Brownian motion

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File:BrownianMotion.png
An example of 1000 simulated steps of Brownian motion in two dimensions. The origin of the motion is at [0,0] and the x and y components of each step are independently and normally distributed with variance 2 and mean 0. The mathematical model posits motion in which the steps are not discrete.

The term Brownian motion (in honor of the botanist Robert Brown) refers to either

  1. The physical phenomenon that minute particles immersed in a fluid move about randomly; or
  2. The mathematical models used to describe those random movements.

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. Another example is the evolution of physical characteristics in the fossil record.

Brownian motion is among the simplest stochastic processes 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 accuracy as models that motivates their use. All three quoted examples of Brownian motion are cases of this:

  1. It has been argued that Lévy flights are a more accurate, if still imperfect, model of stock-market fluctuations.
  2. The physical Brownian motion can be modelled more accurately by a more general diffusion process.
  3. The dust hasn't settled yet on what the best model for the fossil record is, even after correcting for non-Gaussian data.

History

Jan Ingenhousz made some observations of the irregular motion of carbon dust on alcohol in 1765 but Brownian motion is generally regarded as having been discovered by the biologist Robert Brown in 1827. The story goes that Brown was studying pollen particles floating in water under the microscope. He then 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 Louis Bachelier in 1900 in his PhD thesis "The theory of speculation".

At that time the atomic nature of matter was still a controversial idea. Albert Einstein and Marian Smoluchowski 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. Theodor Svedberg made important demonstrations of Brownian motion in colloids and Felix Ehrenhaft, of particles of silver in air. Jean Perrin carried out experiments to test the new mathematical models, and his published results finally put an end to the century-long dispute about the reality of atoms and molecules.

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 that has continuous trajectories. 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 closely related process, geometric Brownian motion.

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.

Modelling the brownian motion using differential equations

The equations governing brownian motion related slightly differently to the each of the two definitions of brownian motion given at the start of this article.

Mathematical brownian motion

For a particle experiencing a brownian motion corresponding to the mathematical definition, the equation governing the time evolution of the probability density function associated to the position of the brownian particle is the diffusion equation, a partial differential equation.

The time evolution of the position of the brownian particle itself can be described approximately by Langevin equation, an equation which involves a random force field representing the effect of the thermal fluctuations of the solvant on the brownian particle. On long timescales, the mathematical brownian motion is well described by Langevin equation. On small timescales, Inertial effects are prevalent in Langevin equation. However the mathematical brownian motion is exempt of such inertial effects. Note that inertial effects have to be considered in Langevin equation, otherwise the equation becomes singular, so that simply removing the inertia term from this equation would not yield an exact description, but rather a singular behavior in which the particle doesn't move at all...

Physical brownian motion

The diffusion equation yields an approximation of the time evolution of the probability density function associated to the position of the particle undergoing a brownian movement under the physical definition. The approximation is valid on long timescales (see Langevin equation for details).

The time evolution of the position of the brownian particle itself is best described using Langevin equation, an equation which involves a random force field representing the effect of the thermal fluctuations of the solvant on the particle.

See also

References

  • Einstein, A. "Über die von der molekularkinetischen Theorie der Wärme geforderte Bewegung von in ruhenden Flüssigkeiten suspendierten Teilchen." Ann. Phys. 17, 549, 1905. [1]
  • Einstein, A. Investigations on the Theory of Brownian Movement. New York: Dover, 1956. ISBN 0486603040

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