Special Issue: The Oosawa Lectures on DIY Statistical Mechanics
Oosawa’s Preface
2021 年 18 巻 Supplemental 号 p. S003-S007
詳細
2021 年 18 巻 Supplemental 号 p. S003-S007
This book is intended to be an introduction to statistical mechanics. Although there are many introductory books and references on this subject, the policy of this book is different from that of other related works in that importance is placed on empirical examples and intuitive argument. It is the author’s hope that, through the use of this book, all readers will be able to learn statistical mechanics with a clear imagination of molecular behavior, allowing them to “enter the gate0.1”, so to speak.
When making arguments in statistical mechanics, we often imagine that numerous particles gather, with each capable of free motion. In comparison, in this book, we begin by considering just a small number of particles. We then attempt to describe the movement of each individual particle, as well as the collective movement of these (small number of) particles as a whole. For that purpose, we make frequent use of the behavior of “dice” to represent the individual movement of particles and “gaming chips” to represent the energy of these particles. According to the rules of the game these chips are able to be exchanged and grouped according to the numbers on the dice. Through observation of how the gaming chips are distributed upon rolling the dice under various settings, many unexpected situations can occur, some of which may surprise even “professional” researchers. The essence of statistical mechanics exists in this unpredictability, and I would like you to carefully observe these outcomes while rolling the dice yourself. “DIY statistical mechanics” can be established as a result.
The basic concepts of statistical mechanics, such as entropy, free energy, and temperature, can be quite simply developed from the dice and gaming chips approach. By examining both the movement of each particle in the group, their collective average movement as a whole, and the rate of change of those average quantities, you will be able to understand these fundamental concepts (entropy, free energy and temperature) intuitively. I recommend that after solving these simple problems (either by playing the described games or making diagrams) that you then compare the approach used here to that adopted in other textbooks on statistical mechanics. It is my belief that the topic of statistical mechanics can be best understood based on the accurate determination of the mechanical behavior of small numbers of particles. Using this approach, interesting viewpoints on difficult (yet important) problems at the front line of biological and physical research, such as the “temperature” of single macromolecules, will naturally emerge. Using such concepts as “local temperature”, one can infer general aspects about the three-dimensional structure of individual macromolecules (tiny rods and bars), such as their general shape and flexibility. It would be gratifying, if the methods described in this book allow you, the reader, to develop a clearer understanding of some of these difficult concepts in current biological and physical research.
The general approaches described herein extend to subjects that appear to have no direct relation to “statistical mechanics”. Such deeper understanding can lead the reader to the great discovery that Brownian motion and harmonic functions in potential theory are directly related. Furthermore, the qualitative differences between one-dimensional, two-dimensional, and three-dimensional Brownian motion, can be understood mathematically, and can also be “realized” from the game involving the rolling of dice.
In summary, using the dice and gaming chips approach, I want you to convince yourself that the presented ideas are correct.
This book presents several exceedingly difficult problems, with Feynman’s ratchet being one of these. Even if you do not fully understand these challenging problems, by understanding their essence, you can develop ideas that may relate to your own “research subject”. After you read this book and put it into practice you will gain an understanding of the basic principles of “statistical mechanics” in a broad sense. Research always begins with the words “interesting” and “surprising”. Science does not have to start with difficult theories and experiments, but you can also begin from easy dice rolls and simple calculations using “your own hands”.
This book was written with this aim, and the mantras are “easy to understand”, “enjoyable”, and “everything with my own hands”. Regardless of the reader’s specialization, this book will appeal to researchers in both the sciences and the humanities as well as also to non-researchers. In fact, in a lecture for the general public (which was partially the basis for this book) non-technical people, unfamiliar with physics and mathematics, were intrigued by the concepts, and proved sufficiently capable to understand the essence of the main ideas.
I hope that this book will be widely read.
Fumio OOSAWA
| Oosawa’s Preface | S003 |
| Introduction | S006 |
| Part I Understanding the Basics of Statistical Mechanics Using Your Own Hands | S008 |
| Chapter 1 Starting Rules | S009 |
| Chapter 2 Consumption Tax in the World of Molecules | S012 |
| 2-1 If the chances are fair, the results will be unfair | S013 |
| 2-2 Increasing the number of exchanges | S014 |
| 2-3 When interacting with each other, differences are emphasized | S015 |
| 2-4 Writing down all the distribution methods | S016 |
| 2-5 Computer simulations | S018 |
| 2-6 It’s because of the others that I have less gaming chips | S020 |
| Chapter 3 Changing the Rules | S025 |
| 3-1 Bankruptcy elimination type: Polymer growth and equilibrium | S025 |
| 3-2 Income tax-type: When there is a restoring force | S027 |
| 3-3 Focusing on multiple boxes | S029 |
| Chapter 4 Always “Now” is the Peak Moment | S035 |
| 4-1 Summary so far | S035 |
| 4-2 It is the same even if you reverse the time | S035 |
| 4-3 “Now” is always the peak moment | S036 |
| 4-4 The reaction progresses at a stroke | S038 |
| 4-5 Solid state physical properties: Consideration of magnetic susceptibility | S038 |
| Chapter 5 Summary of Part I | S041 |
| 5-1 Boltzmann distribution and the order of establishment of the isobaric principle | S041 |
| 5-2 Statistical mechanics can be experienced with a small number of components | S041 |
| 5-3 Time-reversal symmetry and always “now” is the peak moment | S042 |
| Part II Application of Statistical Mechanics to Phenomena in Cellular Systems | S044 |
| Chapter 6 Energy Exchange and Required Time | S045 |
| 6-1 Low viscosity motion (in gas) | S045 |
| 6-2 High viscosity motion (in liquid) | S047 |
| 6-3 Low and high viscosities: Cis/Trans transition of molecules | S048 |
| 6-4 Single molecule measurement of the hydrolysis of ATP | S048 |
| 6-5 What is viscosity? | S050 |
| 6-6 Torsional motion of a mirror suspended in gas | S051 |
| Chapter 7 Local Temperature | S056 |
| 7-1 Measurement of the local temperature from outside the box | S056 |
| 7-2 “Bending motion” of F-actin | S057 |
| 7-3 Feynman’s ratchet | S058 |
| 7-4 Ratchet model of the “sliding” motion of F-actin | S060 |
| 7-5 Measurement of local temperature is possible | S060 |
| 7-6 Energy accumulates in a few degrees of freedom | S062 |
| 7-7 Fluctuation-dissipation theorem | S063 |
| Chapter 8 Brownian Motion | S066 |
| 8-1 Experiencing Brownian motion: The mean square distance is proportional to the time | S066 |
| 8-2 How to catch a school of fish in the sea in a single throw | S070 |
| 8-3 Brownian motion and potential theory: The contact point of micro- and macro-scale | S072 |
| 8-4 Asakura and Oosawa’s force (depletion effect): Force of attraction between colloidal particles | S073 |
| Appendix A: How to Catch a School of Fish in the Sea in a Single Throw | S076 |
| Appendix B: Weyl’s Billiards | S077 |
| Afterword | S079 |
A few years ago, I held a series of seminars on statistical mechanics for the general public. Subsequently, I held a more detailed set of lectures for graduate students. This book is a compilation of those presentations. It is intended not only for undergraduate and graduate students in the fields of biology and physics, but also for other science students. Based on the response that I received to the public lecture series it may also hold some interest for the general public. Therefore, the content appearing in the first half of this book was written to be accessible to those with a high school education.
One of the objectives of this book is to entice the reader into playing simple games that involve the rolling of dice and the passing of gaming chips. In my lectures, the audience members participated in this activity with six people gathering together at a table and rolling dice. According to the numbers appearing on the rolled dice, the six participants exchange gaming chips between them. I would like you to attempt this game with a team of six people. Of course, you can also play the six roles by yourself. In my lectures, the participants from the public lecture series enthusiastically participated. However, many of the college students did not, as they were perhaps too shy. It was quite exciting to observe the active engagement of the general public. The purpose of my lectures was to create a fun atmosphere in which amateurs and professional scientists could enjoy statistical mechanics together. Some readers may already be familiar with this subject. However, please forget all prior knowledge, return to the beginning, and enjoy yourselves.
The results of the dice rolling activities may, at times, seem to defy common sense. However, participation in these games will allow the reader to gain experience with some basic statistical principles. After that, by considering familiar questions such as, “why does the world consist of only a few rich people and many poor people?”, we will gradually enter the world of statistical mechanics. In the second half of this book, we will start our investigations into a number of interesting topics in biophysics.
Both biology and physics are individually fascinating areas of research. The subject of biophysics, which is my area of specialization, is quite ambitious in that it aims to combine both the interest and fascination of its two parent disciplines, as well as tease out the unique synergies that emerge in combination. Since biology and physics are naturally disparate areas of study, for biophysical research to attract the attention of scientists working solely in either biology or physics areas, it must combine the rigor and fascination of both. Even if one tries to perform such a high-level study to attract such scientists, in the majority of occasions neither group of scientists will fully engage with the work due to the fact that biology and physics are very different disciplines. Hence, biophysics is extremely difficult! Modern biology, including molecular biology, regards life as a huge network of genes and molecules, a situation resembling the recent internet society (a huge network of computers and humans). I would say that modern biology is “close” to the modern (IoT) society. On the other hand, our understanding of the world, as it relates to our daily lives, is much closer to classical physics, established over 100 years ago. For instance, the statistical mechanics covered in this book was largely created by Gibbs0.2 well before his death in 1903. Electromagnetics was developed by Maxwell0.3 prior to 1873. Quantum mechanics, although it is a part of modern physics, was created when I was born, with Yukawa0.4 being one of the notable individuals making a great contribution to this area. Physics “seems” classical. Thus, we start our study of statistical mechanics using not modern but classical tools, dice and gaming chips.
Whereas thermodynamics deals with a system containing a large number of molecules at a macroscopic level, statistical mechanics deals with the state of each molecule and attempts to understand the nature of a system in which the molecules have gathered. Generally, when presenting ordinary statistical mechanics lectures within formal university courses, many teachers specify, “a very large collection of molecules” such as, for example, Avogadro’s number0.5. However, even if one approaches the subject using just a small number of molecules, the essence of statistical mechanics can still be understood extremely well. Such matters of small numbers, can be experienced manually by a group of people using dice and gaming chips! (Table 0.1)
Our attitude to statistical mechanics addressed in this book
| · | In general lectures, the phenomena are normally formulated as the property of a large number of molecules |
| · | The basic concept of statistical mechanics can be understood as a system with a small number of molecules |
| · | The phenomena can be manually reproduced (with dice and gaming chips) using various methods. |
In physics, the statistical mechanics of a system consisting of a small number of molecules has been discussed in various ways. Although it is related to part 2 of this book, currently, it has been uncovered that in the molecular mechanism of living organisms, free energy0.6 conversion is performed via a single operation of a single macromolecular machine. This seems obvious, but it is a very difficult idea. As the concept of free-energy originates from the behavior of many particles, it is surprising that a single molecular machine, made up of just a few molecules, performs not just energy conversion, but free-energy conversion [0.1]. Therefore, an important part of this book is the extension of statistical mechanics to describe how such systems with a small number of molecules, convert free-energy to work. Thus, although I stated that physics is a well-developed field, we see that it can be applied in novel ways to address modern and interesting problems in biology.
0.1 Translator’s note: “Enter the gate” in Japanese means entering the school gate to “start studying as a beginner” or studying with “a study guide for beginners”.
0.2 Original note: Josiah Willard Gibbs (1839–1903)—An American mathematician and physicist, discovered the phase rule, and the Gibbs free-energy and the Gibbs-Helmholtz equation is termed after him.
0.3 Original note: James Clerk Maxwell (1831–1879)—A British theoretical physicist. In1864, he established classical electromagnetism by deriving a set of equations termed as “Maxwell’s equations”, and theoretically predicted the existence of electromagnetic waves.
0.4 Original note: Hideki Yukawa (1907–1981)—A Japanese theoretical physicist, emeritus professor at Kyoto University and Osaka University. In 1949, he received the first Nobel Prize to be awarded to a citizen of Japan. He contributed greatly to the development of nuclear and elementary particle physics by advocating the meson theory.
0.5 Original note: This indicates the total number of components in 1 mole of a substance (i.e. if it is carbon, the number of atoms in 12 g of carbon), which is approximately 6×1023.
0.6 Original note: Normally, it is not possible to extract all the energy of matter. Energy that can be (freely) extracted (converted into work) under certain conditions is called the free-energy. Helmholtz free-energy F and Gibbs free-energy G are well-known. The former is the amount of energy that can be extracted under isothermal (constant temperature) conditions and the latter is that under isothermal and isobaric (constant pressure) conditions. Another important quantity, the chemical potential, is equivalent to the Gibbs free energy per mole (or per molecule).
