Special Issue: The Oosawa Lectures on DIY Statistical Mechanics
Chapter 4: Always “Now” is the Peak Moment
2021 年 18 巻 Supplemental 号 p. S035-S040
詳細
2021 年 18 巻 Supplemental 号 p. S035-S040
Reviewing what we have discussed until now, the main point is that the case of exchanging energy (random exchange) is entirely different from that of unilateral energy flow (random distribution) (Table 1.1). When the exchange goes on, energy (chips) is distributed unfairly for each instance. For a molecule, the changes in the direction of losing the energy happen overwhelmingly, and those of gaining the energy happen rarely. Again, for the case of considering a molecule, its energy is extremely high in the higher energy state and almost zero in the lower energy state. However, in terms of the time of experience for a molecule, the higher energy state rarely occurs, but the lower energy state is almost overwhelmingly long. As a result, the average value over time can be described by an intermediate average that is not reflective of its mostly low and rarely high energy reality. The main point is that the exchange causes these phenomena.
Although the reason for this is considered to be due to each distribution state of the gaming chips (energy) with equal probability (principle of equal a priori weights, Section 2-3), I would appreciate it if you could confirm this by your hand (Exercise 2). A distribution state with equal probability means that the smaller the number of gaming chips (energy) a person possesses, the greater the number of ways it can be distributed to others4.1. Hence, it indicates that the fewer gaming chips (energy) you possess, the better for the people except for you (Section 2-6). It is better to have fewer gaming chips (energy), as it is convenient for the other’s profit!
Although the above conclusion for the dice and chip game would be drawn if one only had a single player (degree of freedom), the situation is different if two or more players are considered (Section 3-3). For the case of more than one player, there should always be more than one way to distribute the gaming chips i.e. entropy, and this entropy increases as the number of chips held by the player increases. Moreover, the entropy increases rapidly as the energy increases from 0, 1, 2 and so on4.2. Therefore, if the number of players (degrees of freedom) is two or more, the probability of having finite energy is always higher than zero (Fig. 3.8). The main point is that the peak point of the energy is determined by the balance between actual energy (in this case chips) and entropy (in this case related to the multiplicity of their arrangements).
Much of the story so far will be the same when you reverse the time. Let’s assume that we take a recording for a long time. Since it is useless to consider a situation immediately after the unilateral distribution at the beginning, let us consider the increase or decrease of each person after steady-state4.3. After steady-state, if you select a certain time arbitrarily, and examine before and after that time, it will appear to be the same. In other words, the manner of increase or decrease in the number of chips is symmetrical with respect to the time variation. For Fig. 3.3, a gray dot-plot represents the time variation of the number of chips when the probability of giving out chips does not depend on the number of chips held by each person (this situation means no restoring force). In this case, although the number of chips held fluctuates (as shown in this figure), it looks the same even if it is flipped around the time axis (see Fig. 6.2). In other words, there is time-reversal symmetry4.4. Another aspect is that in the time course of the number of chips, peak-like changes that increase and then decrease, are primarily observed. Once you become rich, it is most likely that you will start losing rather than continuing to be rich. This means that “now” is always the peak moment4.5.
In terms of time reversal symmetry, the frequency of occurrence of changes from a low to high energy state is the same as the frequency of losing energy from the high to low energy state. Since it is easier to consider the case of damping in various systems, this leads to the fact that the time taken to decay is also exactly the time it takes to receive energy by chance in thermal equilibrium. This is what happens when you consider a chemical reaction as an example. Although we are interested in determining the time required to have enough energy to overcome the energy barrier when it is in equilibrium, it is difficult to conceptualize. However, employing the concept of time reversal symmetry we note that after receiving such an amount of energy, the relaxation time of losing the energy (i.e. due to viscosity of solution) can be understood. Since it can be considered that the “relaxation time=arrival time,” the time required to receive such a ‘high’ energy can be estimated. This is the point of the current discussion.
Since it is difficult to appreciate why most of the patterns are in the form of an increase and a decrease, I would like to explain in detail. I introduced the principle of equal a priori probability earlier (Section 2-3). The demonstrated extension of this principle (that all forward and backwards steps are equally likely to occur) was that the fewer the number of chips you have, the greater the number of ways of distributing the existing gaming chips amongst the other players – i.e. a greater system multiplicity of arrangements. Conversely, the more chips you have, the fewer the number of ways of distributing the existing gaming chips amongst the other players (i.e. a lower system multiplicity of arrangements). In statistical mechanics, the total number of available ways of storing a gaming chip is called the total number of microscopic states. When applying the principle of equal a priori probability (such that all microscopic states occur with equal probability) the smaller the number of microscopic states, the less likely the state will occur. Therefore, a situation in which you have a great number of chips, is very unlikely to occur. Thus, at the time of any subsequent exchange, it is more likely that you will have fewer chips compared to now (rather than more chips). Namely, if you are in a position of having a significant number of chips then “now” is always the peak moment. This will be explained next with a practical example.
Next, we consider the case of exchanging six gaming chips each amongst six boxes, a total of 36 gaming chips. When I gave an intensive lecture in Nagoya, a student from the Toyota Institute of Technology created a program to track the passage of time. The program was designed to count the number of times each pattern of ups and downs (in terms of the number of gaming chips held) appeared while recording the passage of time (Fig. 4.1, Fig. 4.2, Fig. 4.3). Although the number of gaming chips in each box will continually change, we place our focus on just one of the boxes and count the number of times each pattern appears. The average number of gaming chips in any box is six. Therefore, when there are six gaming chips in the box of interest, we examine the pattern before and after that: when the previous pattern is five chips, the current pattern is six, and the next pattern is five, we write the pattern of (5, 6, 5) as pattern A. Similarly, we write (5, 6, 7) as pattern B, (7, 6, 5) as pattern C, and (7, 6, 7) as pattern D (Fig. 4.1). There are only these four ways for the before and after of the situation whereby six gaming chips exist presently within the box. In this case, we assume that time is reduced when the dice are hit by another person and there is no change in the box of interest. Omitting all the time for which there is no change, we simply focus on the times when the change occurred and examine how it changed. Incidentally, we define pattern E, F, G, and H in the same way with the center being 9 gaming chips, which is higher than the average (Fig. 4.1). We determine the number of times the following occurred: reaching 9 gaming chips from the bottom and falling (8, 9, 8) as pattern E, going up monotonically (8, 9, 10) as pattern F, falling monotonically (10, 9, 8) as pattern G, and approaching from above and then returning up (10, 9, 10) as pattern H.
Patterns of increase and decrease: Pattern of increase and decrease when 6 players exchange 36 gaming chips. The pattern of increase and decrease before and after a certain number (dotted line); (Left) 6 chips (the average), and (Right) 9 chips (exceeding the average). Patterns A and E: Increase and decrease, patterns B and F: Increase and increase, patterns C and G: Decrease and decrease, and patterns D and H: Decrease and increase. If there is no change, the time is shortened. For example, 5 → 6 → 6 → 5 is also counted as pattern A.
Frequency of increase and decrease patterns4.6: The frequency of appearance of the increase and decrease patterns when 6 people exchange 36 gaming chips 10,000 times. The vertical axis is the number of appearances, and the patterns A-H (horizontal axis) correspond to Fig. 4.1. Left: Increase and decrease of gaming chips before and after the average (6 chips) in all the boxes. Right: Increase and decrease before and after 9 chips (exceeding the average).
Frequency of increase and decrease patterns after 1,000,000 exchanges4.7: The frequency of appearance of the increase and decrease patterns before and after the average (6 chips, Left) and 9 chips (exceeding the average, Right) when 6 people (boxes) exchange 36 gaming chips 1,000,000 times. The axes are same with Fig. 4.2.
The gaming chips are first exchanged 10,000 times (Fig. 4.2), and then 1,000,000 times (Fig. 4.3). Here, we focus on a particular box (box-1 to -6) and count the number of times each pattern is experienced. When the number of exchanges was 10,000, we can see a tendency that pattern-A is the most, B and C are almost the same and middle, and D is the least. In the case of patterns E, F, G, and H, although pattern E is the least whereas the others are the most in box-1 and 6, the other boxes exhibit the same tendency. Please note that there are considerable variations (Fig. 4.2). When we increase the number of exchanges to 1,000,000, box-1 experienced pattern A 4,818 times, B 4,135 times, C 4135 times, and D 3,611 times, respectively (Fig. 4.3). As the number of exchanges is increased, the patterns B and C are always experienced the same number of times. Because of the principle of time-reversal symmetry, they are always the same. As expected, when we increase the number of exchanges, the expected distribution can be obtained in all boxes: the pattern A is the most common, B and C are equal in the middle, and the pattern D is the least. The same is true for E, F, G, and H.
The patterns A, B, C, and D all describe passing through the average. It is interesting to note that when you think you have finally reached the average, the next step is to go down, even though it is the average. We may think that since it is the average, it would be ideal if the probability of going up or down from that point was the same. However, once you finally reach the average, it is more likely to go down next. Because the number of exchanges is not large, the difference is not large but its existence is significant. When the number of gaming chips decreases from 7 to 6, it is very difficult to raise it again to 7. This is quite beautiful data. The point is that a number of gaming chips above the average does not easily appear. In addition, a considerable number of exchanges are necessary to observe such patterns for each box. The relationship between A, B, C, and D does not show the correct tendency in some boxes in the case of only 10,000 exchanges. After approximately 1,000,000 times, the pattern A exhibiting rising and falling appears most frequently, which is another manifestation of the case where “Now” is always the peak moment is properly realized.
Again, let’s consider the case of two players. We use the method that has a restoring force, whereby the probability to give out gaming chips is proportional to the number of chips you have. Let’s say we are exchanging 12 gaming chips with an average of 6 chips, and a player happens to have 8 chips. By rolling the dice, I examined whether the player with 8 chips would go up and stay up (7, 8, 9), or up and down (7, 8, 7), or down and up (9, 8, 9), or down and down (9, 8, 7) (Fig. 4.4). (7, 8, 7) was the most frequent with 75 times, (7, 8, 9) and (9, 8, 7) was moderate with 42 and 39 times respectively, and (9, 8, 9) was significantly lower with just 29 times. Taking (7, 8, 9) and (9, 8, 7) as the reference, (7, 8, 7) occurred the most and (9, 8, 9) occurred the least. Therefore, even for this example it can be said that “now” is always the peak moment.
Frequency of increase and decrease patterns with two players: The frequency of appearance of the increase and decrease patterns when two people exchange 12 gaming chips. Results of actual dice rolls. Pattern of increase and decrease before and after 8 chips (top), and the number of appearances of each (below).
If we turn the time backwards, the number of occurrences of (7, 8, 9) and (9, 8, 7) should be equal, so they look the same. This is an example of two people exchanging when there is a restoring force to bring them closer to the average. Since there is a restoring force, readers will agree with the observed results. On the other hand, the previous case of six people, for which there was no restoring force, is more surprising, or in a sense, uncommon, and quite interesting. I encourage you to experience this for yourself.
In Section 2-3, I introduced the principle of equal a priori probability. This principle implies that the less energy you have, the more ways of exchange the others have. Thus, given that it is more likely to have less energy, it is easy to get to 0 gaming chips. Here is another perspective on why it is easy to have 0 gaming chips when there is no restoring force. It is easy to think about what will happen if there is a player with 0 chips. When the number of players with 0 chips starts to increase, the number of people who can give out chips decreases, and those who have chips must give out chips more often. Although those with 0 gaming chips and those with non-zero chips have equal probability to receive chips, as the population of 0 chips increases, the persons with non-zero chips must give out chips more frequently. Therefore, as the population of 0 gaming chips increases, those who have chips are more likely to lose their chips.
You can understand so far that, when a person becomes rich, the money (the number of gaming chips one has) increases almost at a stroke. This is quite interesting. Chemical reactions occur in the same manner. Not gradually reaching the energy barrier and finally reacting, but rather the energy level of a molecule rapidly meets and then exceeds the energy barrier to undergo reaction. The activation event in a chemical reaction occurs rarely, but when it happens, it is a big event. I don’t know why, but it happens by chance in one stroke. Although I can realize that the process occurs at a stroke when I rolled the dice, I am a little troubled when asked why this should happen. Since it is quite difficult to explain theoretically, please test it via dice rolling and chip exchange by hand.
My approach to the discussion of solid-state physical properties is to also use simple physical experiments that can be carried out by hand. As an example, let’s consider a phase transition. As I won’t go into much detail, for the sake of a brief description, we will reduce the number of atoms as much as possible, and consider a lattice in which 9 atoms are arranged in 3×3. Let’s assume that each atom has either an up-spin or a down-spin. First, let’s write out all the patterns of spin configurations (Fig. 4.5A). Then, we write down all the interaction energies for each configuration and perform a calculation. From the energies, we can determine the probability of each configuration under a given temperature. My opinion is that if you do this by hand, you can get a better idea of what the phase transition is.
An example of application to solid-state physics (change in magnetic susceptibility) by hand. A: Consider a lattice of 9 atoms arranged in 3×3. Assuming that each atom has either an up-spin (●) or a down-spin (○), all the spin configuration patterns are written. There is an energy λ between the adjacent spins ● and ○. Symmetrical configurations are omitted. B: The horizontal axis is the difference between the number of upward and downward spins (| ● - ○ |), and the vertical axis is the probability of each arrangement. C: Magnetic susceptibility (mean of | ● - ○ |) when the temperature (horizontal axis) is changed.
Since the number of atoms is too small with the 9-atom lattice, we will not be able to obtain up to the transition temperature. However, we can see how the magnetic susceptibility changes with temperature (Fig. 4.5C) [4.1]. The reason why I recommend this approach is that it is interesting because we obtain the essence without complicated theory or approximation.
This is not just related to biophysics, but I feel there are roughly three types of characteristics and styles among researchers. There are types of researchers who like “molecular structure”, who like “kinetics”, and who like “equilibrium theory”. Those who like back and forth behavior but slow migration are the “equilibrium type”. Some people who like processes at a stroke are the type who like kinetics. I guess these characteristics are innate. I like the equilibrium theory. I enjoy investigating something that moves gradually as it goes back and forth. I like frequent going back and forth rather than progressing. However, people who like kinetics may be interested in the process, which goes on in a large portion, although it also includes additional back and forth movements. Structural theory is a completely different style of thinking, and they like sticking in one place without moving. In pure physics, whether it is a semiconductor or magnetic body, saying “I like semiconductors” or “I like superconductivity” is probably something that has nothing to do with one’s personality. However, in the case of biophysics, unlike theoretical physics, it is better to make the best use of one’s personality to select the subject of study. After all, the subject is a living organism.
We thank Damien Hall (Kanazawa University) for providing an additional check of exposition within our translated manuscript.
4.1 Translator’s note: Please see the translators note 2.9.
4.2 Translator’s note: Please see Fig 3.7.
4.3 Translator’s note: Let us consider a time window recorded under ‘equilibrium’ conditions.
4.4 Original note: Since the time is short, it is not easy to understand in this figure. See Fig. 6.2 for more details.
4.5 Original note: Please see Fig. 6.2 for this one also.
4.6 Translator’s note: We repeated the calculations and made a new figure to help clarify the original figure presented in the Japanese book.
4.7 Translator’s note: Although the number of exchanges is 100,000 written in the original Japanese book, 1,000,000 is correct number (as specified in the original Oosawa lectures). To check and confirm this point we repeated the calculations and used the results of these new calculations to produce an updated figure.
