted examples of Brownian motion are cases of this: it has been argued that Levy flights are a more accurate, if still imperfect, model of stock-market fluctuations; the physical Brownian motion can be modelled more accurately by more general diffusion process; and the dust hasn't settled yet on what the best model for the fossil record is, even after correcting for non-gaussian data.

History of Brownian motion

Brownian motion was discovered by the biologist Robert Brown in 1827. The story goes that Brown was studying pollen particles floating in water under the microscope, and he observed minute particles within vacuoles in the pollen grains executing the jittery motion that now bears his name. By doing the same with particles of dust, he was able to rule out that the motion was due to pollen being "alive", but it remained to explain the origin of the motion. The first to give a theory of Brownian motion was none other than Albert Einstein in 1905.

At that time the atomic nature of matter was still a controversial idea. Einstein observed that, if the kinetic theory of fluids was right, then the molecules of water would move at random and so a small particle would receive a random number of impacts of random strength and from random directions in any short period of time. This random bombardment by the molecules of the fluid would cause a sufficiently small particle to move in exactly the way described by Brown.

Description of the mathematical model

Mathematically, Brownian motion is a Wiener process in which the conditional probability distribution of the particle's position at time t+dt, given that its position at time t is p, is a Normal distribution with a mean of p+μ dt and a variance of σ2 dt; the parameter μ is the drift velocity, and the parameter σ2 is the power of the noise. These properties clearly establish that Brownian motion is Markovian (i.e. it satisfies the Markov property). Brownian motion is related to the random walk problem and it is generic in the sense that many different stochastic processes reduce to Brownian motion in suitable limits.

In fact, the Wiener process is the only time-homogeneous stochastic process with independent increments and which is continuous in probability. These are all reasonable approximations to the physical properties of Brownian motion.

The mathematical theory of Brownian motion has been applied in contexts ranging far beyond the movement of particles in fluids. For example, in the modern theory of option pricing, asset classes are sometimes modeled as if they move according to a Brownian motion with drift.

It turns out that the Wiener process is not a physically realistic model of the motion of Brownian particles. More sophisticated formulations of the problem have led to the mathematical theory of diffusion processes. The accompanying equation of motion is called the Langevin equation or the Fokker-Planck equation depending on whether it is formulated in terms of random trajectories or probability densities.

See also

osmosis, brownian tree

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