Special Issue: The Oosawa Lectures on DIY Statistical Mechanics
Chapter 5: Summary of Part I
2021 年 18 巻 Supplemental 号 p. S041-S043
詳細
2021 年 18 巻 Supplemental 号 p. S041-S043
I will summarize the points we learnt from the dice and chips game in the first part of this book.
The first point relates to the principle of equal a priori probability. This is one of the most basic principles of statistical mechanics, and it states that each microscopic state (referring to a particular way of distributing the gaming chips) will be realized, on average, the same number of times, if the exchange is continued for a long time. In the following we refer to Figure 2.5 in which 4 players exchange four chips between each other. Some considerable time after the exchange begins, the distribution of the number of chips held by any individual will become exponential-like in nature (even if at that stage each microscopic state is not realized with equal probability) At this point if the four players results are averaged a Boltzmann-like distribution will, in fact, be observed a long time before the principle of equal a priori probability (which is the basis of statistical mechanics) is valid. After the establishment of a Boltzmann-like distribution derived from averaging the chips held by the four individuals, at an even later time an exponential Boltzmann distribution of the number of chips held by an individual person recorded over time, will also be recorded. The overall averaged Boltzmann distribution, each individual person’s Boltzmann distribution (established over time) and then the principle of equal a priori probability will be realized in three stages in time. To summarize, in statistical mechanics, the Boltzmann distribution based on the principle of equal a priori probability has been proven, however in reality, the Boltzmann distribution is gradually established before the principle of equal a priori probability is established (Table 5.1).
| Starting the exchange |
| 1: Someone with 0 chips will appear. |
| 2: Each person experiences 0 chips. At this time, the overall population average will be a Boltzmann distribution. |
| 3: Each person experiences the best. At this time, the time average of each person will be a Boltzmann distribution. |
| 4: All the distribution patterns achieve the same number of times=principle of equal a priori probability. |
These kinds of unexpected things cannot be truly understood without experimenting by oneself. Formal lectures on statistical mechanics often begin with the assumption of the principle of equal a priori probability. I think that it is quite difficult to derive the Boltzmann distribution on the basis of assuming that the principle of equal a priori probability is valid. To overcome these difficulties I wish to emphasize the order of how this occurs i.e. general Boltzmann over all individuals, individual Boltzmann distribution over time for single individuals and then full realization of the principle of equal a priori probability of all microscopic states. In your spare time, please try to play the game of this dice and chips by yourself, and check if this is the case with the passage time.
The second point is that you do not have to care about the small number of boxes and chips (Table 5.2). What I showed you was a game with just four gaming chips in four boxes (persons), which means one chip per box on average. Even with such a simple simulation, the order of the emergence of the three-stages of distribution change (described in the preceding section) can be seen to hold. Of course, in the example given involving four chips in four boxes, it was not an exponential distribution, but a line going down towards the right. However, this downward sloping line towards the right can be realized using an extremely small number of molecules, several degrees of freedom, and even a small total energy. Therefore, you can approximate the line going down towards the right using an exponential function (Table 5.2). It is mathematically more convenient to write expressions using exponential functions.
| Downward sloping curve in the case of 4 chips in 4 boxes (exponential function approximation is possible) → Temperature5.2 can be defined in 4 degrees of freedom |
The fact that the gaming chip distribution experienced in the game approaches an exponential function, that is, the Boltzmann distribution, means that a ‘temperature’ can be defined using only four degrees of freedom. As such, the “kBT” under the energy, which is the exponent in the Boltzmann distribution (Table 2.7), can be determined. This is an important topic. Strictly speaking, it is not exponential with four or five degrees of freedom, and it is just an approximation. However, the temperature can be defined in four degrees of freedom. This implies that the temperature can be measured (Table 5.2). If we insert a fifth player or probe that can exchange energy with the other four, and if we assume that the fifth person can also receive energy, we can measure the temperature. We will discuss an example for that later (Chapter 7).
I encourage you to do this dice and chip game by rolling the dice yourself. Although it takes a bit of time, please try it in order to appreciate and understand that the results of the game will actually appear in the order shown in Table 5.1. If you do it as a team, it will be fun.
Let us assume that you have recorded the exchange of chips over time. Examining the record, you recognize two things. After a certain stage the chip distribution time course will display time-reversal symmetry i.e. even if the passage of time is reversed, it will have a similar time course of the number of chips per person. The second is that “now” is always the peak moment. These two points are very important (Table 5.3).
| · Time-reversal symmetry |
| · “Now” is always the peak moment |
Instead of stating “I did the experiment, then something interesting happened”, it is better if someone asks the question, “Is there a case for which it does not behave similarly if the passage of time is reversed, or a case whereby “now” is not the peak moment?”, this poses a new problem. Let’s consider the case wherein for an earlier part of the time course of the number of chips does not display time-reversal symmetry. When rolling the dice and exchanging chips, time reversal is symmetrical. Further, let us consider the case wherein “now is always the peak moment” does not frequently hold when chips are exchanged. If such questions are converted into one scientific issue, furthermore, if you can consider these questions and find an appropriate example, you can write a research paper. I have published one too [5.1].
For considering such issues, let us examine the network diagrams shown in Figure 2.3 and Figure 3.5. When is it more likely to receive one more gaming chip just after receiving a chip? Let us consider the case when the time course of the number of chips becomes a saw-tooth wave, that is, when it rises, it rises rather quickly, and when it falls, the decay is gradual, without time-reversal symmetry. Then you can think of a way of representing the arrows in the network. Since each line that connects the network has the same probability of giving and receiving for a typical roll of the dice, I implore you to think while asking yourself, “what can be done to cause the probability of giving and receiving to be different?”. This is my wish as a teacher and an educator.
Dear researchers. Please don’t think that these stories about statistical mechanics have nothing to do with your experiments. This kind of talk often unexpectedly leads you back to thinking about your own experiments. There will be times when some kind of interest or questions about your experiments may haunt you day and night. At such times, if you listen to a seemingly unrelated lecture, you may come up with a new idea. In general, as I have experienced many times, research does not progress well when we have discussions with friends or fellow rivals. However, when you discuss with someone who is interested in your talk, but whose area of expertise is slightly different from yours, or when you attempt to get someone from another field interested in your talk, what such a person says will often be the best clue. You will feel that something has opened up. So, when you have any experimental problems or even theoretical problems, sometimes it is good to take a hint from a distant topic.
We thank Damien Hall (Kanazawa University) for providing an additional check of exposition within our translated manuscript.
5.1 Translator’s note: Normally the topics discussed in this chapter are introduced formally in terms of comparing and contrasting the principle of equal a priori probability against the ergodic hypothesis i.e. when the ensemble (collection) average is equal to the long time average for a single case.
5.2 Translator’s note: It is not “temperature” but the “energy” in the original Japanese book. However, we believe it is “temperature.”
