Special Issue: The Oosawa Lectures on DIY Statistical Mechanics
Chapter 8: Brownian Motion
2021 Volume 18 Issue Supplemental Pages S066-S075
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2021 Volume 18 Issue Supplemental Pages S066-S075
The phenomenon of irregular movements of fine particles in a liquid is called Brownian motion, named after Robert Brown, who observed fine particles of pollen moving randomly in water. In this chapter, we will discuss this phenomenon in detail. We will first consider the ease of movement of particles when they randomly collide with solvent particles in a liquid and undergo Brownian motion, without an external force (Fig. 8.1). Some of you may know that the mean square (<x2>) of the position fluctuations, represented by Equation 8.1, can be written in terms of the diffusion constant D and time τ as in Equation 8.2.
| (Eq. 8.1) |
| (Eq. 8.2) |
Furthermore, D can be written as
| (Eq. 8.3) |
where kB is the Boltzmann’s constant; v is the velocity of the Brownian motion, T is the absolute temperature; and ξ is the friction coefficient. Equation 8.3 was derived by Einstein [8.1]: The method of deriving (Equation 8.1), i.e., the mean-square of the position x of a particle, from the path integral of the velocity v of Brownian motion, is explained in Iwanami’s “Kagaku” [8.2]. I recommend that readers read this excellent paper There is a concept called Kubo theory. Although I do not completely understand the quantum mechanical part of the Kubo theory, this paper explains its classical physical part of quite well.
Brownian motion is another case of spontaneous fluctuations—a topic that has been previously discussed. Since ξ is the coefficient of friction, when a movement occurs because of an external force, the spontaneous position fluctuation is inversely proportional to this ξ. Actually, it would be better to derive each of them; however, I will omit those details here.
8-1-1 One-dimensional Brownian motionConsider the one-dimensional Brownian motion. We assume that we take one step in the positive or negative direction each time with the same probability. First, we write down each case by hand, as if you are a child who does not know anything more (Exercise 5). Please try it. If I show you how to do this ahead, you will not be impressed. For example, in the case of two steps, the possible cases are: (plus, plus), (plus, minus), (minus, plus), and (minus, minus). Because (plus, plus) is two steps forward, it is +2, (plus, minus) is 0, (minus, plus) is 0, and (minus, minus) is –2. One method for determining the average is to take the average of the absolute values. The average will be 1 (step) (Equation 8.4). Another method is to find the mean square, which is calculated to be 2. Hence, the root mean square value is
Although it is well-known that the mean square for n steps is proportional to n, most people believe that this will be realized after many steps. I was really surprised when I actually attempted this exercise. Even for two, three, or four steps, the mean square is exactly proportional to n. I have my students do this every year. Please try it. Of course, this can easily be proved using a mathematical formula. However, please be sure to write out everything, such as plus, plus, plus, and plus. Cover all the cases for three steps. If you attempt to take the absolute values and add them, you will obtain some strange value. If we take the squared values and average them, the answer will be 3.
In the case of four steps, if we take the mean square, it will beautifully be 64/16, that is 4. Please attempt this for cases involving five and six steps. You will not be able to appreciate this if you do not attempt it. Gauss, a genius, would have proved this beautifully using a mathematical formula. Since I am not such a genius, I write everything. For everything from the case of one step to two steps, etc. make sure that the mean square of the particle position x is exactly n (Exercise 5).
In a one-dimensional Brownian motion, although the average of the absolute values of n steps is a strange value, the mean square is exactly n. Write + or – for movements of one step in the positive or negative direction, respectively, and enter the distance moved (Note: not the journey) in (). Write out all combinations and calculate the average of the absolute values and the root mean square. Considering the following two-step (n=2) movement as a reference, calculate the same for n=1, 3, 4, and 5.
(when the 1st step is + and 2nd step is +) + + (+2)
(when the 1st step is + and 2nd step is −) + − (0)
(when the 1st step is − and 2nd step is +) − + (0)
(when the 1st step is − and 2nd step is −) − − (−2)
| (Eq. 8.4) |
| (Eq. 8.5) |
1 step
+ (+1), – (–1)
2 steps
++ (+2), +– (0), –+ (0), –– (–2)
3 steps
+++ (+3), ++– (+1), +–+ (+1), +–– (–1)
–++ (+1), –+– (–1), ––+ (–1), ––– (–3)
4 steps
++++ (+4), +++– (+2), ++–+ (+2), ++–– (0),
+–++ (+2), +–+– (0), +––+ (0), +––– (–2)
–+++ (+2), –++– (0), –+–+ (0), –+–– (–2),
––++ (0), ––+– (–2), –––+ (–2), –––– (–4)
5 steps
+++++ (+5), ++++– (+3), +++–+ (+3), +++–– (+1),
++–++ (+3), ++–+– (+1), ++––+ (+1), ++––– (–1)
+–+++ (+3), +–++– (+1), +–+–+ (+1), +–+–– (–1),
+––++ (+1), +––+– (–1), +–––+ (–1), +–––– (–3)
–++++ (+3), –+++– (+1), –++–+ (+1), –++–– (–1),
–+–++ (+1), –+–+– (–1), –+––+ (–1), –+––– (–3)
––+++ (+1), ––++– (–1), ––+–+ (–1), ––+–– (–3),
–––++ (–1), –––+– (–3), ––––+ (–3), ––––– (–5)
Now, let us consider a two-dimensional motion, which includes up–down and left–right movements (Fig. 8.2). Similar to the one-dimensional case, please attempt to write out all the cases, namely one step, two steps, etc. Then, you will discover that the mean square of the particle position r is n. Please perform this task so that you can see it for yourself. We cover all the cases by classifying them into up–down and left–right. However, we do not go diagonally. Please try this and you will be impressed (Fig. 8.2).
Place dots on the graph sheet as shown in Fig. 8.3. First, place dots with a spacing of 1 cm on a horizontal line. On the horizontal lines above (and below), shift 0.5 cm to the right and left, and place dots every 1 cm. Thus, for each point, there will be adjacent points in six directions, as shown in the figure. Therefore, you can decide which way to go starting from one point based on the roll of a dice (for example, 1 is right, 2 is upper right, 3 is upper left, and so on). A point is marked at each particle position starting from a certain point up to the n-th time. Then, the points in a circle encompassing this graph paper and the moving point will have almost the same density.
By the way, do you know how to illustrate Brownian motion on graph paper? Most may not know this approach. We use a graph paper and a dice. As shown in Fig. 8.3, place a dot every 1 cm on the graph paper. At a position 1 cm above, place dots at a spacing of 0.5 cm between two dots. Once this is done, although a little distorted, regular hexagons are obtained. Starting from a point, we decide the direction for each dice roll; for example, when we roll a 1, we go one way, and when we roll a 2, we go another way, and so on. As you continue to roll the dice, the “particles” start to move with a beautiful motion. Please try this as well. Some people may find this to be a boring exercise; however, if you do it, you can precisely understand the peculiarities of a two-dimensional Brownian motion. Although related to the
Now, let us assume that the particles have reached a point, where the average position is
Let us apply the same approach to three dimensions, wherein, typically, an overwhelmingly large number of vacant places exist. Even in three dimensions, assuming that the average position of a particle after n steps is
Do you know how to catch all the fish in the sea? Assume that there is an infinitely large sea and many fish are swimming in it, and that we want to catch all the fish in a net. The way to catch them is to erect a cylindrical net of an arbitrary size in the sea. There is a theorem that implies that if we wait patiently casting a net such that fish that enter the net cannot leave it, eventually, all the fish will enter the net. This is same as the theorem of Brownian motion. An infinite number of fish in the infinitely large sea will eventually enter this limited space. This mathematical theorem impressed me greatly. I think that this theorem was discovered by Shizuo Kakutani8.2, a mathematician at Osaka University [8.3]. Although the commentator is a different individual, a wonderful commentary can be found in “Scientific American” [8.4]. I request you to read it.
Although the subject of Brownian motion may not seem to be related to statistical mechanics, it is a key point in statistical mechanics. “Fundamentals of Contemporary Physics”, volume 2 of classical physics, published by Iwanami-Shoten [8.5], explains that this issue is the logical point of contact that connects the micro and macro of physics. Although I think that the author’s claim is overemphasized in the book, because it has something to do with statistical mechanics, please read it if you are interested in this issue. I will provide a brief explanation because this is a typical mathematical theory that connects the micro and macro.
8-2-1 One-dimensional Brownian motionFirst, let us consider a one-dimensional space from r to R, and start the Brownian motion of a particle from point x in it (Fig. 8.5). Finding the probability of reaching a point r in the left, we obtain Equation 8.68.3.
| (Eq. 8.6) |
In other words, when calculating the equation in the space of ∞, (R→∞), because the probability of reaching point r becomes 1, it means that we will always reach the point r mathematically.
This is similar to the game of two people rolling a dice and exchanging chips, particularly without a restoring force. As such, when only two people exchange chips, one of them would soon reach 0, and the game would end. The probability of winning the game is proportional to the amount of money you initially have; this can be easily understood by playing the game by yourself. The probability of bankruptcy is, of course, higher if you have less money to start with. This is a one-dimensional diffusion problem analogous to Motoo Kimura’s population genetics8.4.
8-2-2 Two-dimensional Brownian motionNow, let us consider the two-dimensional case (Fig. 8.6). We draw two circles, an inner circle with radius r and an outer circle with radius R. We start the Brownian motion of a particle from a point x between the two circles. We assume that the probabilities of stepping up, down, left, and right are equal. In this case, the particle will eventually arrive at either r or R and then end its motion. The particle cannot remain in between forever. At this time, the formula for the probability of arriving at the inner circle (of radius r), is given by Equation 8.7, which consists of logarithmic functions.
| (Eq. 8.7) |
Here, if the outer circle radius R is made infinitely large, the probability of reaching the inner circle is 1, indicating that the particle will certainly reach the inner circle at some point in time. Therefore, it represents a logic that states “all the fish in the sea of an infinite size will eventually enter into the finite cylinder”.
Although the probability of entering the inner circle is 1/2 when starting from the center in the one-dimensional case, it is 1/2 when starting from the geometric mean (
The above conclusion is not applicable to three-dimensional scenarios (Fig. 8.7). The probability of reaching the inner area (sphere) in the case of three dimensions is given by Equation 8.8.
| (Eq. 8.8) |
In this case, if the outer spherical shell radius R is assumed to be infinity, when starting from a point between the inner spherical shell of radius r and outer spherical shell of radius R, the probability of being sucked into the inner sphere is r/x. It is noteworthy that this probability is finite. If we consider the density of points on the possible trajectory of a Brownian particle, the space can be filled almost uniformly in two dimensions. However, in three dimensions, the gap between the possible trajectories, which means the places the particle cannot reach, will continue to increase with increasing R. Therefore, only a limited number of fish will enter the inner spherical shell, leaving the others that will escape to infinitely farther distances. The reason why I introduce this issue is that there is a wide variety of statistical mechanics and Brownian motion, and I wanted to highlight the fact that Brownian motion is very different under different dimensional spaces.
I would like to briefly explain why differences exist in Brownian motions in different dimensional spaces, by bringing in the topic of “Brownian Motion and Potential Theory”8.6 [8.3]. The question is, what is the probability P(x; R) that a Brownian particle, starting at a point x, will reach a point R? We start from the point x. Since the first step has equal probability in all directions, it goes around the point x with equal probability. Given that it has equal probability for all steps, after completing the first step, the probability will always be equal to the average of the probabilities on the circumference of x+dx around the point x. Therefore, the character of this function is that the value at a certain point is always equal to the mean value of the function at points on the differential sphere in the vicinity of the point x, or on the circumference of the circle around the point x. The fact that the value at a point is always equal to the average value of the circumference is expressed by a harmonic function. If we write the Laplace equation for a certain potential ψ using Δ(Laplacian operator)8.7, we can obtain a harmonic function as the solution of Δψ=0.
Because this probability always has the property of a harmonic function, in which the value of one point is the average of its surroundings, there would never be a peak or valley anywhere else. For example, we know that the potential distribution, when there is no true charge8.8 is Δψ=0, and the temperature distribution when there is no heat source is ΔT=0. Therefore, the fact that the potential at some points, when there is no true charge is the average value of the surrounding potentials, and that the temperature at a certain point, when there is no heat source is the average of the surrounding temperatures is exactly the same as the situation, in which the probability of a fish being inside a cylindrical or spherical net must be the average of those in the surroundings.
Thus, the problem of Brownian motion now becomes the same as that of finding the solution to the potential theory equation Δψ=0. In electromagnetics, we have learned that when there is an electric charge in the center, the potential becomes 1/r (corresponding to Fig. 8.7 and Equation 8.8) in a 3-dimensional space, and log r (corresponding to Fig. 8.6 and Equation 8.7) in a 2-dimensional space. Using the 1/r potential or log r potential to solve the problem of Brownian motion, we find the probability of reaching the inner radius r. From this argument, it is evident that there is a qualitative difference between two- and three-dimensional spaces, which is what the manual of Brownian motion and potential theory states. This is an intriguing finding.
Let us discuss an example of how to use this in practice. Consider a three-dimensional uneven object. Let us assume that we know the temperature of each point of its surface; however, because we cannot insert a thermometer inside the object, we do not know the internal temperature. Assuming that the material properties of the object are known and that the material is uniform, we can determine the temperature at any point x inside (Fig. 8.8). We consider this by using a computer. Starting with the Brownian motion from point x, we obtain for the first time that it reaches point s and the temperature at point s is T(s). The second time, it reaches a point s’ and the temperature at point s’ is T(s’). This process is repeated. After several repetitions, the average of the temperatures of the points that are reached is used to obtain the temperature of the internal point x.
I heard that this has been applied in practice for designing a nuclear reactor. They often use this type of analysis because this method is faster when the solution cannot be obtained analytically owing to the difficult boundary conditions being imposed. Performing Brownian motion is interesting, and you obtain the correct solution. Taking the average value of the temperatures of all the locations reached gives the temperature T(x) at point x inside. When the potential theory cannot be utilized, simulating the Brownian motion allows us to obtain a mathematical solution of the potential theory. In contrast, the example of the arrival probability described earlier employed a method that replaced the problem of Brownian motion with that of the potential theory—a macro theory that yields the desired answer.
What impresses me most is that Brownian motion, which seems to have no relation to the potential theory of electromagnetics or thermal science at first glance, is suddenly related to it in a very simple and clear manner at essential points. I think that this is a great accomplishment of our predecessors. Such a theory, wherein the macro-and microscales suddenly coalesce in an intuitively easy-to-understand manner, is rare. Moreover, because physics and mathematics are intertwined, I always refer to this in my talks on statistical mechanics. I want you to believe that when unexpected fields of study come together in unexpected ways, there will be tremendous progress; or rather, our eyes will be opened.
Finally, I would like to introduce my paper on Brownian motion [8.6]. This is related to the interaction between two colloidal particles immersed in a polymer solution. We assume that the two colloidal particles float in the solution with a polymer chain between them. Although polymers undergo Brownian motion, it is not same as that of spherical particles, but rather a Brownian motion, in which parts of the polymer chain change their arrangement (Fig. 8.9). When a part of the chain undergoes Brownian motion, the space between the colloidal particles is cramped, and the polymer attempts to leave the space. When the polymer exits, only water will be left between the colloidal particles, and the outside becomes a polymer solution. Therefore, the space between the colloidal particles will be pushed and drawn back owing to the osmotic pressure. Thus, this study focuses on the emergence of an attractive force between the colloidal particles. This problem is also examined in Husimi’s book [8.7]. Although this was not fully solved, it was to some extent, and I felt great about his work.
No active or direct interaction exists between the colloidal particles. Likewise, neither an attractive force nor a repulsive force exists between the polymer and the colloidal particles. The relationship between water and the polymer is just an ideal solution8.9. This means that, although there is no energy interaction, a relatively strong attractive force exists. This theory was demonstrated approximately 20 years ago based on a direct measurement of forces between colloidal particles [8.8]. It is still being measured under various conditions. Approximately 25 years ago, it was called “Asakura-Oosawa force;” however, nowadays it goes by the name “depletion effect.” I am very grateful to the editor, as it was published approximately 35 years prior to other papers in the field.
I now stop my lecture. I know I have repeated this several times; however, please try it yourself, as studies in this field are both physiological and intuitive. Although thermodynamics and statistical mechanics are said to be very difficult subjects (while quantum mechanics is really difficult), statistical mechanics is more intuitive than the other subjects. Because there are many things in the world that will cause you to say “Oh!” when looking at various phenomena, please try moving your hands.
When I got my first job at Nagoya University, initially, I was working in an earthquake laboratory. My professor8.10 said to me “Since earthquake is a bit out of the field for you, and hence is difficult, try doing mud research in the laboratory.” Following his advice, I started researching on mud. This work involved experiments, in which we scraped mud from a schoolyard, put it in a test tube, and performed careful observations. However, depending on the weather conditions (fine or cloudy), the way the mud behaved was different. This was a significant finding for me. It has been more than 60 years; but I have not yet published a paper on it. There was no one in the laboratory at that time, except for a few desks. I think this was an experiment I performed in 1945, right after the war. If a test tube is set up near the window of an empty room on a sunny day, a large temperature difference would exist between the window side and the other side of the test tube. On a cloudy day, the temperature difference would be smaller. Hence, the convection in the test tube would change.
In 1900, a popular experiment was performed called Bénard convection [8.9–8.11]8.11. This was the first experiment that considered dissipative structure formation. If the temperature difference is small, a convection tube is formed instead of the whole mud undergoing convection. A honeycomb pattern can be observed from above. That was actually my experiment. I observed that a slight temperature difference between the two sides caused a layered structure to form sideways in the horizontal directions, and I thought this might be the case. Subsequently, whatever I did, I could not determine the composition of the mud, and I moved to polymer research, thinking that I could study a polymer with a known composition.
8.1 Translator’s note: A question mark is missing in the title of 2.2 in the Table of Contents at the beginning of the book.
8.2 Original note: Shizuo Kakutani (1911–2004). Mathematician and Professor at Yale University. His work, Kakutani’s fixed-point theorem, is famous in game theory and economics.
8.3 Original note: See Appendix A at the end of the book for the solution.
8.4 Original note: Population genetics describes the process of genetic mutations within and among populations of organisms. This study utilizes the diffusion theory. Reference [8.12] is a commentary written by the author for beginners on this subject.
8.5 Original note: Geometric mean. The commonly used mean, called arithmetic mean, divides the sum of all the terms by the number of terms. The geometric mean, on the other hand, is the n-th root of the product of all the terms, where n is the number of terms.
8.6 Original note: Potential theory is a general theory in the field of mathematics, and is used for the solution of the Laplace equation. Because the exact explanation is mathematical, I do not go into the details here. Briefly, the force at each point in space is obtained by differentiating the amount of potential with respect to space. Potential theory is often used to analytically solve potential energy, electrostatic field, and heat conduction problems.
8.7 Original note: A mathematical operator named after P. S. Laplace (1749–1827), a French mathematician. It is usually denoted by “Δ”. In a three-dimensional space represented by (x, y, z), it is given as
8.8 Original note: This is also called the free charge. Atoms and molecules usually exist in an electrically neutral state. The true charge is the charge that appears at the macro level when electrons are separated from atoms and molecules.
8.9 Original note: Generally, it refers to a solution that follows Raoult’s law of vapor pressure. Briefly, it is a solution, in which the solvent, water, and solute polymers are independent and do not interact.
8.10 Original note: Professor Naomi Miyabe, Department of Physics, Faculty of Science, Nagoya University.
8.11 Original note: This is also called the Rayleigh–Bénard convection. When the lower surface of a fluid is heated and the upper surface is cooled, convection occurs in segregated zones, rather than in the entire fluid. For example, when milk is added to black tea or miso soup, a honeycomb-like pattern may be observed owing to this convection.
