What does Brownian motion study? Brownian motion in physics. Integration of deterministic fractals and chaos

What is Brownian motion

This movement is characterized by the following features:

continues indefinitely without any visible changes,
the intensity of motion of Brownian particles depends on their size, but does not depend on their nature,
intensity increases with increasing temperature,
the intensity increases with decreasing viscosity of the liquid or gas.

Brownian motion is not molecular motion, but serves as direct evidence of the existence of molecules and the chaotic nature of their thermal motion.

The essence of Brownian motion

The essence of this movement is as follows. A particle together with molecules of a liquid or gas form one statistical system. In accordance with the theorem on the uniform distribution of energy over the degree of freedom, each degree of freedom accounts for 1/2kT of energy. The energy of 2/3kT per three translational degrees of freedom of the particle leads to the movement of its center of mass, which is observed under a microscope in the form of particle trembling. If a Brownian particle is sufficiently rigid, then another 3/2kT of energy falls on its rotational degrees of freedom. Therefore, when it trembles, it also experiences constant changes in orientation in space.

Brownian motion can be explained this way: the cause of Brownian motion is pressure fluctuations that are exerted on the surface of a small particle by the molecules of the medium. The force and pressure changes in magnitude and direction, as a result of which the particle is in random motion.

The motion of a Brownian particle is a random process. The probability (dw) that a Brownian particle, located in a homogeneous isotropic medium at the initial moment of time (t=0) at the origin of coordinates, will move along an arbitrarily directed (at t$>$0) Ox axis so that its coordinate will lie in the interval from x to x+dx, is equal to:

where $\triangle x$ is a small change in the particle coordinate due to fluctuation.

Let us consider the position of a Brownian particle at some fixed time intervals. Let's place the origin of coordinates at the point where the particle was at t=0. Let's denote $\overrightarrow(q_i)$ - a vector that characterizes the movement of a particle between (i-1) and i observations. After n observations, the particle will move from the zero position to a point with radius vector $\overrightarrow(r_n)$. Wherein:

\[\overrightarrow(r_n)=\sum\limits^n_(i=1)(\overrightarrow(q_i))\left(2\right).\]

The particle moves along a complex broken line throughout the observation period.

Let's find the average square of the particle's distance from the beginning after n steps in a large series of experiments:

\[\left\langle r^2_n\right\rangle =\left\langle \sum\limits^n_(i,j=1)(q_iq_j)\right\rangle =\sum\limits^n_(i=1) (\left\langle (q_i)^2\right\rangle )+\sum\limits^n_(i\ne j)(\left\langle q_iq_j\right\rangle )\left(3\right)\]

where $\left\langle q^2_i\right\rangle $ is the mean square of the particle displacement at the i-th step in a series of experiments (it is the same for all steps and is equal to some positive value a2), $\left\langle q_iq_j\ right\rangle $- is the average value dot product at i-th step to move when jth step in various experiments. These quantities are independent of each other; both positive and negative values of the scalar product are equally common. Therefore, we assume that $\left\langle q_iq_j\right\rangle $=0 for $\ i\ne j$. Then we have from (3):

\[\left\langle r^2_n\right\rangle =a^2n=\frac(a^2)(\triangle t)t=\alpha t=\left\langle r^2\right\rangle \left( 4\right),\]

where $\triangle t$ is the time interval between observations; t=$\triangle tn$ - time during which the average square of the particle's removal became equal to $\left\langle r^2\right\rangle .$ We get that the particle is moving away from the beginning. It is important that the average square of the distance increases in proportion to the first power of time. $\alpha \ $- can be found experimentally, or theoretically, as will be shown in Example 1.

A Brownian particle moves not only translationally, but also rotatingly. The average value of the rotation angle $\triangle \varphi $ of a Brownian particle over time t is equal to:

\[(\triangle \varphi )^2=2D_(vr)t(5),\]

where $D_(vr)$ is the rotational diffusion coefficient. For a spherical Brownian particle of radius - and $D_(vr)\ $ is equal to:

where $\eta $ is the viscosity coefficient of the medium.

Brownian motion limits accuracy measuring instruments. The limit of accuracy of a mirror galvanometer is determined by the vibration of the mirror, like a Brownian particle that is subject to impacts from air molecules. The random movement of electrons causes noise in electrical networks.

Example 1

Assignment: In order to mathematically fully characterize Brownian motion, it is necessary to find $\alpha $ in the formula $\left\langle r^2_n\right\rangle =\alpha t$. Assume that the liquid viscosity coefficient is known and equal to b, and the liquid temperature is T.

Let us write the equation of motion of a Brownian particle in projection onto the Ox axis:

where m is the mass of the particle, $F_x$ is the random force acting on the particle, $b\dot(x)$ is the term of the equation characterizing the friction force acting on the particle in the liquid.

Equations for quantities related to other coordinate axes have a similar form.

Let us multiply both sides of equation (1.1) by x, and transform the terms $\ddot(x)x\ and\ \dot(x)x$:

\[\ddot(x)x=\ddot(\left(\frac(x^2)(2)\right))-(\dot(x))^2,\dot(x)x=(\frac (x^2)(2)\)(1.2)\]

Then we reduce equation (1.1) to the form:

\[\frac(m)(2)(\ddot(x^2))-m(\dot(x))^2=-\frac(b)(2)\left(\dot(x^2) \right)+F_xx\ (1.3)\]

Let us average both sides of this equation over an ensemble of Brownian particles, taking into account that the average of the derivative with respect to time is equal to the derivative of average size, since this is averaging over an ensemble of particles, and, therefore, we will rearrange it using the operation of differentiation with respect to time. As a result of averaging (1.3) we obtain:

\[\frac(m)(2)\left(\left\langle \ddot(x^2)\right\rangle \right)-\left\langle m(\dot(x))^2\right\rangle =-\frac(b)(2)\left(\dot(\left\langle x^2\right\rangle )\right)+\left\langle F_xx\right\rangle \ \left(1.4\right). \]

Since deviations of a Brownian particle in any direction are equally probable, then:

\[\left\langle x^2\right\rangle =\left\langle y^2\right\rangle =\left\langle z^2\right\rangle =\frac(\left\langle r^2\right \rangle )(3)\left(1.5\right)\]

We use $\left\langle r^2_n\right\rangle =a^2n=\frac(a^2)(\triangle t)t=\alpha t=\left\langle r^2\right\rangle $, we get $\left\langle x^2\right\rangle =\frac(\alpha t)(3)$, therefore: $\dot(\left\langle x^2\right\rangle )=\frac(\alpha ) (3)$, $\left\langle \ddot(x^2)\right\rangle =0$

Due to the random nature of the force $F_x$ and the particle coordinate x and their independence from each other, the equality $\left\langle F_xx\right\rangle =0$ must be satisfied, then (1.5) reduces to the equality:

\[\left\langle m(\dot(\left(x\right)))^2\right\rangle =\frac(\alpha b)(6)\left(1.6\right).\]

According to the theorem about the uniform distribution of energy over degrees of freedom:

\[\left\langle m(\dot(\left(x\right)))^2\right\rangle =kT\left(1.7\right).\] \[\frac(\alpha b)(6) =kT\to \alpha =\frac(6kT)(b).\]

Thus, we obtain a formula for solving the problem of Brownian motion:

\[\left\langle r^2\right\rangle =\frac(6kT)(b)t\]

Answer: The formula $\left\langle r^2\right\rangle =\frac(6kT)(b)t$ solves the problem of Brownian motion of suspended particles.

Example 2

Assignment: Spherical gum particles of radius r participate in Brownian motion in a gas. Density of gummigut $\rho$. Find the root-mean-square speed of gum particles at temperature T.

The root mean square speed of molecules is:

\[\left\langle v^2\right\rangle =\sqrt(\frac(3kT)(m_0))\left(2.1\right)\]

A Brownian particle is in equilibrium with the matter in which it is located, and we can calculate its root mean square velocity using the formula for the speed of gas molecules, which, in turn, move, causing the Brownian particle to move. First, let's find the mass of the particle:

\[\left\langle v^2\right\rangle =\sqrt(\frac(9kT)(4\pi R^3\rho ))\]

Answer: The speed of a particle of gum suspended in a gas can be found as $\left\langle v^2\right\rangle =\sqrt(\frac(9kT)(4\pi R^3\rho ))$.

Brownian motion- random movement of microscopic visible particles of a solid substance suspended in a liquid or gas, caused by the thermal movement of the particles of the 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 clear experimental confirmation of the concepts of molecular kinetic theory about the chaotic thermal motion of atoms and molecules. If the observation period is large enough for the forces acting on the particle from the molecules of the medium to 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). Formula for coefficient D is based on the application of Stokes' law for hydrodynamic resistance to the motion of a sphere of radius A in a viscous fluid. The relationships for A and D were experimentally confirmed by measurements by J. Perrin and T. Svedberg. From these measurements, the Boltzmann constant was experimentally determined k and Avogadro's constant N A. In addition to translational Brownian motion, there is also rotational Brownian motion - the random rotation of a Brownian particle under the influence of impacts of molecules of the medium. For rotational Brownian motion, the root mean square angular displacement of the 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.

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Brownian motion occurs due to the fact that all liquids and gases consist of atoms or molecules - tiny particles that are in constant chaotic thermal motion, and therefore continuously push the Brownian particle with different sides. It was found that large particles with sizes greater than 5 µm practically do not participate in Brownian motion (they are stationary or sediment), smaller particles (less than 3 µm) move forward along very complex trajectories or rotate. When a large body is immersed in a medium, the shocks occurring in huge quantities 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 varying force, leading to oscillations of the particle. Brownian particles usually do not sink or float, but are suspended in the medium.

Opening

Brownian motion theory

Construction of the classical theory

D = R T 6 N A π a ξ , (\displaystyle D=(\frac (RT)(6N_(A)\pi a\xi )),)
Where D (\displaystyle D)- diffusion coefficient, R (\displaystyle R)- universal gas constant, T (\displaystyle T)- absolute temperature, N A (\displaystyle N_(A))- Avogadro's constant, a (\displaystyle a)- particle radius, ξ (\displaystyle \xi )- dynamic viscosity.

Experimental confirmation

Einstein's formula was confirmed by the experiments of Jean Perrin and his students in 1908-1909. As Brownian particles, they used grains of resin from the mastic tree and gum, the thick milky sap of trees of the genus Garcinia. The validity of the formula was established for various particle sizes - from 0.212 microns to 5.5 microns, for various solutions (sugar solution, glycerin) in which the particles moved.

Brownian motion as a non-Markov random process

The theory of Brownian motion, well developed over the last century, is an approximate one. And although in most practically important cases 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 Biological Laboratory in Heidelberg (under the leadership of S. Jeney) showed the difference in the behavior of the Brownian particle from that theoretically predicted by the Einstein-Smoluchowski theory, which was especially noticeable when increasing particle sizes. The studies also touched upon the analysis of the movement of surrounding particles of the medium and showed significant mutual influence the movement of a Brownian particle and the movement of particles of the medium towards each other caused by it, that is, the presence of “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 a more accurate description it is necessary to use integral stochastic equations.

Brownian motion is a continuous, constant chaotic movement of particles suspended in a liquid (or gas). The name currently used for the phenomenon was given in honor of its discoverer, the English botanist R. Brown. In 1827, he conducted an experiment, as a result of which Brownian motion was discovered. The scientist also drew attention to the fact that particles not only move along environment, but also rotate around its own axis. Since at that time the molecular theory of the structure of matter had not yet been created, Brown was not able to fully analyze the process.
Modern representations
It is currently believed that Brownian motion is caused by the collision of particles suspended in a liquid or gas with the molecules of the substance surrounding them. The latter are in constant motion, called thermal. They cause the chaotic movement of the particles that make up any substance. It is important to note that two others are associated with this phenomenon: the Brownian motion we describe and diffusion (the penetration of particles of one substance into another). These processes should be considered together, since they explain each other. So, due to collisions with surrounding molecules, particles suspended in the medium are in continuous motion, which is also chaotic. Chaoticity is expressed in inconstancy, both direction and speed.
From a thermodynamic point of view
It is known that as the temperature increases, the speed of Brownian motion also increases. This dependence is easily explained by the equation for describing the average kinetic energy of a moving particle: E=mv 2 =3kT/2, where m is the mass of the particle, v is the speed of the particle, k is Boltzmann’s constant, and T is the external temperature. As we see, the square of the speed of movement of a suspended particle is directly proportional to the temperature, therefore, with increasing temperature external environment speed also increases. Note that the basic principle on which the equation is based is the equality of the average kinetic energy moving particle kinetic energy of the particles that make up the medium (that is, the liquid or gas in which it is suspended). This theory was formulated by A. Einstein and M. Smoluchowski at approximately the same time, independently of each other.
Movement of Brownian particles
Particles suspended in a liquid or gas move along a zigzag path, gradually moving away from the point of origin of movement. Again, Einstein and Smoluchowski came to the conclusion that to study the motion of a Brownian particle, what is of primary importance is not the distance traveled or the actual speed, but its average displacement over a certain period of time. The equation proposed by Einstein is as follows: r 2 =6kTBt. In this formula, r is the average displacement of a suspended particle, B is its mobility (this value, in turn, is inversely dependent on the viscosity of the medium and the size of the particle), t is time. Consequently, the lower the viscosity of the medium, the higher the speed of movement of a suspended particle. The validity of the equation was experimentally proven by the French physicist J. Perrin.

Brownian motion - random movement of microscopic visible particles of a solid substance suspended in a liquid or gas, caused by the thermal movement of the particles of the 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 clear experimental confirmation of the concepts of molecular kinetic theory about the chaotic thermal motion of atoms and molecules. If the observation period is large enough for the forces acting on the particle from the molecules of the medium to 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 coefficient D is based on the application of Stokes' law for hydrodynamic resistance to the motion of a sphere of radius a in a viscous fluid. The relationships for and D were experimentally confirmed by measurements by J. Perrin and T. Svedberg. From these measurements, Boltzmann's constant k and Avogadro's constant NA were experimentally determined. In addition to translational Brownian motion, there is also rotational Brownian motion - the random rotation of a Brownian particle under the influence of impacts of molecules of the medium. For rotational Brownian motion, the root mean square angular displacement of the 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 - tiny particles that are in constant chaotic thermal motion, and therefore continuously push the Brownian particle from different directions. It was found that large particles with sizes greater than 5 µm practically do not participate in Brownian motion (they are stationary or sediment), smaller particles (less than 3 µm) move forward along very complex trajectories or rotate. When a large body is immersed in a medium, the shocks occurring in huge quantities are averaged and form a constant pressure. If a large body is surrounded by the environment 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 varying force, leading to oscillations of the particle. Brownian particles usually do not sink or float, but are suspended in the medium.

Brownian motion theory

In 1905, Albert Einstein created the molecular kinetic theory to quantitatively describe Brownian motion. In particular, he derived a formula for the diffusion coefficient of spherical Brownian particles:

Where D- diffusion coefficient, R- universal gas constant, T- absolute temperature, N A- Avogadro's 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 an approximate one. And although in most practically important cases 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 Biological Laboratory in Heidelberg (under the leadership of S. Jeney) showed the difference in the behavior of the Brownian particle from that theoretically predicted by the Einstein-Smoluchowski theory, which was especially noticeable when increasing particle sizes. The studies also touched upon the analysis of the movement of 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 “memory” of the Brownian particle, or, in other words, the dependence of its statistical characteristics in the future on the entire prehistory her past behavior. 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 a more accurate description it is necessary to use integral stochastic equations.

Thermal movement

Any substance consists of tiny particles - molecules. Molecule- is the smallest particle of a given substance that retains all of it Chemical properties. Molecules are located discretely in space, i.e. at certain distances from each other, and are in a state of continuous disorderly (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 one or another molecule will experience from others. Therefore, they say that the position of the molecule and its speed at each moment of time are random. However, this does not mean that the movement of molecules does not obey certain laws. In particular, although the speeds of molecules at some point in time are different, most of them have speed values close to some specific value. Usually, when speaking about the speed of movement of molecules, they mean average speed (v$cp).

It is impossible to single out any specific direction in which all molecules move. The movement of molecules never stops. We can say that it is continuous. Such 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 body temperature. The more average speed movement of body molecules, the higher its temperature. Conversely, the higher the body temperature, the greater the average speed of molecular movement.

Brownian motion

The movement of liquid molecules was discovered by observing Brownian motion - the movement of very small particles of solid matter suspended in it. Each particle continuously makes abrupt movements in arbitrary directions, describing trajectories in the form of a broken line. This behavior of particles can be explained by considering that they experience impacts from liquid molecules simultaneously from different sides. The difference in the number of these impacts from opposite directions leads to the movement 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.

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