Chapter 2: Consumption Tax in the World of Molecules

2-1: If the chances are fair, the results will be unfair

Readers, did you try the game out? What was the result? Here, I will show you the initial condition by the random distribution of the first 30 gaming chips and the subsequent results of the random exchange game that was performed by the 12 groups during the lecture (Table 2.1).

Table 2.1  Results of dice and gaming chips

Groups	Before exchange (Random distribution)	After exchange
1st	5, 6, 5, 5, 2, 7	0, 4, 3, 5, 4, 14
2nd	3, 6, 4, 6, 5, 6	3, 1, 5, 12, 7, 2
3rd	3, 6, 5, 2, 10, 4	2, 9, 9, 0, 7, 3
4th	5, 5, 3, 3, 3, 11	11, 1, 2, 2, 6, 8
5th	8, 4, 5, 6, 2, 5	10, 1, 9, 1, 0, 9
6th	7, 3, 6, 6, 2, 6	7, 5, 10, 2, 1, 5
7th	5, 2, 6, 4, 5, 8	6, 0, 9, 1, 4, 10
8th	3, 8, 8, 3, 4, 4	0, 16, 6, 7, 1, 0
9th	4, 5, 4, 6, 6, 5	4, 0, 4, 7, 12, 3
10th	3, 5, 2, 8, 7, 5	2, 4, 11, 8, 4, 1
11th	5, 3, 4, 3, 7, 8	5, 2, 6, 3, 7, 7
12th	3, 5, 7, 3, 5, 7	0, 6, 6, 6, 4, 8

The numerals are the numbers of gaming chips of each person.

The initial condition of Group 4 is remarkably unfair. Sometimes, this can happen. The distribution did not change significantly even after the exchange. Group 9 is nearly ideal. There is not much change in Group 11 before and after the exchange. Although the number of exchanges during the lecture were insufficient, the results resembled the theoretically expected distributions.

Let me tell you the conclusion first, as we keep exchanging, someone will eventually have just 0 or 1 gaming chip, and someone else will tend to have a large number of gaming chips. After the exchange rather than before, the distribution tends to be generally like that. Before the exchange process commenced, the number of people having the expected average, i.e. 5 gaming chips, was the observed maximum of the histogram, while the shape of the histogram was close to that of a normal (bell-shaped) distribution (cf. Fig. 2.1A vs B white bars). After multiple exchange events, the distribution was such that there were many people with 0, 1 or 2 gaming chips with a much fewer number of people having a larger number of gaming chips (cf. Fig. 2.1A vs B black bars). Indeed, there were only 4 people with a lot of gaming chips—approximately 12 gaming chips or more. In summary, the basic pattern is that people with 0 gaming chips appear quickly. Most of the gaming chips accumulate to someone. Thus, the key result is, even if you perform fairly, this “unfair” distribution will be the result. Although people familiar with statistical mechanics will already know the result, even if you understand it logically, it seems quite strange and unconvincing from the perspective of “why does it become so unfair, even though I am doing it fairly?” If you were to continue this task endlessly (since there is no bankruptcy) you may spend most of your time with 0 or 1 gaming chip, and occasionally have a lot of gaming chips for a short period. Therefore, the story is that if you do it for a long time, it becomes fair (Table 2.2).

Figure 2.1 

Before and after exchange: A: Distribution of the results in Table 2.1. B: Normal distribution and exponential distribution. The horizontal axis is the number of gaming chips held and the vertical axis is the number of people with the corresponding number of gaming chips.

Table 2.2  Dice and Gaming chips: Interpretation of results^2.1

· First, the gaming chips are distributed randomly from a source (the distribution is almost uniform)

· When the exchange is started, the state of distribution changes completely. Many persons will get 0 or 1 gaming chips while just a few will have many chips.

· After the exchanges go on for a long time, everyone will experience all the cases equally; spending most of their time with 0, 1, .... and occasionally having most of many gaming chips for a short time.

· Although it seems to be unfair if we focus for each instance for a short period, it appears to be fair after a long time.

This “unfair” distribution represents the reality of the world. In a consumption tax system, the probability of payment at each exchange is equal for everyone, regardless of their being rich or poor, and so the distribution looks like this (Fig. 2.1A Black bars). Thus, even though the average should be 5, most people spend the majority of their time with just 0 or 1, and the duration for which they are above the average is approximately only one-third of their life. In terms of time, it is interesting to note that the time spent below the average is significantly longer than the length of time above it, despite the “fair” nature of the exchange process. Instead, people only become rich occasionally. Since it is a matter of time, if they live longer, i.e. if they can exchange gaming chips for an infinitely long time, then they can certainly become rich once. However, in reality this is not the case, as time is limited.

When you study this concept in statistical mechanics textbooks and classrooms at the university level, you find that this distribution eventually becomes an exponential distribution after a long enough time (Fig. 2.1B Black bars). In the exponential distribution, the probability of 0 or 1 is high, whilst the probability keeps decreasing for a higher number of held gaming chips increases. The probability of being equal to or above the average is approximately one-third, which can be calculated from theory^2.2. Please attempt to calculate this on your own.

The results shown are from the games that were stopped too early, therefore, I was unable to obtain sufficient data. However, I really like it this way. Sometimes, I play the roles of all the six individuals by myself.

2-2: Increasing the number of exchanges

Here I will show results obtained from a different occasion. These are the results that I prepared by myself, rolling the dice to exchange chips between 6 boxes (corresponding to 6 people) (Table 2.3). A slightly more formal way of stating this is that there are N boxes and M gaming chips (so here N represents 6 people). In all I prepared 18 gaming chips (M=18) so that the average is 3 each. The dice was rolled and the exchange reaction was performed 350 times. The results were recorded at each stage (by me) and then compiled into a table. To summarize, A, B, C, D, E, and F performed the exchange 350 times, during which the number of times they respectively existed with 0 gaming chips was 18 times, 77 times, 82 times, 91 times, 31 times, and 50 times. Although you may live a busy life, please conduct this experiment and experience it by yourself. It is quite difficult to learn only by reading books and listening to other people’s talks. One learns differently (and often much better) by conducting an experiment, so I request that all the readers should roll the dice.

Table 2.3  Dice and Gaming chips: An example of 6 persons exchanging an average of 3 gaming chips per person

Exchange M gaming chips amongst N boxes (persons)
N=6, M=18 (Average number of chips per person is 3, Number of trials is approximately 350)
Persons	A	B	C	D	E	F
Number of times of being 0 chip	18	77	82	91	31	50
Number of times of being top	160	0	6	15	109	38

Examining Table 2.3, on average, all 6 people have experienced the situation of having zero gaming chips. We see that, while A was on the top for as many as 160 times, B was never on top. Indeed, the number of times of being at the top seems to be very uneven. Since 350 times was a relatively short time, attainment of the top position was not “averaged out” amongst the six people. If I want to give everyone the opportunity of being on the top, then I must exchange gaming chips more times. Being a 0 can be easily experienced on average but being the top can take more time!

As mentioned earlier, this exchange of gaming chips represents molecules colliding with each other to undergo energy exchange. Amongst any group of molecules that exchange energy, even when the average energy of the system is quite high, each molecule will tend to experience a low energy state, i.e. a 0 or 1, for a relatively long time. In other words, even if the temperature is quite high^2.3, the time duration for which the energy of each molecule is 0, 1 or, 2 will be relatively great. Although some molecules will have high energy for an “instant”, such a high energy state will occur relatively rarely, thereby requiring a long waiting time for its occurrence. In the world of molecules, since energy is exchanged at a tremendous rate, they will experience the opportunity to have high energy in a much shorter time compared to the time scales of our game. Nevertheless, the take home message is that having high energy is a rare phenomenon for molecules in comparison to their most probable, much lower energy states. I will talk about this later in detail. The probability of receiving the transition (activation) energy, E, that is required to enter the transition state in a chemical reaction is calculated by

  

$$\left(\mathrm{T}\mathrm{h}\mathrm{e}\mathrm{ }\mathrm{p}\mathrm{r}\mathrm{o}\mathrm{b}\mathrm{a}\mathrm{b}\mathrm{i}\mathrm{l}\mathrm{i}\mathrm{t}\mathrm{y}\mathrm{ }\mathrm{o}\mathrm{f}\mathrm{ }\mathrm{r}\mathrm{e}\mathrm{c}\mathrm{e}\mathrm{i}\mathrm{v}\mathrm{i}\mathrm{n}\mathrm{g}\mathrm{ }\mathrm{t}\mathrm{h}\mathrm{e}\mathrm{ }\mathrm{e}\mathrm{n}\mathrm{e}\mathrm{r}\mathrm{g}\mathrm{y}\mathrm{ }E\mathrm{ }\mathrm{r}\mathrm{e}\mathrm{q}\mathrm{u}\mathrm{i}\mathrm{r}\mathrm{e}\mathrm{d}\mathrm{ }\mathrm{t}\mathrm{o}\mathrm{ }\mathrm{e}\mathrm{n}\mathrm{t}\mathrm{e}\mathrm{r}\mathrm{ }\mathrm{t}\mathrm{h}\mathrm{e}\mathrm{ }\mathrm{t}\mathrm{r}\mathrm{a}\mathrm{n}\mathrm{s}\mathrm{i}\mathrm{t}\mathrm{i}\mathrm{o}\mathrm{n}\mathrm{ }{\mathrm{s}\mathrm{t}\mathrm{a}\mathrm{t}\mathrm{e}}^{2.4}\right)\mathrm{ }\propto \mathrm{ }\mathrm{e}\mathrm{x}\mathrm{p}\mathrm{ }(-E/RT)$$(Eq. 2.1)

Thus, receiving a high transition (activation) energy means that a rare event has occurred. This kind of description is used because attaining a high energy state rarely occurs whereas low energy states, by comparison, occur all the time.

I will show you another example. N is 6 and the total amount of gaming chips M is 24, i.e., the average number of gaming chips each person will have is 4 (Table 2.4). The result looks the same: the number of times of becoming 0 is averaged for everyone, which is similar. However, in terms of the number of times of being at the top, B is overwhelmingly the leader. D and F are also, comparatively, in a good position.

Table 2.4  Dice and Gaming chips: An example of 6 persons exchanging an average of 4 gaming chips per person

Exchange M gaming chips by N boxes (persons)
N=6, M=24 (Average number of chips per person is 4, Number of trials is approximately 350)
Persons	A	B	C	D	E	F
Number of times of being 0 chip	72	21	37	46	50	4
Number of times of being top	1	184	0	85	17	40

Next, I increased the total number of gaming chips M to 36, which with N still fixed at 6 sets the average number of gaming chips per person to 6. That is, I increased the temperature, or rather, the average energy of the system (Table 2.5). Still, everyone experienced 0. The point is that all low energies are experienced by everyone relatively equally. However, with respect to those with high energy, i.e., 18 gaming chips or more, there is a clear distinction between the number of molecules (persons) that have experienced it versus the number of molecules (persons) that haven’t. Again, if you run the game for a sufficiently long time, everyone will experience a high energy state equally. However, within the given relatively short number of times that this game was run, the results seem again to not be “fair”.

First, we distributed the gaming chips, that is, we received the energy that flowed from one large source. Next, we exchanged the energy. Since each person’s probabilities of giving and receiving are the same during each exchange, the exchange may be described as a diffusion process. Another way of saying this is that exchange occurs via Brownian motion^2.5. I will discuss Brownian motion later (in Chapter 8). When the current number of gaming chips held by an individual person is on average, n, the number of times required to change the number to m is (n–m)². If the number of gaming chips held by a person is the expected average i.e. n=M/N=6, for 6 to become 0, this requires (6–0)²=36 interactions, during which 6 may become 0. For each of the six people to experience holding zero gaming chips, requires approximately a further six times greater number of exchange events. However, those with larger number of gaming chips are much further away from the average. If you want to go up to 15, it will be 9 more than the average of 6, and the square of (15–6) is 81, which will be overwhelmingly time-consuming. Since the change in situation happens in proportion to the second power of the situation differential, a rich state is very rare. This is the reality of the world.

Table 2.5  Dice and Gaming chips: An example of 6 persons exchanging an average of 6 gaming chips per person

Exchange M gaming chips by N persons
N=6, M=36 (Average number of chips per person is 6, Number of trials is approximately 500)
Persons	A	B	C	D	E	F
Number of times of being 0 chip	66	56	82	14	23	96
Number of times with chips ≧15	58	94	0	70	2	0
Number of times with chips ≧18	15	19	0	0	0	0

In this section, we assume that both the poor and rich have the same probability of giving a chip (tax payment) during the exchange, regardless of how much money they have (consumption tax type model). Assuming that what they obtain and what they give are equal, the result will become an exponential distribution. This is the reality of molecules, rather than the reality of humans. If we change the rule to “give the gaming chips in proportion to the number of gaming chips you have”, it becomes an income tax type of model. If we do this, the result of gaming chip distribution will change completely, which we will see later (Section 3-2).

2-3: When interacting with each other, the diversity is increased

Here I will make an analogy between this game and education. The first point I would like to make is that when energy packets are distributed randomly from a big source to the players, the resulting diversity of the distributed packets becomes small. The second point is that when the energy packets are exchanged between the players, the diversity is increased (Table 2.6).

Table 2.6  Instructive message and lesson from the dice and chips game

· Random distribution from a big source decreases the diversity

· Exchange between the players increases the diversity

When I lecture on this topic at art colleges, I explain it in the following manner. If you learn unilaterally from the same teacher, everyone will tend to become the same. The works of art will be similar. If there is a talented teacher in the design department, the art works of the students at that department will all appear very similar to the untrained eye. However, when all the students interact with each other and have discussions, they begin to generate dissimilarity. These are important points. From an educational point of view, people are often reluctant to discuss with each other, because they believe that discussion leads to people having the same opinion. However, I believe that the opposite is true. When we have discussions with many people, the differences are emphasized. Although this is a “human story” and different from the exchange of gaming chips, I think the analogy between them is plausible. When people interact, the differences between them become evident. Students tend to feel that a situation is unfair when one person is lucky and the others are not. So I explain as follows. If they participate in the game once with red gaming chips, a person will emerge who has a lot of red gaming chips, while the others will have less. However, if you play once more amongst the same people with blue gaming chips, a different person is likely to get a lot more of the blue gaming chips. In these situations everyone seems to be convinced by my explanation. This implies that each person has a different probability to get “lucky.”

A research group cultured bacteria that had the same DNA, and they found that when cultured for several weeks, differences in the cloned bacteria became apparent. Instead of increasing the similarity in the amount of purified enzyme, the differences were greatly increased [2.1][2.2]. Therefore, perhaps, the bacteria are themselves exchanging something. In general, when interactions take place, the differences become emphasized.

Let us once again return to the important points of gaming chips and dice games, placing our focus on the exchange of energy. As individuals interact, the system moves between different distribution states. If it is possible to move from one distribution state to another, the reverse is also possible with equal probability. Over a long time, all the distribution states are realized with equal probability. In terms of statistical mechanics, the distribution is in a microscopic state. Thus, assuming that going and returning have the same probability, the microscopic states are realized with equal probability. In statistical mechanics, this is called the “Principle of equal a priori probability”.

2-4: Writing down all the distribution methods

Once again, I would like the readers to perform some manual tasks. As the concept of microscopic states with equal probability (principle of equal a priori probability) seems quite natural, you might be tempted to think that there is no need to try this exercise. However, since concepts can be best illuminated by experience, I request that you please try drawing the distributions of three gaming chips in three boxes on a paper (Exercise 2–Fig. 2.2A). Three gaming chips in three boxes represent a unique case that can be drawn in a plane, that is, it is the only case you can draw in two dimensions. If you have 4 gaming chips in 4 boxes, you will not be able to draw on a planar surface without crossing lines, and you may even have to transform it into three dimensions. Please consider various ways of describing the distribution and draw it by yourself. Experience is important.

Exercise 2: Write down all the distributions when exchanging three gaming chips in three boxes or players (when the probability of win or lose gaming chips for each box is equal)

Figure 2.2 

A: Diagram of all the cases of distribution of three chips amongst three persons A, B, and C (please fill out the missing states). B: Shows all the possible transitions and their probabilities theoretically available to the state where A has 3 chips, while B and C have 0 chips. i: (A gives 1 gaming chip to B)=(moves to lower left), j: (B gives 1 gaming chip to A)=(moves to upper right), k: (B gives 1 gaming chip to C)=(moves to the right), l: (C gives 1 gaming chip to B)=(moves to the left), m: (C gives 1 gaming chip to A)=(moves to upper left), n: (A gives 1 gaming chip to C)=(lower right).

The Figure 2.2 refer to the case where three people (A, B, and C) exchange three gaming chips. Enter all possible numbers of gaming chips held by A, B and C in the unmarked space shown as ( , , ) in Fig. 2.2A. The state (3, 0, 0) corresponds to the situation where A has 3 gaming chips, B has 0 gaming chips and C has 0 gaming chips. In principle, during the exchange process, each person can either give or receive a chip. For a three-person system (A, B, C) this means each state has up to six possible exchange transitions available to it. For example, when A gives a gaming chip to B, this corresponds to moving to the state of (2, 1, 0) in the lower left of the diagram of Fig. 2.2B. To better understand both the number of states and the number of possible ways of forming each state via an exchange transition please work through the following six steps.

• 1.   Write down all the states in Fig. 2.2A
• 2.   All individual paths available to form a particular state (via exchange) are indicated in Fig. 2.2B. Although a person with 0 gaming chips cannot donate a chip (as there is no debt), and a person with three chips (the maximum in this game) cannot receive another chip, both transitions are counted as one trial and that route is counted as one potential method of formation by a curved returning arrow “↷”. In addition to these closed routes of exchange there are two real routes for the formation of (3, 0, 0) with these transitions shown by a black straight arrow, “→”, between (2, 1, 0) and (2, 0, 1). So, in all there are 4 closed and two open routes of exchange transition (gray curved arrows) in Fig. 2.2B). Thus, altogether there are 6 theoretical ways to form the (3, 0, 0) state.
• 3.   Count the number of routes for the exchange of each state, z(A, B, C). Add these numbers up to determine the total number of routes, Ztot, available in the formation of all states in the system (in this case Ztot=10×6=60). Now estimate the probability, P(A, B, C), of each particular state existing which is the number of ways it can be formed (via exchange transition) divided by the total number of states available to the system i.e. P(A, B, C)=z(A, B, C)/Ztot. Using this approach calculate the probability, p, of player A having 3 gaming chips p=P(3, 0, 0), the probability, q, of player A having 2 gaming chips and player B having 1 gaming chip q=P(2, 1, 0), and the probability, r, of each one having one gaming chip each r=P(1, 1, 1). For this system there are 10 available states. We can evaluate the probabilities of the remaining seven states individually using the P(A, B, C)=z(A, B, C)/Ztot trick.
• 4.   Determine the total probability of person C having either 0, 1, 2, or 3 gaming chips.
• 5.   Consider an alternative condition in which debt is not allowable and the trial is not counted. And redo the steps 1 to 4 above.
• 6.   Try doing 1) to 4) above for the case of 4 gaming chips in 4 boxes.

The case of three gaming chips in three boxes can be drawn neatly on a flat surface. At one of my lectures, I asked the participants to write the case of four gaming chips in four boxes as homework. Although researchers don’t do this kind of homework, a participant appeared two weeks later, and they had drawn the answer on paper. It was a very difficult diagram. In the case of four gaming chips in four boxes, one has to create a tetrahedron. It can be readily drawn in 3D. However, because most people cannot come up with this solution, they attempt to draw it in 2D, which is difficult. After two weeks, the participant handed over the drawing to me and said, “I have been discussing this with my family every night when we had dinner, and I drew it”. I was very much impressed and issued a certificate of commendation at the end of my lecture saying “To so-and-so, for scoring an excellent grade”. It is a nostalgic story. If you are free, you may find it fun to similarly create a diagram of four gaming chips in four boxes in 3D. Even for this case, you will still be able to draw an exchange network.

Figure 2.3 

Diagram of all the cases of distributing 3 gaming chips to 3 boxes: The arrow indicates the transition due to the exchange (Answers to Exercise 2).

Now let’s return to the main subject. In the case just considered of three gaming chips in three boxes, the exercises considered two situations where debt either was, or wasn’t, allowed. Whether debt was or wasn’t allowed, each state can be formed in six ways and thus the probability of realizing each state will be equal (3 of Exercise 2 for the no debt allowed case) and the ratio of probabilities in the case of person C having either 0, 1, 2, or 3 gaming chips would be 4, 3, 2, and 1 (4 of Exercise 2, Fig. 2.4A-no debt allowed situation). As such, the probability of possessing a large number of gaming chips decreases significantly. Although it may be obvious, please try writing the diagram by hand. Also, I would like everyone to come up with good ideas for drawing four gaming chips in four boxes. Unfortunately, the case of three gaming chips in three boxes might be the most straightforward with respect to drawing on a flat sheet of paper. As the number of gaming chips and boxes grows, the diagram must be drawn by increasing the dimensions in various ways.

Figure 2.4 

A: Distribution of 3 gaming chips held by person C in 3 boxes: The horizontal axis represents the number of gaming chips in a box of interest, and the vertical axis represents the number of possible cases shown in Fig. 2.3 (debt allowable case). B: What will be the state of distribution of 4 chips in 4 boxes?

2-5: Computer simulations

Using a computer, a participant of my lecture attempted to simulate the dice and gaming chips experiment we played earlier. The results will be presented shortly. This sort of thing often happens when I give a lecture. Some students will instantly create a program on a computer and show it to me. Although a computer is a useful tool, it is more exciting and more intuitive to play the game manually. I would suggest that a deep realization that the number of times of obtaining 0 gaming chips is almost equal for everyone, and that becoming rich is rare event, is best achieved by manual rolling of the dice rather than by visual inspection of the output of a computer program. Even rolling the dice a thousand times does not require much time. To learn statistical mechanics, or maybe physics in general, instead of working by just thinking, it is better to put into practice. Although it is difficult to study Maxwell’s equation through dice rolling, I believe that statistical mechanics can be well-understood intuitively by such means. Therefore, roll the dice by hand, use your head, and create your drawings. Although such an exercise takes time, I want suggest all readers to roll the dice using their hands.

However, with this said let me explain a little about the computer programs^2.6. The first program divides the four gaming chips among four persons, and counts all patterns of how many gaming chips the persons A, B, C, and D will have, and how many times each pattern will appear during the exchange (Fig. 2.5). For the moment, let’s attempt this using 100 exchanges, 1,000 exchanges and 10,000 exchanges. For the initial distribution, we put one gaming chip in each of the four boxes and then begin the exchange process. The number of times that the gaming chips are held by A, B, C, and D is respectively 0, 1, 2, 3, and 4 (as written on each figure). Furthermore, all the possible distribution combinations among A, B, C, and D (microscopic state^2.7) are listed horizontally, and the numbers of observed microscopic states are accumulated vertically in the lower figures. Ideally, the numbers in the upper figures must decrease towards the right and the numbers of all the distributions in the lower figure must be even. Although the distribution of microscopic states in the figure below is not so equal for 100 times and 1,000 times, the distribution trends of the number of times each person experiences each number of gaming chips (upper figure) are quite similar (Fig. 2.5A). When this becomes 10,000 times, the distribution of the microscopic state in the figure below also becomes smooth (Fig. 2.5C). However, same as we did before, this simulation obeys the rule that when a person with 0 gaming chips receives a command to give gaming chips, they ’cannot do so (as there is no debt), but the number of trials is incremented (Fig. 2.5 A, B, and C).

Next, we change the rule to the following: when a person with 0 gaming chips receives a command to give gaming chips, it is not counted as a trial, that is, there was no exchange and no increase in the number of trials (Fig. 2.5 D, E, and F). Except for the change in th is rule for the person with 0 gaming chips, the other rules remain the same. It takes longer to equalize the distribution of the number of microscopic states. Since the distribution of the number of times each person has each number of gaming chips is almost the same for 0 and 1 gaming chip, it is not like the exponential distribution (Fig. 2.5F Upper part). Because all the microscopic states do not have equal probability and the probabilities of giving and taking are not equal (5 of Exercise 2), such a difference appears.

Figure 2.5 

4 people exchanged 4 gaming chips^2.6: The distribution of the number of gaming chips for each person (top of each figure) and the distribution of how many gaming chips each person holds (bottom of each figure) is shown in the case that 4 persons exchanged 4 gaming chips. The number of trials is 100 (A, D), 1,000 (B, E), 10,000 (C, F). The rules are as follows.

·　Distribute 4 gaming chips in 4 boxes

·　Even if there are persons with 0 gaming chips, continue as is (if the person gets the dice roll to receive gaming chips, he can receives)

·　When the persons with 0 gaming chips gets the dice roll to give gaming chips, they don’t give the gaming chip. This case is counted as a trial in A, B, and C and it is not counted as a trial in D, E, and F.

The following program^2.6 has a total population of 100 people and a total of 2,000 gaming chips and it performs exchange on a one-to-one basis as we did previously (Fig. 2.6). First, the gaming chips are distributed randomly (Fig. 2.6A). In the figure on the left, the horizontal axis shows all 100 people, and the vertical axis shows the number of gaming chips each person has. In the first random distribution, it appears to be close to a normal distribution with sharp peaks, as shown on the lower panel of Fig. 2.6A. With everything set, let us start the exchange. This time, since we have 100 people and a total of 2,000 gaming chips, we perform the trial 50,000 times^2.8. Exactly at 50,000, a little less than 20 people have 0 gaming chips (Fig. 2.6 C lower panel). Although there are several people with 80 gaming chips or more, most people have 0 to 3 gaming chips. Since this program calculates each time separately with different initial distribution states, the answer will be slightly different in every simulation.

Figure 2.6 

2,000 gaming chips in 100 boxes^2.6: Distribution before exchange (A), after exchanging 5,000 times (B), and after exchanging 50,000 times (C). The horizontal axis in the upper panels represents each of the 100 persons, and the vertical axis represents the number of their gaming chips. Lower panels are snapshots of the distribution of the number of persons (vertical) with respective to the number of gaming chips (horizontal axis). The rules are as follows:

·　Distribute 2,000 gaming chips in 100 boxes

·　Even if there are persons with 0 gaming chips, continue as is (if the person gets the dice roll to receive gaming chips, he receives)

·　When the persons with 0 gaming chips get the dice roll to give gaming chips, they don’t give the gaming chip but it is counted as a trial

·　The probability to get the dice roll is equal to all the boxes (consumption tax type).

2-6: It’s because of the others that I have less gaming chips

I will provide a slightly more formal (yet approximate) explanation of the appearance of these “unfair” distributions that you have observed in the dice and chip game. If we were to consider a large number of N boxes and divide them into two groups consisting of one particular box and all the other (N–1) boxes combined (which we will call the environment). In distributing the M number of gaming chips amongst all boxes we intuitively realize that the likelihood of observing chips appearing in one’s particular box depends heavily on the number of other boxes (for the case where all particular distributions can be equally realized when all microscopic states can be accessed with equal probability). For example, if you possess 3 of M gaming chips, the number of ways to distribute the remaining M–3 gaming chips to the remaining N–1 boxes will increase only when the number 3 is reduced, that is, when more gaming chips are left for the other persons. The less a person has, the more possible states will exist for others. Hence, only when you have less, will the overall number of states increase (total states for you and others). Thus, the reason why the probabilities of 0 and 1 are higher is due to the number of others (the environment) (Fig. 2.7).

Figure 2.7 

Due to the others (environment), most of the time my chips are less!! The fewer the number of gaming chips (energy) one has, the more gaming chips are distributed to the others (environment), and the greater the number of ways to distribute them (number of cases)^2.9.

Readers with formal training in statistical mechanics will know that this is a canonical distribution^2.10. When there is a certain number of particles, if we consider one of them, the probability that this particle has an energy ε is given by the Boltzmann distribution formula written as

  

$$\left(\mathrm{P}\mathrm{r}\mathrm{o}\mathrm{b}\mathrm{a}\mathrm{b}\mathrm{i}\mathrm{l}\mathrm{i}\mathrm{t}\mathrm{y}\mathrm{ }\mathrm{o}\mathrm{f}\mathrm{ }\mathrm{h}\mathrm{a}\mathrm{v}\mathrm{i}\mathrm{n}\mathrm{g}\mathrm{ }\mathrm{e}\mathrm{n}\mathrm{e}\mathrm{r}\mathrm{g}\mathrm{y}\mathrm{ }\epsilon \right)\mathrm{ }\propto \mathrm{ }\mathrm{e}\mathrm{x}\mathrm{p}\mathrm{ }(-\epsilon /k_{B}T)$$(Eq 2.2)

Here, k_B is the Boltzmann constant and T is the absolute temperature. ε corresponds to the number of gaming chips. When ε increases, the probability decreases in a geometric progression. That is, the smaller ε, the higher the probability. The main point is that the probability of having energy ε (called Boltzmann factor) arises not because of one’s fault, but due to the contribution of the others i.e. this kind of distribution is due to the presence of others. The most important point in the origin of the Boltzmann factor is that one refrains from improving oneself to improve others.

I forgot to mention something important. When we exchanged gaming chips in earlier sections, six people performed the exchange directly. This time, pile up the gaming chips in the middle of the six people, roll the dice, and if it hits one, the first person must give out one of his gaming chips to the pile of gaming chips in the middle. Roll the dice again, and the next person whose number hits on the dice takes a gaming chip from the pile of gaming chips in the middle. Even in this case, the result is the same. Hence, instead of interacting with your neighboring opponents, even if you interact with the pile of gaming chips, i.e., a “store” of energy (energy bath) in the middle, the result is the same. If the time-averaged number of exchanges is the same, even if it is not necessarily “give-take, give-take,” but rather “give-give, take-take,” the results are the same. This is an important point. This implies that the exponential distribution and the downward sloping distribution indicate that the other person represents the “store”. Just consideration of a “store” is acceptable, without necessarily having knowledge about your neighboring opponents. The original idea of Gibbs’s canonical ensemble is that when you are exchanging energy with your opponent via a “store”, your energy will be described by a Boltzmann distribution. It would be ideal if we could deal with the “store” rather than directly with each other. The idea of statistical mechanics can account for the distribution even if the five out of six people other than me are not real but five virtual people.

When we learn from a statistical mechanics textbook, we tend to develop examples by distributing a very large number of M gaming chips to a very large number of N boxes, and the question will be related to the probability that a particular box has m number of gaming chips, for which we calculate the number of distributions. When you receive m gaming chips, only M–m gaming chips will be distributed to the other boxes. Calculating the number for each distribution, if M and N are sufficiently large, and if m is not just 1, 2, or 3, but a sufficiently large number, it can be proven that it approximates a geometric series of mth power. The geometric series is the same as the exponential function. This is how we obtain Eq. 2.3, and this is what is written in typical textbooks.

  

$$\mathrm{P}\mathrm{r}\mathrm{o}\mathrm{b}\mathrm{a}\mathrm{b}\mathrm{i}\mathrm{l}\mathrm{i}\mathrm{t}\mathrm{y}\mathrm{ }\mathrm{t}\mathrm{h}\mathrm{a}\mathrm{t}\mathrm{ }\mathrm{t}\mathrm{h}\mathrm{e}\mathrm{ }\mathrm{n}\mathrm{u}\mathrm{m}\mathrm{b}\mathrm{e}\mathrm{r}\mathrm{ }\mathrm{o}\mathrm{f}\mathrm{ }\mathrm{g}\mathrm{a}\mathrm{m}\mathrm{i}\mathrm{n}\mathrm{g}\mathrm{ }\mathrm{c}\mathrm{h}\mathrm{i}\mathrm{p}\mathrm{s}\mathrm{ }\mathrm{i}\mathrm{n}\mathrm{ }\mathrm{a}\mathrm{ }\mathrm{b}\mathrm{o}\mathrm{x}\mathrm{ }\mathrm{i}\mathrm{s}\mathrm{ }\mathrm{m}:\mathrm{ }P\left(m\right)\mathrm{ }\propto \mathrm{ }{\mathrm{\lambda }}^{\mathrm{m}}\mathrm{ }(\mathrm{\lambda }<1)$$(Eq. 2.3)

where λ is the ratio of decrease of the probability when m increases by 1.

However, what I’ve shown you here is that even if it is only three gaming chips in three boxes, the distribution of gaming chips is 4, 3, 2, and 1, from the left. Another important point is that a decline towards the right is sufficiently achieved by only a few numbers. If there are five molecules, it is worthwhile to consider that the interaction between them gives a distribution that can be approximated by an exponential function (since exponential functions are mathematically simpler), or expressed by a geometric series (Figure 2.8). Similar to the molecular machine mentioned in the introduction, even if there are only a few degrees of freedom and very few molecules, we may want to apply the Boltzmann probability (exponential distribution). At that time, this point becomes very important.

Figure 2.8 

Geometric series distribution: When M gaming chips are distributed to N boxes and exchanges are repeated, the probability (proportion of time) that the number of gaming chips in a box is m is given by P(m) ∝ λ^m (λ<1) (Eq. 2.3), which decreases in a geometric series. The probability decreases with ratio λ as m increases by 1, and the ratio approaches to 1 as the average number of gaming chips per box increases.

Although λ in Eq. 2.3 (the probability P(m) of the decrease in the number of gaming chips m in a box with each increment of m) is smaller than 1, as the average number of gaming chips per box becomes larger, λ becomes closer to 1. Therefore, λ^m in Eq. 2.3 becomes similar to the middle of Eq. 2.4.

  

$${\mathrm{\lambda }}^m\mathrm{ }~\mathrm{ }\mathrm{e}\mathrm{x}\mathrm{p}(-m/<m>)\mathrm{ }~\mathrm{ }\mathrm{e}\mathrm{x}\mathrm{p}(-\epsilon /k_{B}T)$$(Eq. 2.4)

Here, <m> corresponds to the average number of gaming chips per box, which is the average energy, that is, the temperature. Therefore, the formulas for the canonical distribution and the Boltzmann distribution are as shown on the right of Eq. 2.4 (Eq. 2.2).

One of the key points is, in a small system, for example, even if there are four or five degrees of freedom, the probability of the energy per degree of freedom, can be described by Eq. 2.4. Another point is that you can consider any box. It does not have to be the box of the same companion. In summary, since Eq. 2.4 is determined by the surrounding boxes, it does not matter even if the box of interest is unique. Since all that is required is to exchange its energy, and the probability will be determined by the surroundings, a Boltzmann distribution implies that a box of interest can be unique (Table 2.7).

Table 2.7  Probability that the number of the gaming chips (energy) in a box (molecule) is m (ε)

The probability that the number of the gaming chips in a box is m, or that a molecule has energy ε.

${\mathrm{\lambda }}^m\mathrm{ }~\mathrm{ }\mathrm{e}\mathrm{x}\mathrm{p}(-m/<m>)\mathrm{ }~\mathrm{ }\mathrm{e}\mathrm{x}\mathrm{p}(-\epsilon /k_{B}T)$   (Eq. 2.4)

Here, <m> is the average number of the gaming chips per box (average energy of each molecule=temperature).

ε is the energy of the molecule, kB is the Boltzmann constant, and T is the absolute temperature.

· Small system: The formula approximately holds if the number of boxes (molecules / degrees of freedom) is from 4 to 5.

· Box of interest: Any box can be chosen. The preceding formula is determined based on the surrounding boxes (environment).

For entertainment, I searched for the billionaire’s list, thinking that it might be an exponential distribution. Although the portions of high-income people and the low-income people were slightly out of fit, the distribution of middle-income people had a shape that was close to a Boltzmann distribution (Fig. 2.9). In the real world, since we are not simply exchanging, there will be such a deviation, but overall, it is quite similar to a Boltzmann distribution. For both distributions, the number of people with income above the average is much smaller than half of the total.

Figure 2.9 

Comparison of income distribution of 2004 in Japan and Boltzmann distribution: The distribution is close to the Boltzmann distribution except for the high-income and low-income region. The percentage (40.3%) of people with incomes above the average income of all households (5,797,000 yen) is less than half of the total. (Created based on Fig. 8 “Relative frequency distribution of the number of households by income level” of the “Ministry of Health, Labor and Welfare: Overview of 2004 Comprehensive Survey of Living Conditions” https://www.mhlw.go.jp/toukei/saikin/hw/k-tyosa/k-tyosa04/2-1.html).

Coffee break

It is generally known that if you keep many frogs in a box, some frogs will grow fatter and some will become thinner. The difference increases rapidly. Carefully controlled experiments were conducted to determine how individual difference emerges within a genetically identical group, and it was concluded that the difference in the body size increased [2.3][2.4][2.5]. This phenomenon is known as the density effect, and many researchers are investigating this subject. In accounting for the increase in the difference, the most popular explanation is as follows; once the frogs start to gain weight, they become increasingly powerful and eat more food. By contrast, once other frogs start getting leaner, they become weaker and cannot obtain food. However, even if this is not the case, the difference is observed. So, I interpret that if they just swim together, live together, and interact with each other, naturally, some will become fat and some will become thin by chance. Irrespective of the mechanism or the reason why they become big or small, just by living together and interacting with each other is sufficient to generate diversity naturally. This interpretation is reasonable, isn’t it?

Acknowledgements

We thank D. Hall for providing an additional check of exposition within our translated manuscript.

Notes

2.1 Translator’s note: We combined the original Table 2.2 and 2.7 to make Table 2.2 and delete the original Table 2.7 because these tables were almost the same.

2.2 Original note: To be exact, the probability is 1/e.

2.3 Original note: A high temperature corresponds to a high overall average energy.

2.4 Original note: Arrhenius equation: T is the absolute temperature, R is the gas constant, and ∝ indicates proportionality relation.

2.5 Original note: Brownian motion is named after Robert Brown. He determined that the motion of tiny particles from the pollen grains in fluid (observed under a microscope) was driven by the environment. We now know that this motion is due to random collisions of the solvent molecules with the pollen grains.

2.6 Original note: Shoichi Toyabe (Tohoku Univ.) made the program, although the original programs were made by another person. http://bpwakate.net/Oosawa/simulator.html

2.7 Original note: Distribution method describing which person has gaming chips and how many gaming chips do they have. For example, the leftmost figure in the lower part of A becomes 4, 0, 0, 0 from the top, which means that person “A” has 4 gaming chips and the other three have 0 gaming chips. In statistical mechanics, this corresponds to “How much energy is stored in each molecule?” The detail about each molecular state is called a “microscopic state”. On the contrary, “macroscopic state” is a characteristic of overall state of a system such as average energy, temperature, and so on.

2.8 Original note: Based on the discussion so far, after 400 trials, that is the average value of 20 squared, there are some persons who have 0 gaming chips. All the 100 persons experience having 0 gaming chips once after approximately 40,000 trials, i.e., 100 persons times 400 trials.

2.9 Translator’s note: We have added a diagram at the close of this chapter to more fully explain this point.

2.10 Original note: Canonical distribution: One of the probability distributions to describe microstates in statistical mechanics. The probability distribution describes the distribution of particles or systems with the energy ε equilibrated under exchange of energy with energy storage (heat bath). Canon comes from the Greek word “kanon”, which means laws, rules, and standards, and the same applies to canon in the music format. It is also called the Boltzmann distribution.

References

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