Brownian motion

2-dimensional " random walk " of a silver adatom on a silver (111) surface.

The Brownian motion is the by the Scottish botanist Robert Brown in 1827 under the microscope discovered irregular and jerky thermal motion of small particles in liquids and gases . The name Brownian molecular movement , which is also in use, stems from the fact that the word molecule was still used generally to denote a small body. Molecules in today's sense are much smaller than the particles visible in the microscope and remain completely invisible here. But the molecules of the surrounding matter produce Brownian motion. According to the explanation given by Albert Einstein in 1905 and Marian Smoluchowski in 1906 , the displacement of the particles visible in the microscope is caused by the fact that, due to their disordered heat movement, the molecules constantly hit the particles in large numbers from all directions, and one of them is purely random Direction, sometimes the other direction is more important. This idea was quantitatively confirmed in the following years by the experiments and measurements of Jean Baptiste Perrin . The successful explanation of the Brownian movement is considered a milestone on the way to the scientific proof of the existence of the molecules and thus of the atoms .

Brownian motion of fluorescent latex beads (diameter approximately 20 nm) in water observed with an SPI microscope .

History of exploration

Brown observed under the microscope in 1827 that elongated particles about 6–8 micrometers in size existed within a drop of water floating pollen grains of Clarkia , which when released made irregular jerky movements (the pollen grains of around 100 micrometers were too large to even close to Brownian motion discover). It is now known that these particles are organelles such as amyloplasts and spherosomes. Brown originally assumed that this was an indication of a life force inherent in pollen , as it has long been assumed by scientists to exist (see organic chemistry ). However, he was then able to observe the same movement in certainly inanimate grains of dust in water.

Jan Ingenhousz reported a very similar phenomenon in soot particles on alcohol as early as 1784. He stated that the cause was the evaporation of the liquid. Ingenhousz mentioned this phenomenon only in passing as an example of avoidable interference in the study of microbes if the drop under the microscope is not covered with a cover slip. His observation was then forgotten until the 20th century. Nevertheless, Ingenhousz is sometimes referred to as the real discoverer of the Brownian movement. As David Walker noted, however, the particles of coal suspended in alcohol that Ingenhousz described were too large to study Brownian motion in them, and the motion was completely superimposed by convection motions due to evaporation, which Ingenhousz correctly suspected as the likely source of motion. Ingenhousz described in the same book the covering of the drops with glass plates, which restricted the evaporation movements to the edge areas, if these were not sealed. Here, too, Brownian motion could only be observed in the smallest coal particles (with diameters of around 5 micrometers or less) and this case was not described by Ingenhousz. A significant source of error was also vibrations that were caused by the breath of the observer alone.

After Brown's publication, detailed experiments, particularly by Christian Wiener in 1863, increasingly revealed the certainty that Brownian motion is a general and fundamental phenomenon that is caused by the motion of invisible small particles of liquid. The Brownian movement thus provided the first proof of the general heat movement of all particles assumed in the molecular theory of heat (see also History of Thermodynamics , Phlogiston ).

Einstein came in 1905 on a purely theoretical route, based on the molecular theory of heat, to a quantitative "prediction" of Brownian motion. He considered it “possible” that the theoretically derived movement coincided with the Brownian movement , but found the information available to him to be too “imprecise” to be able to “form a judgment”. According to his formula, the square of the distance traveled by a particle increases on average proportionally to the time span and to the (absolute) temperature , as well as inversely proportional to the radius of the particle and the viscosity of the liquid. This formula was confirmed in the following years by the experiments of Jean Baptiste Perrin , who received the Nobel Prize in Physics for this in 1926. Also, diffusion , osmosis and thermophoresis based on the thermal motion of the molecules.

Physical model

For particles in a viscous medium that move away from their starting point due to irregular collisions, Albert Einstein (1905), Marian Smoluchowski (1906) and Paul Langevin (1908) were able to show that the mean square distance from their starting point increases proportionally with time. For movement in one dimension applies

{\ displaystyle x ^ {2} = {\ frac {k _ {\ mathrm {B}} T} {3r \ pi \ eta}} \, t \.}

This is the mean of the squares of the coordinates that the particles reach in time , starting from the location . is the Boltzmann constant , the absolute temperature , the radius of the particles and the viscosity of the liquid or gas. An important aspect of the formula is that here the Boltzmann constant is linked to macroscopically measurable quantities. This enables the direct experimental determination of this size and thus the Avogadro constant and also the number, size and mass of the molecules, which are invisible because of their small size. ${\ displaystyle x ^ {2}}$ ${\ displaystyle x}$ ${\ displaystyle x = 0}$ ${\ displaystyle t}$ ${\ displaystyle k _ {\ mathrm {B}}}$ ${\ displaystyle T}$ ${\ displaystyle r}$ ${\ displaystyle \ eta}$ ${\ displaystyle k _ {\ mathrm {B}}}$

The simplest derivation comes from Langevin:

A particle of mass follows the equation of motion (here only in -direction) ${\ displaystyle m}$ ${\ displaystyle x}$

{\ displaystyle m {\ frac {\ mathrm {d} ^ {2} x} {\ mathrm {d} t ^ {2}}} = F_ {x} -f \, {\ frac {\ mathrm {d} x} {\ mathrm {d} t}} \,}

if, in addition to a force from the medium, a frictional force is exerted. After multiplying by , this can be transformed into ${\ displaystyle F_ {x}}$ ${\ displaystyle F_ {R} = - f \, \ mathrm {d} x / \ mathrm {d} t}$ ${\ displaystyle x}$

{\ displaystyle {\ frac {1} {2}} m \ cdot {\ frac {\ mathrm {d}} {\ mathrm {d} t}} \ left ({\ frac {\ mathrm {d} x ^ { 2}} {\ mathrm {d} t}} \ right) -m \ cdot \ left ({\ frac {\ mathrm {d} x} {\ mathrm {d} t}} \ right) ^ {2} = x \ cdot F_ {x} - {\ frac {1} {2}} f {\ frac {\ mathrm {d} x ^ {2}} {\ mathrm {d} t}} \.}

The mean value is formed from this over many particles (or over many repeated observations on the same particle). On the left side of the equation, the size in the 1st term becomes the mean square distance of the particle from the point , i.e. the variance of the statistical distribution formed by many particles. The 2nd term on the left is the mean kinetic energy and is given by the uniform distribution theorem : ${\ displaystyle x ^ {2}}$ ${\ displaystyle x = 0}$ ${\ displaystyle V}$

{\ displaystyle {\ frac {1} {2}} m \ cdot \ left ({\ frac {\ mathrm {d} x} {\ mathrm {d} t}} \ right) ^ {2} = {\ overline {E _ {\ text {kin, x}}}} = {\ frac {1} {2}} k _ {\ mathrm {B}} T \.}

The mean value of the term disappears if the forces result from disordered collisions of the molecules, which on average push the particle neither after nor after . The following remains for the average values: ${\ displaystyle x \ cdot F_ {x}}$ ${\ displaystyle F_ {x}}$ ${\ displaystyle + x}$ ${\ displaystyle -x}$

{\ displaystyle {\ frac {1} {2}} m \ cdot {\ frac {\ mathrm {d}} {\ mathrm {d} t}} \ left ({\ frac {\ mathrm {d} V} { \ mathrm {d} t}} \ right) = k _ {\ mathrm {B}} T - {\ frac {1} {2}} f {\ frac {\ mathrm {d} V} {\ mathrm {d} t}} \.}

This is a differential equation according to which the variance increases with time, with the speed tending towards a saturation value. After reaching this steady state the left side of the equation disappears and it remains: ${\ displaystyle V (t)}$

{\ displaystyle {\ frac {1} {2}} f {\ frac {\ mathrm {d} V} {\ mathrm {d} t}} = k _ {\ mathrm {B}} T \.}

${\ displaystyle V (t)}$ then grows proportionally to time. Inserting ( Stokes' law ) finally gives the formula given above for the mean square distance from the starting point. ${\ displaystyle f = 6 \ pi r \ eta}$

Mathematical model

In mathematics, Brownian motion is a centered Gaussian process with a covariance function for everyone . The resulting stochastic process is today in honor of Norbert Wiener , the probabilistic same existence in 1923 proved as Wiener process known. ${\ displaystyle B = (B_ {t}) _ {t \ in [0, \ infty]}}$ ${\ displaystyle \ operatorname {Cov} (B_ {t}, B_ {s}) = \ operatorname {min} (t, s)}$ ${\ displaystyle t, s \ geq 0}$

There are several ways to mathematically construct a Brownian motion:

the abstract construction based on Kolmogorow's scheme , whereby one then gets problems with the continuity of the path.

the Lévy - Ciesielski -construction: this is Brownian motion with the help of by the hair system on induced Schauder basis already constructed as a stochastic process with continuous paths. ${\ displaystyle C ([0,1])}$

Be , ... and independent standard normal distribution , ${\ displaystyle Z_ {0}}$ ${\ displaystyle Z_ {1}}$

then

{\ displaystyle S (t) = Z_ {0} t + \ sum _ {k = 1} ^ {\ infty} Z_ {k} {\ frac {{\ sqrt {2}} \ sin (k \ pi t)} {k \ pi}}}

a Brownian movement.

Brownian motion also plays in the simulation of stock price movements play a role, they also serve as the basis of the study of queues .

literature

Rüdiger Bessenrodt: Brownian Movement: Hundred Years of Theory of the Most Important Bridge between Micro- and Macrophysics , Physik Journal, 1977, Volume 33, Pages 7-16, doi : 10.1002 / phbl.19770330104

Web links

Individual evidence

↑ Teaching the Growth, Ripening, and Agglomeration of Nanostructures in Computer Experiments, Jan Philipp Meyburg and Detlef Diesing, Journal of Chemical Education, (2017) , 94, 9, 1225–1231, doi : 10.1021 / acs.jchemed.6b01008
↑ Robert Brown: "A brief account of microscopic observations made in the months of June, July and August, 1827, on the particles contained in the pollen of plants; and on the general existence of active molecules in organic and inorganic bodies." In: Philosophical Magazine . tape 4 , 1905, pp. 161-173 .
↑ Philip Pearle, Brian Collett, Kenneth Bart, David Bilderback, Dara Newman and Scott Samuels: "What Brown saw and you can too" . In: Am. J. Phys. tape 78 , 2010, p. 1278 , doi : 10.1119 / 1.3475685 , arxiv : 1008.0039 .
↑ Jan Ingenhousz: Comments on the use of the magnifying glass. In: NC Molitor (Ed.): Mixed writings by Ingen-Housz. 2nd edition, Volume II, Vienna 1784, pp. 123–124 ( limited preview in the Google book search).
^ First, PW van der Pas The Early History of the Brownian Motion , XIIe Congrès International d'Histoire des Sciences, Paris 1968, Actes Tome VIII, Histoire des Sciences Naturelles et de la Biologie, Paris: Blanchard 1971, pp. 143–158, van der Pas The Discovery of the Brownian Motion , Scientiarum Historia, Volume 13, 1971, pp. 27-35. And the article on Ingehousz by van der Pas in the Dictionary of Scientific Biography .
↑ David Walker: Did Jan Ingenhousz in 1784 unwittingly report Brownian motion / movement in an inert material to give him priority over Brown? A review of the evidence with videos. Retrieved February 6, 2018 .
↑ The Svedberg : The Existence of Molecules . Akad. Verlagsgesellschaft, Leipzig 1828, p. 161-173 .
↑ Jean Perrin: Mouvement brownien et réalité moléculaire . In: Annales de chimie et de physique. ser. 8, 18, 1909, pp. 5-114.
↑ A. Einstein: About the motion of particles suspended in liquids at rest, required by the molecular kinetic theory of heat . In: Annals of Physics . tape 322 , no. 8 , 1905, pp. 549-560 ( digitized version ( PDF; 733 kB)).

↑ M. Smoluchowski: On the kinetic theory of Brownian molecular motion and suspensions . In: Annals of Physics . tape 326 , no. 14 , 1906, pp. 756–780 ( digital version ; PDF; 1.4 MB).

^ P. Langevin: Sur la théorie du mouvement Brownien . In: Comptes Rendues . tape 146 , 1908, pp. 530 ( digitized at Gallica ).

↑ The math of queuing “We are like molecules when we wait”. on sueddeutsche.de, May 17, 2010.

[1] Teaching the Growth, Ripening, and Agglomeration of Nanostructures in Computer Experiments, Jan Philipp Meyburg and Detlef Diesing, Journal of Chemical Education, (2017) , 94, 9, 1225–1231, doi : 10.1021 / acs.jchemed.6b01008

[2] Robert Brown: "A brief account of microscopic observations made in the months of June, July and August, 1827, on the particles contained in the pollen of plants; and on the general existence of active molecules in organic and inorganic bodies." In: Philosophical Magazine . tape 4 , 1905, pp. 161-173 .

[3] Philip Pearle, Brian Collett, Kenneth Bart, David Bilderback, Dara Newman and Scott Samuels: "What Brown saw and you can too" . In: Am. J. Phys. tape 78 , 2010, p. 1278 , doi : 10.1119 / 1.3475685 , arxiv : 1008.0039 .

[4] Jan Ingenhousz: Comments on the use of the magnifying glass. In: NC Molitor (Ed.): Mixed writings by Ingen-Housz. 2nd edition, Volume II, Vienna 1784, pp. 123–124 ( limited preview in the Google book search).

[5] First, PW van der Pas The Early History of the Brownian Motion , XIIe Congrès International d'Histoire des Sciences, Paris 1968, Actes Tome VIII, Histoire des Sciences Naturelles et de la Biologie, Paris: Blanchard 1971, pp. 143–158, van der Pas The Discovery of the Brownian Motion , Scientiarum Historia, Volume 13, 1971, pp. 27-35. And the article on Ingehousz by van der Pas in the Dictionary of Scientific Biography .

[6] David Walker: Did Jan Ingenhousz in 1784 unwittingly report Brownian motion / movement in an inert material to give him priority over Brown? A review of the evidence with videos. Retrieved February 6, 2018 .

[7] The Svedberg : The Existence of Molecules . Akad. Verlagsgesellschaft, Leipzig 1828, p. 161-173 .

[8] Jean Perrin: Mouvement brownien et réalité moléculaire . In: Annales de chimie et de physique. ser. 8, 18, 1909, pp. 5-114.

[9] A. Einstein: About the motion of particles suspended in liquids at rest, required by the molecular kinetic theory of heat . In: Annals of Physics . tape 322 , no. 8 , 1905, pp. 549-560 ( digitized version ( PDF; 733 kB)).

[10] M. Smoluchowski: On the kinetic theory of Brownian molecular motion and suspensions . In: Annals of Physics . tape 326 , no. 14 , 1906, pp. 756–780 ( digital version ; PDF; 1.4 MB).

[11] P. Langevin: Sur la théorie du mouvement Brownien . In: Comptes Rendues . tape 146 , 1908, pp. 530 ( digitized at Gallica ).

[12] The math of queuing “We are like molecules when we wait”. on sueddeutsche.de, May 17, 2010.