Brownian motion Brownian motion
(Brownian motion), the random movement of tiny particles suspended in a liquid or gas, under the influence of molecular impacts environment; discovered by R. Brown.
BROWNIAN MOTIONBROWNIAN MOVEMENT (Brownian motion), the random movement of the smallest particles suspended in a liquid or gas, occurring under the influence of impacts of environmental molecules; discovered by R. Brown (cm. BROWN Robert (botanist) in 1827
When observing a suspension of flower pollen in water under a microscope, Brown observed a chaotic movement of particles that arises "not from the movement of the liquid and not from its evaporation." Suspended particles 1 µm or less in size, visible only under a microscope, performed disordered independent movements, describing complex zigzag trajectories. Brownian motion does not weaken with time and does not depend on chemical properties medium, its intensity increases with increasing temperature of the medium and with a decrease in its viscosity and particle size. Even a qualitative explanation of the causes of Brownian motion was possible only 50 years later, when the cause of Brownian motion began to be associated with the impacts of liquid molecules on the surface of a particle suspended in it.
The first quantitative theory of Brownian motion was given by A. Einstein (cm. EINSTEIN Albert) and M. Smoluchovsky (cm. SMOLUKHOVSKY Marian) in 1905-06 based on molecular kinetic theory. It was shown that random walks of Brownian particles are associated with their participation in thermal motion along with the molecules of the medium in which they are suspended. Particles have on average the same kinetic energy, but due to the greater mass they have a lower speed. The theory of Brownian motion explains the random motion of a particle by the action of random forces from molecules and friction forces. According to this theory, the molecules of a liquid or gas are in constant thermal motion, and the impulses of different molecules are not the same in magnitude and direction. If the surface of a particle placed in such a medium is small, as is the case for a Brownian particle, then the impacts experienced by the particle from the surrounding molecules will not be exactly compensated. Therefore, as a result of the “bombardment” by molecules, a Brownian particle begins to move randomly, changing the magnitude and direction of its speed approximately 10 14 times per second. It followed from this theory that, by measuring the displacement of a particle over a certain time and knowing its radius and the viscosity of the liquid, one can calculate the Avogadro number (cm. AVOGADRO CONSTANT).
The conclusions of the theory of Brownian motion were confirmed by the measurements of J. Perrin (cm. PERRIN Jean Baptiste) and T. Svedberg (cm. SWEDBERG Theodor) in 1906. Based on these relations, the Boltzmann constant was experimentally determined (cm. BOLTZMANN CONSTANT) and the Avogadro constant.
When observing Brownian motion, the position of a particle is fixed at regular intervals. The shorter the time intervals, the more broken the particle's trajectory will look.
The patterns of Brownian motion serve as a clear confirmation of the fundamental provisions of the molecular kinetic theory. It was finally established that the thermal form of the motion of matter is due to the chaotic motion of atoms or molecules that make up macroscopic bodies.
The theory of Brownian motion played important role in the justification of statistical mechanics, it is based on the kinetic theory of coagulation aqueous solutions. In addition, it also has practical significance in metrology, since Brownian motion is considered as the main factor limiting the accuracy of measuring instruments. For example, the limit of accuracy of readings of a mirror galvanometer is determined by the trembling of the mirror, like a Brownian particle bombarded by air molecules. The laws of Brownian motion determine the random movement of electrons, causing noise in electrical circuits. Dielectric losses in dielectrics are explained by random movements of the dipole molecules that make up the dielectric. Random movements of ions in electrolyte solutions increase their electrical resistance.
encyclopedic Dictionary. 2009 .
See what "Brownian motion" is in other dictionaries:
- (Brownian motion), random movement of small particles suspended in a liquid or gas, occurring under the influence of impacts of environmental molecules. Investigated in 1827 by the English. scientist R. Brown (Brown; R. Brown), who observed through a microscope ... ... Physical Encyclopedia
BROWNIAN MOTION- (Brown), the movement of the smallest particles suspended in a liquid, occurring under the influence of collisions between these particles and the molecules of the liquid. It was first seen under a microscope. botanist Brown in 1827. If in sight ... ... Big Medical Encyclopedia
- (Brownian motion) random movement of the smallest particles suspended in a liquid or gas, under the influence of impacts of environmental molecules; discovered by R. Brown ... Big Encyclopedic Dictionary
BROWNIAN MOVEMENT, disordered, zigzag movement of particles suspended in a stream (liquid or gas). It is caused by uneven bombardment of larger particles from different sides by smaller molecules of a moving stream. This… … Scientific and technical encyclopedic dictionary
Brownian motion- - oscillatory, rotational or translational motion of the particles of the dispersed phase under the action of the thermal motion of the molecules of the dispersion medium. general chemistry: textbook / A. V. Zholnin ... Chemical terms
BROWNIAN MOTION- random movement of the smallest particles suspended in a liquid or gas, under the influence of impacts of environmental molecules that are in thermal motion; plays an important role in some physical. chem. processes, limits accuracy… … Great Polytechnic Encyclopedia
Brownian motion- — [Ya.N. Luginsky, M.S. Fezi Zhilinskaya, Yu.S. Kabirov. English Russian Dictionary of Electrical Engineering and Power Industry, Moscow, 1999] Topics in electrical engineering, basic concepts of EN Brownian motion ... Technical Translator's Handbook
This article or section needs revision. Please improve the article in accordance with the rules for writing articles ... Wikipedia
The continuous chaotic movement of microscopic particles suspended in a gas or liquid, due to the thermal movement of the molecules of the environment. This phenomenon was first described in 1827 by the Scottish botanist R. Brown, who studied under ... ... Collier Encyclopedia
Brownian motion is more correct, random motion of small (several microns or less in size) particles suspended in a liquid or gas, occurring under the action of shocks from the molecules of the environment. Discovered by R. Brown in 1827. ... ... Great Soviet Encyclopedia
Books
- Brownian motion of a vibrator, Yu.A. Krutkov. Reproduced in the original author's spelling of the 1935 edition (publishing house `Proceedings of the Academy of Sciences of the USSR`). IN…
Brownian motion
From Brownian motion (encyclopedia Elements)
In the second half of the 20th century, a serious discussion about the nature of atoms flared up in scientific circles. On one side were irrefutable authorities such as Ernst Mach (cm. shock waves), who argued that atoms are simply mathematical functions, which successfully describe the observed physical phenomena and do not have a real physical basis. On the other hand, scientists of the new wave - in particular, Ludwig Boltzmann ( cm. Boltzmann constant) - insisted that atoms are physical realities. And neither of the two sides was aware that already decades before the start of their dispute, received experimental results, once and for all resolving the issue in favor of the existence of atoms as a physical reality - however, they were obtained in the discipline of natural science adjacent to physics by botanist Robert Brown.
Back in the summer of 1827, Brown, while studying the behavior of pollen under a microscope (he studied an aqueous suspension of plant pollen Clarkia pulchella), suddenly discovered that individual spores make absolutely chaotic impulsive movements. He determined for certain that these movements were in no way connected with either the eddies and currents of water, or with its evaporation, after which, having described the nature of the movement of particles, he honestly signed his own impotence to explain the origin of this chaotic movement. However, being a meticulous experimenter, Brown found that such a chaotic movement is characteristic of any microscopic particles, be it plant pollen, mineral suspensions, or any crushed substance in general.
Only in 1905, none other than Albert Einstein, for the first time realized that this mysterious, at first glance, phenomenon serves as the best experimental confirmation of the correctness of the atomic theory of the structure of matter. He explained it something like this: a spore suspended in water is subjected to constant “bombardment” by randomly moving water molecules. On average, molecules act on it from all sides with equal intensity and at regular intervals. However, no matter how small the dispute is, due to purely random deviations, it first receives an impulse from the side of the molecule that hit it from one side, then from the side of the molecule that hit it from the other, and so on. As a result of averaging such collisions, it turns out that that at some point the particle "twitches" in one direction, then, if on the other side it was "pushed" by more molecules, in the other, etc. Using the laws mathematical statistics and molecular kinetic theory of gases, Einstein derived an equation describing the dependence of the mean square displacement of a Brownian particle on macroscopic parameters. ( Interesting fact: in one of the volumes of the German journal "Annals of Physics" ( Annalen der Physik) in 1905, three articles by Einstein were published: an article with a theoretical explanation of Brownian motion, an article on the foundations of the special theory of relativity, and, finally, an article describing the theory of the photoelectric effect. It was for the latter that Albert Einstein was awarded Nobel Prize in physics in 1921.)
In 1908 the French physicist Jean-Baptiste Perrin (Jean-Baptiste Perrin, 1870-1942) conducted a brilliant series of experiments that confirmed the correctness of Einstein's explanation of the phenomenon of Brownian motion. It became finally clear that the observed "chaotic" motion of Brownian particles is a consequence of intermolecular collisions. Since “useful mathematical conventions” (according to Mach) cannot lead to observable and completely real movements of physical particles, it became finally clear that the debate about the reality of atoms is over: they exist in nature. As a “bonus game”, Perrin got the formula derived by Einstein, which allowed the Frenchman to analyze and estimate the average number of atoms and / or molecules colliding with a particle suspended in a liquid over a given period of time and, using this indicator, calculate the molar numbers of various liquids. This idea was based on the fact that at each given moment of time the acceleration of a suspended particle depends on the number of collisions with the molecules of the medium ( cm. Newton's laws of mechanics), and hence on the number of molecules per unit volume of liquid. And this is nothing but Avogadro's number (cm. Avogadro's law) is one of the fundamental constants that determine the structure of our world.
From Brownian motion In any medium there are constant microscopic pressure fluctuations. They, acting on the particles placed in the medium, lead to their random displacements. This chaotic movement of the smallest particles in a liquid or gas is called Brownian motion, and the particle itself is called Brownian.Small suspension particles move randomly under the influence of impacts of liquid molecules.
In the second half of the 19th century, a serious discussion about the nature of atoms flared up in scientific circles. On one side were irrefutable authorities such as Ernst Mach ( cm. Shock waves), who argued that atoms are simply mathematical functions that successfully describe observable physical phenomena and have no real physical basis. On the other hand, scientists of the new wave - in particular, Ludwig Boltzmann ( cm. Boltzmann constant) - insisted that atoms are physical realities. And neither of the two sides was aware that already decades before the start of their dispute, experimental results had been obtained that once and for all decided the question in favor of the existence of atoms as a physical reality - however, they were obtained in the discipline of natural science adjacent to physics by the botanist Robert Brown.
Back in the summer of 1827, Brown, while studying the behavior of pollen under a microscope (he studied an aqueous suspension of plant pollen Clarkia pulchella), suddenly discovered that individual spores make absolutely chaotic impulsive movements. He determined for certain that these movements were in no way connected with either the eddies and currents of water, or with its evaporation, after which, having described the nature of the movement of particles, he honestly signed his own impotence to explain the origin of this chaotic movement. However, being a meticulous experimenter, Brown found that such a chaotic movement is characteristic of any microscopic particles, be it plant pollen, mineral suspensions, or any crushed substance in general.
Only in 1905, none other than Albert Einstein, for the first time realized that this mysterious, at first glance, phenomenon serves as the best experimental confirmation of the correctness of the atomic theory of the structure of matter. He explained it something like this: a spore suspended in water is subjected to constant “bombardment” by randomly moving water molecules. On average, molecules act on it from all sides with equal intensity and at regular intervals. However, no matter how small the dispute is, due to purely random deviations, it first receives an impulse from the side of the molecule that hit it from one side, then from the side of the molecule that hit it from the other, and so on. As a result of averaging such collisions, it turns out that that at some point the particle “twitches” in one direction, then, if on the other side it was “pushed” by more molecules, to the other, etc. Using the laws of mathematical statistics and the molecular-kinetic theory of gases, Einstein derived an equation describing dependence of the rms displacement of a Brownian particle on macroscopic parameters. (Interesting fact: in one of the volumes of the German journal "Annals of Physics" ( Annalen der Physik) three articles by Einstein were published in 1905: an article with a theoretical explanation of Brownian motion, an article on the foundations of the special theory of relativity, and, finally, an article describing the theory of the photoelectric effect. It was for the latter that Albert Einstein was awarded the Nobel Prize in Physics in 1921.)
In 1908 the French physicist Jean-Baptiste Perrin (Jean-Baptiste Perrin, 1870-1942) conducted a brilliant series of experiments that confirmed the correctness of Einstein's explanation of the phenomenon of Brownian motion. It became finally clear that the observed "chaotic" motion of Brownian particles is a consequence of intermolecular collisions. Since “useful mathematical conventions” (according to Mach) cannot lead to observable and completely real movements of physical particles, it became finally clear that the debate about the reality of atoms is over: they exist in nature. As a “bonus game”, Perrin got the formula derived by Einstein, which allowed the Frenchman to analyze and estimate the average number of atoms and / or molecules colliding with a particle suspended in a liquid over a given period of time and, using this indicator, calculate the molar numbers of various liquids. This idea was based on the fact that at each given moment of time the acceleration of a suspended particle depends on the number of collisions with the molecules of the medium ( cm. Newton's laws of mechanics), and hence on the number of molecules per unit volume of liquid. And this is nothing but Avogadro's number (cm. Avogadro's law) is one of the fundamental constants that determine the structure of our world.
Brownian motion
Pupils 10 "B" class
Onischuk Ekaterina
The concept of Brownian motion
Patterns of Brownian motion and application in science
The concept of Brownian motion from the point of view of Chaos theory
billiard ball movement
Integration of deterministic fractals and chaos
The concept of Brownian motion
Brownian motion, more correctly Brownian motion, thermal motion of particles of matter (with dimensions of several micron and less) suspended in liquid or gas particles. The reason for Brownian motion is a series of uncompensated impulses that a Brownian particle receives from surrounding liquid or gas molecules. Discovered by R. Brown (1773 - 1858) in 1827. Suspended particles, visible only under a microscope, move independently of each other and describe complex zigzag trajectories. Brownian motion does not weaken with time and does not depend on the chemical properties of the medium. The intensity of the Brownian motion increases with an increase in the temperature of the medium and with a decrease in its viscosity and particle size.
A consistent explanation of Brownian motion was given by A. Einstein and M. Smoluchowski in 1905-06 on the basis of molecular kinetic theory. According to this theory, the molecules of a liquid or gas are in constant thermal motion, and the impulses of different molecules are not the same in magnitude and direction. If the surface of a particle placed in such a medium is small, as is the case for a Brownian particle, then the impacts experienced by the particle from the surrounding molecules will not be exactly compensated. Therefore, as a result of the "bombardment" by molecules, a Brownian particle begins to move randomly, changing the magnitude and direction of its velocity approximately 10 14 times per second. When observing Brownian motion is fixed (see Fig. . 1) the position of the particle at regular intervals. Of course, between observations, the particle does not move in a straight line, but the connection of successive positions by straight lines gives a conditional picture of movement.
Brownian motion of gum particles in water (Fig.1)
Regularities of Brownian motion
The patterns of Brownian motion serve as a clear confirmation of the fundamental provisions of the molecular kinetic theory. The overall picture of Brownian motion is described by Einstein's law for the mean square of particle displacement
along any x direction. If during the time between two measurements there is enough big number collisions of a particle with molecules, then it is proportional to this time t: = 2DHere D- diffusion coefficient, which is determined by the resistance exerted by a viscous medium to a particle moving in it. For spherical particles of radius a, it is equal to:
D = kT/6pha, (2)
where k is the Boltzmann constant, T -absolute temperature, h - dynamic viscosity of the medium. The theory of Brownian motion explains the random motion of a particle by the action of random forces from molecules and friction forces. The random nature of the force means that its action for the time interval t 1 is completely independent of the action for the interval t 2 if these intervals do not overlap. The force averaged over a sufficiently long time is zero, and the average displacement of the Brownian particle Dc also turns out to be zero. The conclusions of the theory of Brownian motion are in excellent agreement with the experiment, formulas (1) and (2) were confirmed by the measurements of J. Perrin and T. Svedberg (1906). On the basis of these relations, the Boltzmann constant and the Avogadro number were experimentally determined in accordance with their values obtained by other methods. The theory of Brownian motion has played an important role in the foundation of statistical mechanics. In addition, it also has practical significance. First of all, Brownian motion limits the accuracy of measuring instruments. For example, the limit of accuracy of readings of a mirror galvanometer is determined by the trembling of the mirror, like a Brownian particle bombarded by air molecules. The laws of Brownian motion determine the random movement of electrons, causing noise in electrical circuits. Dielectric losses in dielectrics are explained by random movements of the dipole molecules that make up the dielectric. Random movements of ions in electrolyte solutions increase their electrical resistance.
The concept of Brownian motion from the point of view of Chaos theory
Brownian motion is, for example, the random and chaotic movement of dust particles suspended in water. This type of movement is perhaps the most practical aspect of fractal geometry. Random Brownian motion produces a frequency pattern that can be used to predict things including large quantities data and statistics. A good example is wool prices, which Mandelbrot predicted using Brownian motion.
Frequency diagrams created by plotting from Brownian numbers can also be converted to music. Of course, this type of fractal music is not musical at all and can really tire the listener.
By randomly plotting Brownian numbers, you can get a Dust Fractal like the one shown here as an example. In addition to using Brownian motion to create fractals from fractals, it can also be used to create landscapes. Many science fiction films, such as Star Trek, have used the Brownian motion technique to create alien landscapes such as hills and topological pictures of high plateaus.
These techniques are very effective and can be found in Mandelbrot's book The Fractal Geometry of Nature. Mandelbrot used Brownian lines to create bird's eye view of fractal coastlines and maps of islands (which were really just randomly drawn dots).
MOVEMENT OF THE BILLIARD BALL
Anyone who has ever picked up a pool cue knows that accuracy is the key to the game. The slightest mistake in the angle of the initial impact can quickly lead to a huge error in the position of the ball after only a few collisions. This sensitivity to initial conditions, called chaos, presents an insurmountable barrier to anyone hoping to predict or control the ball's trajectory after more than six or seven collisions. And do not think that the problem lies in the dust on the table or in an unsteady hand. In fact, if you use your computer to build a model containing a pool table that doesn't have any friction, inhuman control over cue positioning accuracy, you still won't be able to predict the ball's trajectory long enough!
How long? This depends partly on the accuracy of your computer, but more on the shape of the table. For a perfectly round table, up to about 500 collision positions can be calculated with an error of about 0.1 percent. But it is worth changing the shape of the table so that it becomes at least a little irregular (oval), and the unpredictability of the trajectory can exceed 90 degrees after only 10 collisions! The only way to get a picture of the general behavior of a billiard ball bouncing off a blank table is to plot the angle of rebound, or the length of the arc corresponding to each hit. Here are two successive magnifications of such a phase-spatial pattern.
Each individual loop or scatter represents the ball's behavior resulting from one set of initial conditions. The area of the picture that displays the results of a particular experiment is called the attractor area for a given set of initial conditions. As can be seen, the shape of the table used for these experiments is the main part of the attractor regions, which are repeated sequentially on a decreasing scale. Theoretically, such self-similarity should continue forever, and if we increase the drawing more and more, we would get all the same forms. This is called very popular today, the word fractal.
INTEGRATION OF DETERMINISTIC FRACTALS AND CHAOS
It can be seen from the above examples of deterministic fractals that they do not exhibit any chaotic behavior and that they are in fact very predictable. As you know, chaos theory uses a fractal to recreate or find patterns in order to predict the behavior of many systems in nature, such as, for example, the problem of bird migration.
Now let's see how this actually happens. Using a fractal called the Pythagorean Tree, not considered here (which, by the way, is not invented by Pythagoras and has nothing to do with the Pythagorean theorem) and Brownian motion (which is chaotic), let's try to make an imitation of a real tree. The ordering of leaves and branches on a tree is quite complex and random, and probably not something simple enough that a short 12-line program can emulate.
First you need to generate the Pythagorean Tree (on the left). It is necessary to make the trunk thicker. At this stage Brownian motion is not used. Instead, each line segment has now become a line of symmetry for the rectangle that becomes the trunk, and the branches outside.
Brownian motion - random movement of microscopic visible particles suspended in a liquid or gas solid caused by the thermal motion of particles of a liquid or gas. Brownian motion never stops. Brownian motion is related to thermal motion, but these concepts should not be confused. Brownian motion is a consequence and evidence of the existence of thermal motion.
Brownian motion is the most obvious experimental confirmation of the ideas of the molecular kinetic theory about the chaotic thermal motion of atoms and molecules. If the observation interval is large enough so that the forces acting on the particle from the molecules of the medium change their direction many times, then the average square of the projection of its displacement on some axis (in the absence of other external forces) is proportional to time.
When deriving Einstein's law, it is assumed that particle displacements in any direction are equally probable and that the inertia of a Brownian particle can be neglected compared to the effect of friction forces (this is acceptable for sufficiently long times). The formula for the coefficient D is based on the application of Stokes' law for the hydrodynamic resistance to the motion of a sphere of radius a in a viscous fluid. The relationships for and D were experimentally confirmed by the measurements of J. Perrin and T. Svedberg. From these measurements, the Boltzmann constant k and the Avogadro constant NA are experimentally determined. In addition to the translational Brownian motion, there is also a rotational Brownian motion - random rotation of a Brownian particle under the influence of impacts of the molecules of the medium. For rotational Brownian motion, the rms angular displacement of a particle is proportional to the observation time. These relationships were also confirmed by Perrin's experiments, although this effect is much more difficult to observe than translational Brownian motion.
The essence of the phenomenon
Brownian motion occurs due to the fact that all liquids and gases consist of atoms or molecules - the smallest particles that are in constant chaotic thermal motion, and therefore continuously push the Brownian particle from different sides. It was found that large particles larger than 5 µm practically do not participate in Brownian motion (they are immobile or sediment), smaller particles (less than 3 µm) move forward along very complex trajectories or rotate. When a large body is immersed in the medium, the shocks that occur in large numbers are averaged and form a constant pressure. If a large body is surrounded by a medium on all sides, then the pressure is practically balanced, only the lifting force of Archimedes remains - such a body smoothly floats up or sinks. If the body is small, like a Brownian particle, then pressure fluctuations become noticeable, which create a noticeable randomly changing force, leading to oscillations of the particle. Brownian particles usually do not sink or float, but are suspended in a medium.
Brownian motion theory
In 1905, Albert Einstein created a molecular kinetic theory for a quantitative description of Brownian motion. In particular, he derived a formula for the diffusion coefficient of spherical Brownian particles:
Where D- diffusion coefficient, R is the universal gas constant, T is the absolute temperature, N A is the Avogadro constant, A- particle radius, ξ - dynamic viscosity.
Brownian motion as non-Markovian
random process
The theory of Brownian motion, well developed over the last century, is approximate. And although in most cases of practical importance the existing theory gives satisfactory results, in some cases it may require clarification. Thus, experimental work carried out at the beginning of the 21st century at the Polytechnic University of Lausanne, the University of Texas and the European Molecular Biology Laboratory in Heidelberg (under the direction of S. Dzheney) showed the difference in the behavior of a Brownian particle from that theoretically predicted by the Einstein-Smoluchowski theory, which was especially noticeable when increase in particle size. The studies also touched upon the analysis of the motion of the surrounding particles of the medium and showed a significant mutual influence the motion of a Brownian particle and the motion of particles of the medium caused by it against each other, that is, the presence of a “memory” of a Brownian particle, or, in other words, the dependence of its statistical characteristics in the future on the entire prehistory of its behavior in the past. This fact was not taken into account in the Einstein-Smoluchowski theory.
The process of Brownian motion of a particle in a viscous medium, generally speaking, belongs to the class of non-Markov processes, and for its more accurate description it is necessary to use integral stochastic equations.
