Brownian motion

As Brownian motion (or Brownian molecular motion ) the rediscovered by Scottish botanist Robert Brown in 1827 thermal motion of particles in liquids and gases is called. In this case, each atom or molecule describes a movement whose intensity is dependent on temperature. Less well known is that already in January 1785 Ingenhousz described the movement of charcoal dust on alcohol.

Brown observed under the microscope, such as pollen made ​​irregular twitching movements in a water drop. Originally, while mistakenly Brown assumed that this was an indication of the life force, which has long been suspected by scientists to exist (see organic chemistry). But this effect he could see the end in further empirical experiments also clearly inanimate dust grains.

A provable explanation deliver the molecules of the water droplet, the bump permanently from all sides against the larger, more visible Pollenteilchen, as could be described by the Maxwellian velocity distribution also mathematically exact 1860. The fact that it is a consequence of the motion of the liquid molecules and not other causes, demonstrated first Christian Wiener in 1863 in experiments.

A colloid encounters per second about 1021- times together with a solvent molecule. Characterized each time it receives a force, resulting in a random motion, a so-called random walk. Without outer influences the likelihood of a change in motion is the same in each direction. Therefore obtained by prolonged viewing of the sum of the change in direction is zero.

Diffusion, osmosis and thermophoresis are based on the movement of the particles.

Mathematical Model

If one uses the Wiener process as a mathematical model for the movement, as can be understood as mean-square displacement of a particle per unit time and estimate experimentally. Albert Einstein ( 1905) and Marian Smoluchowski (1906 ) showed that

Holds, where the universal gas constant, the absolute temperature, the Avogadro constant, the radius of a Brownian particle and the strength ( internal friction ) of the liquid or gas. This allows the Boltzmann constant determined experimentally. Among other things, this Jean Baptiste Perrin was awarded the 1926 Nobel Prize in Physics.

In mathematics, a Brownian motion is a centered Gaussian process with covariance function for all. The resulting stochastic process is now in honor of Norbert Wiener, the same in 1923 proved the existence of probability theory, known as the Wiener process.

There are several ways to construct a Brownian movement:

• The abstract design of the schema of Kolmogorov, which you then get problems with the path continuity.
• The Lévy- Ciesielski construction: Here, the Brownian motion is using the already constructed by the hair system induced Schauder basis as a stochastic process with continuous paths.
• Be, ... independent and standard normally distributed,

Then

A Brownian motion.

The Brownian motion plays in the simulation of stock price movements a role, she also serves as the basis for exploration of queues.