Appendix

Appendix A: How to catch a school of fish in the sea in a single throw

This section explains the derivation of Eq. 8.6, Eq. 8.7, and Eq. 8.8 [A.1].

In case of one-dimension

Take three points $r$, $\xi$, and $R$ in one-dimensional space such that $r<\xi <R$. Let’s assume that $\xi$ is the point from where the particle emerges (source point), and $r$ and $R$ are points where the particle is absorbed (absorption points). Then, the stationary distribution of particle density $C(x,t)$ under boundary conditions

  

$$C(r,t)=C(R,t)=0,C(\xi ,t)=C_0$$(Eq.A.1)

is given as a steady state solution of Fick’s law

  

$$\frac{\partial C}{\partial t}=D\mathrm{\Delta }C,$$(Eq.A.2)

where $D$ is the diffusion coefficient. Solving this, we get,

  

$$C(x,t)=\frac{C_0}{\xi -r}(x-r)$$(Eq.A.3.1)

for $r<x<\xi$, and

  

$$C(x,t)=\frac{C_0}{\xi -R}(x-R)$$(Eq.A.3.2)

for $\xi <x<R$.

Now, consider the diffusion flow $J$ from the source point $\xi$. For simplicity, let us consider the flow per unit area. Since it is given by

  

$$J=-D\nabla C,$$(Eq.A.4)

with Eq. A.3, we obtain

  

$$J=-D\frac{C_0}{\xi -r}$$(Eq.A.5.1)

i.e. $\left|J\right|=D\frac{C_0}{\xi -r}$ in the direction of $x<0$ when $r<x<\xi$, and

  

$$J=-D\frac{C_0}{\xi -R}$$(Eq.A.5.2)

i.e. $\left|J\right|=D\frac{C_0}{R-\xi }$ in the direction of $x>0$ when $\xi <x<R$.

That is, at the source point $\xi$,

· Diffusion flow to the inner absorption point $r$ is $J_{in}=\frac{{DC}_0}{\xi -r}$

· Diffusion flow to the outer absorption point $R$ is $J_{out}=\frac{{DC}_0}{R-\xi }$

Therefore, the probability that a particle leaving the source point ξ is absorbed at the inner absorption point r is given by

  

$$\frac{J_{\mathrm{in}}}{J_{\mathrm{in}}+J_{\mathrm{out}}}=\frac{\frac{1}{(\xi -r)}}{\left\{\frac{1}{(\xi -r)}\right\}+\left\{\frac{1}{(R-\xi )}\right\}}=\frac{R-\xi }{R-r}.$$(Eq.A.6)

Since

  

$$\underset{R\to \infty }{lim}\frac{J_{\mathrm{in}}}{J_{\mathrm{in}}+J_{\mathrm{out}}}=\underset{R\to \infty }{lim}\frac{R-\xi }{R-r}=\underset{R\to \infty }{lim}\frac{1-\frac{\xi }{R}}{1-\frac{r}{R}}=1$$(Eq.A.7)

in the limit of $R\to \infty$, all the particles are absorbed into the inner absorption point $r$.

In the case of two-dimensions and three-dimensions

Similarly, in the case of two- and three-dimensions, solving $\frac{\partial C}{\partial t}=D\mathrm{\Delta }C=0$ under the boundary conditions $C(r,t)=C(R,t)=0$ and $C(\xi ,t)=C_0$, we obtain the stationary distribution of particle densities, and also the diffusion flow $J=-D\nabla C$.

Hint: The Laplacian in polar coordinates, considering symmetry, is given as follows.

Two-dimensional case: Since $\mathrm{\Delta }=\frac{{\partial }^2}{{\partial r}^2}+\frac{1}{r^2}\frac{{\partial }^2}{{\partial \theta }^2}+\frac{1}{r}\frac{\partial }{\partial r}$,

  

$$\mathrm{\Delta }=\frac{{\partial }^2}{{\partial r}^2}+\frac{1}{r}\frac{\partial }{\partial r}=\frac{1}{r}\frac{\partial }{\partial r}\left(r\frac{\partial }{\partial r}\right)$$

Three-dimensional case: Since $\mathrm{\Delta }=\frac{{\partial }^2}{{\partial r}^2}+\frac{1}{r^2}\frac{{\partial }^2}{{\partial \theta }^2}+\frac{1}{r^2{sin}^2\theta }\frac{{\partial }^2}{{\partial \phi }^2}+\frac{2}{r}\frac{\partial }{\partial r}+\frac{1}{r^{2}tan\theta }\frac{\partial }{\partial \theta }$,

  

$$\mathrm{\Delta }=\frac{1}{r^2}\frac{\partial }{\partial r}\left(r^2\frac{\partial }{\partial r}\right)$$

Appendix B: Weyl’s billiards

Ergodicity

Here, we will perform an exercise to experience the ergodicity through the problem called Weyl’s billiards [A.2]. Ergodicity is widely known as “the property of a (stochastic) process in which the average over time of a certain quantity matches with the average over phase space”^A.1.

When one wants to find the average value of the dice rolled many times, for example, the value obtained by a person rolling a dice 1,000 times and that obtained by 1,000 people rolling each dice at once will be the same.

Then, what conditions are necessary for the time average and the average over phase space to coincide? When considering a stochastic process, we know that after a large number of iterations of trials, regardless of the initial state, a finite number of all states should be reached, and there is no state that can be returned to with probability 1 after a certain period of time. In other words, there should be no states that are unreachable (or rarely reachable) because it goes back and forth between the same states.

Problem setting

Consider tracking a point $(x,y)$ defined by^A.2

  

$$\begin{array}{c}x\left(t\right)=e\times t,y\left(t\right)=\pi \times t.(e=2.72\dots ,\pi =3.14\dots )\hfill \\ x=[0,5),y=[0,5).t=0,1,2,3,\dots \hfill \end{array}$$(Eq.B.1)

and it will be reflected at the boundary. Here we manually calculate the coordinates for t=0, 1, …, 24, and observe their distribution in space ($e$ and $\pi$ are selected at random as examples of irrational numbers. The following calculation will give the same result up to t=24, even if the calculation is performed with at least 2 decimal places for $e$ and $\pi$.

Operating procedure

1. In Table B.1 Column A, calculate the value of $x=e\times t$ and $y=\pi \times t$ for each time ($t$) (Determine the coordinates that do not consider the boundary conditions).

2. In column B, enter a value that is less than or equal to the 1’s place of column A. (By ignoring the 10’s place, we create a periodic boundary condition of $x,y=[0,10)$.)

3. If $x$ or $y$ is 5 or more in column B, enter $9-x$ and $9-y$, respectively, in column C. (By doing this, the reflective boundary condition of $x,y=[0,5)$ will be realized).

4. Plot the values in Column C of Table B.1 on Fig. B.1.

5. In the delimited small squares in Fig. B.1, color the places where the points $(x,y)$ are present inside.

Table B.1 Exercise: Weyl’s billiards

t	A		B		C
	x	y	x	y	x	y
0	0	0	0	0	0	0
1	2.72	3.14	2.72	3.14	2.72	3.14
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24

Figure B.1  Grid for exercises: The space (squares) of x, y=0 to 5 is divided into meshes of 1 unit. The space is divided into 5×5=25 small squares.

Figure B.2  Answers for Weyl’s billiards exercise: Out of the 25 small squares, 20 are colored. Although this may be difficult to understand as the space and time are too small in this example, we can see that the points (x, y) are evenly distributed in the space.

Afterword

I wrote this book based on a videotape of my lecture at a summer school^A.3 of the Society of Young Scientists in Biophysics^A.4 in 1996. In the fall of 2009, and I published the predecessor of this book on the website of the Society of Young Scientists in Biophysics, which is limited to the members of the Biophysical Society of Japan. Since the web limited edition was well-received, I decided to thoroughly review the content and to publish it as a book from the University of Nagoya Press.

The editorial work for both the first web-only edition and this book was assisted by the volunteer members of the Society of Young Scientists in Biophysics. They assisted in changing the colloquial expressions into formal language, transforming the two-day seminar into a single flow, and redrawing all the figures etc. to produce a single book. The main assistants were Kiyoshi Ohnuma (Nagaoka University of Technology), Noritaka Masaki (Hamamatsu Medical University), Masako Ohtaki (formerly Waseda University), Taro Toyota (The University of Tokyo), and Masayo Inoue (Osaka University). I also received significant assistant from Takayuki Ariga (The University of Tokyo), Yasunobu Igarashi (Olympus Software Technology Co., Ltd.), Takehiko Inaba (RIKEN), Kei-ichi Okazaki (Waseda University), Akiko Kondow (Fujita Health University), Mieko Tamura (Nomura Research Institute), Yuichi Togashi (Kobe University), Shoichi Toyabe (Chuo University, Simulation program creationp^A.5), Rumi Negishi (Tokyo Institute of Technology), and Masaomi Hatakeyama (University of Zurich), in creating the web edition. I also received great support from Kazushi Tamura (formerly Hokkaido University), Rina Kagawa (Keio University), Keisuke Kamba (Nagoya University), Yohei Kondo (The University of Tokyo), Mayu Suzuki (Kyoto University), Mineyuki Tsuzuki (Nagoya University), Naoto Hori (Kyoto University), and Akihisa Yamamoto (Kyoto University) in re-editing the content to realize this book (information inside the parenthesis are as of May 2011). A lot of people spent a great deal of effort (physical effort, knowledge) and time in the process of finalizing this title.

I am very grateful to all for their assistance. I hope that “DIY Statistical Mechanics” will be widely received and that the reader will enjoy moving their hands, and at times, be impressed. I will be extremely happy if these two aspirations are achieved. There may be a few mistakes or omissions in a few instances. I would appreciate if you would contact me with corrections or supplements^{A.6, A.7}.

May 2011

Fumio OSAWA

Footnotes

A.1 Translator’s note: In simpler expression, “a long-time average of the statistical properties of a (stochastic) process can be represented by a collection of sufficiently large random samples from the process”.

A.2 Translator’s note: $e$ is the Napier’s constant.

A.3 Original note: This is a large-scale research exchange event held every summer for two nights and three days, where hundreds of members of the Society of Young Scientists in Biophysics^A.4 from all over Japan gather. Branches from all over Japan take turns overseeing the event. Students encourage research exchange by inviting lecturers for special talks, discussing their own research, and participating in a banquet.

A.4 Original note: An organization for young researchers such as graduate students and postdocs of the Biophysical Society of Japan. Usually, each branch has small-scale research societies and study groups ranging from a few members to tens of members. https://bpwakate.net/

A.5 Original note: It can be accessed from the website of simulators page in the Society of Young Scientists in Biophysics. https://bpwakate.net/Oosawa/simulator.html

A.6 Original note: “The Oosawa Lectures on DIY Statistical Mechanics” page in Website of the Society of Young Scientists in Biophysics (in Japanese) https://bpwakate.net/Oosawa/

A.7 Translator’s note: please contact: [e-mail]

References

• [A.1]   Berg,  H.C. Random walks in biology. Princeton University Press, Princeton (1993).
• [A.2]   Husimi,  K. Theory of Probability and statistics. Modern Engineering Publishing (Gendai Kogaku Sha), (1998). (in Japanese, *Title translated by editorial team)