thermal motion
Any substance consists of the smallest particles - molecules. Molecule is the smallest particle of a given substance that retains all of its chemical properties. Molecules are located discretely in space, i.e., at certain distances from each other, and are in a state of continuous erratic (chaotic) movement .
Since bodies consist of a large number of molecules and the movement of molecules is random, it is impossible to say exactly how many impacts this or that molecule will experience from others. Therefore, they say that the position of the molecule, its speed at each moment of time is random. However, this does not mean that the movement of molecules does not obey certain laws. In particular, although the velocities of the molecules at some point in time are different, most of them have velocities close to some definite value. Usually, when speaking about the speed of movement of molecules, they mean average speed (v$cp).
It is impossible to single out any particular direction in which all molecules move. The movement of molecules never stops. We can say that it is continuous. Such a continuous chaotic movement of atoms and molecules is called -. This name is determined by the fact that the speed of movement of molecules depends on the temperature of the body. The greater the average speed of movement of the body's molecules, the higher its temperature. Conversely, the higher the temperature of the body, the greater the average speed of the molecules.
Brownian motion
The movement of liquid molecules was discovered by observing Brownian motion - the movement of very small solid particles suspended in it. Each particle continuously makes jumps in arbitrary directions, describing the trajectory in the form of a broken line. This behavior of particles can be explained by assuming that they experience impacts of liquid molecules simultaneously from different sides. The difference in the number of these impacts from opposite directions leads to the motion of the particle, since its mass is commensurate with the masses of the molecules themselves. The movement of such particles was first discovered in 1827 by the English botanist Brown, observing pollen particles in water under a microscope, which is why it was called - Brownian motion.
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 the chemical properties of the 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 impact of liquid molecules on the surface of a particle suspended in it.
The first quantitative theory of Brownian motion was given by A. Einstein and M. Smoluchowski 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.
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 an important role in substantiating statistical mechanics; it is the basis for the kinetic theory of coagulation of 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.
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 a sufficiently large number of collisions of a particle with molecules occurs, then proportionally 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 involving large amounts of 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 island maps (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 discussed 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.
« Physics - Grade 10 "
Recall the diffusion phenomenon from the basic school physics course.
How can this phenomenon be explained?
Previously, you learned what diffusion, i.e., the penetration of molecules of one substance into the intermolecular space of another substance. This phenomenon is determined by the random movement of molecules. This can explain, for example, the fact that the volume of a mixture of water and alcohol is less than the volume of its components.
But the most obvious evidence of the movement of molecules can be obtained by observing under a microscope the smallest particles of any solid substance suspended in water. These particles move randomly, which is called Brownian.
Brownian motion- this is the thermal movement of particles suspended in a liquid (or gas).
Observation of Brownian motion.
The English botanist R. Brown (1773-1858) first observed this phenomenon in 1827, examining the moss spores suspended in water under a microscope.
Later, he considered other small particles, including particles of stone from the Egyptian pyramids. Now, to observe Brownian motion, particles of gummigut paint, which is insoluble in water, are used. These particles move randomly. The most striking and unusual thing for us is that this movement never stops. We are accustomed to the fact that any moving body sooner or later stops. Brown initially thought that the spores of the club moss showed signs of life.
Brownian motion is thermal motion, and it cannot stop. As the temperature increases, its intensity increases.
Figure 8.3 shows the trajectories of Brownian particles. The positions of the particles marked with dots are determined at regular intervals of 30 s. These points are connected by straight lines. In reality, the particle trajectory is much more complicated.
Explanation of Brownian motion.
Brownian motion can be explained only on the basis of molecular-kinetic theory.
“Few phenomena can captivate the observer as much as Brownian motion. Here the observer is allowed to look behind the scenes of what happens in nature. A new world opens before him - a non-stop hustle and bustle of a huge number of particles. The smallest particles fly quickly into the field of view of the microscope, almost instantly changing the direction of movement. Larger particles move more slowly, but they also constantly change direction. Large particles practically jostle in place. Their protrusions clearly show the rotation of particles around their axis, which constantly changes direction in space. Nowhere is there a trace of system or order. The dominance of blind chance - that's what a strong, overwhelming impression this picture makes on the observer. R. Paul (1884-1976).
The reason for the Brownian motion of a particle is that the impacts of liquid molecules on the particle do not cancel each other out.
Figure 8.4 schematically shows the position of one Brownian particle and the molecules closest to it.
When molecules move randomly, the impulses they transmit to a Brownian particle, for example, from the left and from the right, are not the same. Therefore, the resulting pressure force of liquid molecules on a Brownian particle is nonzero. This force causes a change in the motion of the particle.
The molecular-kinetic theory of Brownian motion was created in 1905 by A. Einstein (1879-1955). The construction of the theory of Brownian motion and its experimental confirmation by the French physicist J. Perrin finally completed the victory of the molecular-kinetic theory. In 1926, J. Perrin received the Nobel Prize for his study of the structure of matter.
Perrin's experiments.
The idea behind Perrin's experiments is as follows. It is known that the concentration of gas molecules in the atmosphere decreases with height. If there were no thermal motion, then all the molecules would fall to the Earth and the atmosphere would disappear. However, if there was no attraction to the Earth, then due to thermal motion, the molecules would leave the Earth, since the gas is capable of unlimited expansion. As a result of the action of these opposite factors, a certain distribution of molecules along the height is established, i.e., the concentration of molecules decreases rather quickly with height. Moreover, the larger the mass of molecules, the faster their concentration decreases with height.
Brownian particles participate in thermal motion. Since their interaction is negligible, the totality of these particles in a gas or liquid can be considered as an ideal gas of very heavy molecules. Consequently, the concentration of Brownian particles in a gas or liquid in the Earth's gravitational field must decrease according to the same law as the concentration of gas molecules. This law is known.
Perrin, using a microscope of high magnification and a small depth of field (small depth of field), observed Brownian particles in very thin layers of liquid. Calculating the concentration of particles at different heights, he found that this concentration decreases with height according to the same law as the concentration of gas molecules. The difference is that due to the large mass of Brownian particles, the decrease occurs very quickly.
All these facts testify to the correctness of the theory of Brownian motion and to the fact that Brownian particles participate in the thermal motion of molecules.
Counting Brownian particles at different heights allowed Perrin to determine Avogadro's constant in a completely new way. The value of this constant coincided with the previously known one.
Brownian motion - Random movement of microscopic particles of a solid substance, visible, suspended in a liquid or gas, caused by the thermal movement 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 any 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 influence 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 progressively 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 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, but- 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 refinement. 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 movement of the surrounding particles of the medium and showed a significant mutual influence of the movement of the Brownian particle and the movement of the particles of the medium caused by it on each other, that is, the presence of a "memory" in the Brownian particle, or, in other words, the dependence of its statistical characteristics in the future on the entire prehistory her 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.
