What is Brownian motion

This movement is characterized by the following features:

• continues indefinitely without any visible change,
• 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 for 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 liquid or gas molecules form one statistical system. In accordance with the theorem on the uniform distribution of energy over degrees of freedom, each degree of freedom accounts for 1/2kT of energy. The energy 2/3kT per three translational degrees of freedom of a particle leads to the motion 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 is accounted for by its rotational degrees of freedom. Therefore, with its trembling, it also experiences constant changes in orientation in space.

It is possible to explain Brownian motion in the following way: the cause of Brownian motion is fluctuations of pressure, which is exerted on the surface of a small particle by the molecules of the medium. Force and pressure change in modulus 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, which was in a homogeneous isotropic medium at the initial time (t=0) at the origin of coordinates, will shift along an arbitrarily directed (at t$>$0) axis Ox 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's coordinate due to fluctuation.

Consider the position of a Brownian particle at some fixed time intervals. We place the origin of coordinates at the point where the particle was at t=0. Let $\overrightarrow(q_i)$ denote the vector that characterizes the movement of the particle between (i-1) and i observations. After n observations, the particle will move from the zero position to the point with the radius vector $\overrightarrow(r_n)$. Wherein:

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

The movement of the particle occurs along a complex broken line all the time of observation.

Let's find the average square of the removal of the particle 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 average 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 on displacement at the j-th 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=$\triangle tn$ - the time during which the mean 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 origin. It is essential that the average square of the removal grows in proportion to the first power of time. $\alpha \ $- can be found experimentally, or theoretically, as will be shown in example 1.

The Brownian particle moves not only forward, but also rotating. The average value of the rotation angle $\triangle \varphi $ of a Brownian particle over time t is:

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

where $D_(vr)$ is the rotational diffusion coefficient. For a spherical Brownian particle of radius - a $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 trembling of the mirror, like a Brownian particle that is hit by air molecules. The random movement of electrons causes noise in electrical networks.

Example 1

Task: In order to mathematically fully characterize Brownian motion, you need to find $\alpha $ in the formula $\left\langle r^2_n\right\rangle =\alpha t$. Consider the viscosity coefficient of the liquid known and equal to b, the temperature of the liquid T.

Let us write down 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 fluid.

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

We 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 equation (1.1) is reduced 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 parts of this equation over an ensemble of Brownian particles, taking into account that the average of the time derivative is equal to the derivative of medium size, since this is averaging over an ensemble of particles, and, therefore, we will rearrange it by 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 the deviations of a Brownian particle in any direction are equally probable, then:

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

Using $\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 $\left\langle x^2\right\rangle =\frac(\alpha t)(3)$, hence: $\dot(\left\langle x^2\right\rangle )=\frac(\alpha ) (3)$, $\left\langle \ddot(x^2)\right\rangle =0$

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 hold, 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 on 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 the Brownian motion of suspended particles.

Example 2

Task: Gummigut particles of spherical shape with radius r participate in Brownian motion in gas. Density of gummigut $\rho $. Find the root-mean-square velocity of gum particles at temperature T.

The root-mean-square velocity 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 velocity of the gas molecules, which, in turn, move the Brownian particle. 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 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 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 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.

« 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 Nobel Prize for the 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

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 chemical properties environment. 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: = 2D

Here 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 aspect of fractal geometry that has the most practical use. Random Brownian motion produces a frequency pattern that can be used to predict things involving large amounts of data and statistics. good example are the wool prices predicted by Mandelbrot 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 absolutely round table, you can calculate up to approximately 500 collision positions 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 is a continuous, constant chaotic motion of particles suspended in a liquid (or gas). The name now used was given to the phenomenon 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 could not fully analyze the process.

Modern views

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. It is they who 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 are describing and diffusion (penetration of particles of one substance into another). These processes should be considered as a whole, since they explain each other. So, due to collisions with the surrounding molecules, particles suspended in the medium are in continuous motion, which is also chaotic. Chaoticity is expressed in inconstancy, both in direction and speed.

From the point of view of thermodynamics

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 the Boltzmann constant, and T is the external temperature. As we can see, the square of the speed of a suspended particle is directly proportional to the temperature, therefore, with an increase in the temperature of the external environment, the speed also increases. Note that the basic principle on the basis of which the equation is composed is the equality of the average kinetic energy of a moving particle to the 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 about the same time, independently of each other.

Motion of Brownian particles

Particles suspended in a liquid or gas move along a zigzag trajectory, gradually moving away from the starting point of motion. Again, Einstein and Smoluchowski came to the conclusion that for studying the motion of a Brownian particle, it is not the distance traveled or the actual speed that is of primary importance, 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 related to the viscosity of the medium and particle size), t is time. Consequently, the speed of the suspended particle is the higher, the lower the viscosity of the medium. The validity of the equation was experimentally proven by the French physicist J. Perrin.