Everything about Brownian Motion totally explained
» This article is about the physical phenomenon; for the stochastic process, see Wiener process. For the sports team, see Brownian Motion (Ultimate).
Brownian motion (named in honor of the botanist
Robert Brown) is the random movement of particles suspended in a liquid or gas or the mathematical model used to describe such random movements, often called a
particle theory.
The mathematical model of Brownian motion has several real-world applications. 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 continuous-time
stochastic processes, and it's 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's often mathematical convenience rather than the accuracy of the models that motivates their use.
History
The Roman
Lucretius's scientific poem
On the Nature of Things (c. 60 BC) has a remarkable description of Brownian motion of dust particles. He uses this as a proof of the existence of atoms: "Observe what happens when sunbeams are admitted into a building and shed light on its shadowy places. You will see a multitude of tiny particles mingling in a multitude of ways... their dancing is an actual indication of underlying movements of matter that are hidden from our sight... It originates with the atoms which move of themselves [for examplespontaneously]. Then those small compound bodies that are least removed from the impetus of the atoms are set in motion by the impact of their invisible blows and in turn cannon against slightly larger bodies. So the movement mounts up from the atoms and gradually emerges to the level of our senses, so that those bodies are in motion that we see in sunbeams, moved by blows that remain invisible." Although the mingling motion of dust particles is caused largely by air currents, the glittering, tumbling motion of small dust particles is indeed caused chiefly by true Brownian dynamics.
Jan Ingenhousz had described the irregular motion of
coal dust particles on the surface of
alcohol in 1785. Nevertheless Brownian motion is traditionally regarded as discovered by the botanist
Robert Brown in 1827. It is believed that Brown was studying
pollen particles floating in water under the microscope. He then observed minute particles within the vacuoles of the pollen grains executing a jittery motion. By repeating the experiment with particles of dust, he was able to rule out that the motion was due to pollen particles being 'alive', although the origin of the motion was yet to be explained.
The first person to describe the mathematics behind Brownian motion was
Thorvald N. Thiele in 1880 in a paper on the method of least squares. This was followed independently by
Louis Bachelier in 1900 in his PhD thesis "The theory of speculation", in which he presented a stochastic analysis of the stock and option markets. However, it was
Albert Einstein's (in his
1905 paper) and
Marian Smoluchowski's (1906) independent research of the problem that brought the solution to the attention of physicists, and presented it as a way to indirectly confirm the existence of atoms and molecules.
Intuitive metaphor for Brownian motion
Consider a large balloon of 10 meters in diameter. Imagine this large balloon in a football stadium or any widely crowded area. The balloon is so large that it lies on top of many members of the crowd. Because they're excited, these fans hit the balloon at different times and in different directions with the motions being completely random. In the end, the balloon is pushed in random directions, so it shouldn't move on average. Consider now the force exerted at a certain time. We might have 20 supporters pushing right, and 21 other supporters pushing left, where each supporter is exerting equivalent amounts of force. In this case, the forces exerted from the left side and the right side are imbalanced in favor of the left side; the balloon will move slightly to the left. This type of imbalance exists at all times, and it causes random motion. If we look at this situation from above, so that we can't see the supporters, we see the large balloon as a small object animated by erratic movement.
Now return to Brown’s pollen particle swimming randomly in water. One molecule of water is about 0.1 to 0.2 nm, (a hydrogen-bonded cluster of 300 atoms has a diameter of approximately 3 nm) where the pollen particle is roughly 1 µm in diameter, roughly 10,000 times larger than a water molecule. So, the pollen particle can be considered as a very large balloon constantly being pushed by water molecules. The Brownian motion of particles in a liquid is due to the instantaneous imbalance in the force exerted by the small liquid molecules on the particle.
An
animation of the Brownian motion concept
is available as a
Java applet.
Modelling the Brownian motion using differential equations
The equations governing Brownian motion relate slightly differently to each of the two definitions of
Brownian motion given at the start of this article.
Mathematical Brownian motion
for main article, see Wiener process.
In
mathematics, the
Wiener process is a continuous-time
stochastic process named in honor of
Norbert Wiener. It is one of the best known
Lévy processes (
càdlàg stochastic processes with
stationary independent increments) and occurs frequently in pure and applied mathematics,
economics and
physics.
The Wiener process
Wt is characterized by three facts:
- W0 = 0
- Wt is almost surely continuous
- Wt has independent increments with distribution
where [gij] = [gij]−1 in the sense of the inverse of a square matrix.
Cultural uses
The awareness of Brownian motion as a stochastic process is referred to in science fiction. In Douglas Adams's The Hitchhiker's Guide to the Galaxy, Brownian motion is used to create (or rather calculate) the Infinite Improbability Drive that powers the spaceship Heart of Gold. The Brownian motion generator is a really hot cup of tea. In Murray Leinster's short story, A Logic Named Joe, the logic (computer) suggests building a perpetual motion machine using Brownian motion.
It also appears in other novels. In Julio Cortazar's novel Rayuela, Brownian motion is used to describe travelers in Paris at night.
It also appears in a famous/notorious essay by Constance Penley, "Brownian Motion: Women, Tactics, and Technology".
It also appears in John Crowley's short SF story 'Snow'. Brownian Motion is used to explain why a device that stores memories at a molecular level can only recall stored memories in a random fashion.
Further Information
Get more info on 'Brownian Motion'.
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