In reference to discussions received on our paper, “Self-Organization and Self-Similarity in Boiling Systems,” published in the February issue of the Journal of Heat Transfer1, here, we would like to summarize our responses to these comments as well as our own understanding of them so that some key differences can be understood more explicitly.

During the process of studying boiling heat transfer over the past few years, we discovered self-organized and cooperative or competitive phenomena among sites or bubbles in boiling systems 2,3. Based on this discovery, we further developed and formulated the present analyses 1. In whole, the paper 1 deals with self-organized and self-similar phenomena in boiling systems and other open dissipative systems, which are based mainly on statistical thermodynamics analyses, and, to our best knowledge, not Constructal Theory originated by Professor Bejan. With this in mind, there is no point to debate whether or not we have inexplicably misrepresented Constructal Theory as our own. Since the beginning of 1995, we have shown an interest in self-organization theory and nonlinear sciences theory 2. However, many excellent literatures in this field 4,5,6,7,8,9,10,11,12,13,14, including Professor Bejan’s textbook 15, played an important role on the primary ideas of this paper 1. In fact, in August 2000, when we began to write our paper 1, we developed a good understanding of Professor Bejan’s Constructal Theory. Surprisingly, we found that our final results and conclusions were in good agreement with results from Professor Bejan’s Constructal Theory. Even though our method was different from Constructal Theory, some similar industrial examples were surely inevitable. Most of the comments that we received have paid much attention to the language and not the content of our paper 1, however, the content is surely more important. As non-native English speakers, we learned English from different literature and excellent English textbooks. Because of this, we used very common language in our paper 1, which may be found in many books on self-organization theories and nonlinear science theories. These theories were created and gradually formulated by the contributions of many scientists, such as those of Professor Bejan.

In response to the comments that we have received, the following sections give our detailed explanations on a few primary features in our paper that distinguish it from those found in Professor Bejan’s Constructal Theory 15.

## Nonlinear Non-Equilibrium Statistical Thermodynamics Perspective

Let us begin with the “assumption” that appeared in the paper 1 in which the authors’ propose from the view of statistical mechanics 1. In fact, statistical mechanics shows that the same kind of hypothesis as equilibrium systems is required for describing open non-equilibrium complex systems. More importantly, we obtained probability distribution functions by giving constraints ranging from one-order driving forces to four-order driving forces, which was a way of statistical thinking. This assumption is very common. A similar description can be found in Professor Bejan’s Constructal Theory 15. Of course, we can also find similar statements in much earlier literature 16. In many books, this idea has been proposed as common sense, however, general law or mathematical formulations has not been available.

Though thermodynamics is a rather old discipline of physics, it is surely not old-fashioned. For the universality of thermodynamics, such modern topics as the big bang model, the theory of black holes, and the theory of biological or information systems show that thermodynamics is going through a renaissance. Currently, many scientists are looking for the fourth law of thermodynamics. At present, there are many similar and different descriptions on possible fourth law. For example, according to 17 many scientists at the Santa Fe Institute are eager for the appearance of fourth law or new second law. In our paper 1, we started our discussion from driving forces and generalized fluxes borrowed from classical irreversible processes where thermodynamics impression was evident. Furthermore, by assuming the nonlinear relation between driving forces and generalized fluxes
$J=η+∑iγixi+∑ijγijxixj+∑ijkγijkxixjxk+∑ijklγijklxixjxkxl+Λ$
(1)
We could deduce that our analysis was a kind of nonlinear non-equilibrium thermodynamics.

## Perspective of Renormalization Group Transformation (RGT)

With the constant victories of exploring the ultimate structure of matter from the process of molecules to atoms, atoms to electrons and nuclei, to nucleons, to elementary particles, and to quarks, another mainstream of modern physics has been concentrated on the understanding of structural organization of complex systems, which are more often encountered in our common experience.

Complex systems represent hierarchies, i.e., increasing complexity. This means that they can be divided into different levels, each representing a subsystem, which consists of relatively uniform elements that interact in some way with each other. These interactions may be responsible for autonomous pattern formation at any given hierarchical level. The higher subsystems in the hierarchy provide control over the processes of pattern formation at the lower sub-ordinary levels. The curiosity on the nature of increasing complexity has provided an incentive for numerous investigations over past decades. A substantial number of such efforts have been devoted to understanding and modeling it. A plethora of investigations are available in the literature. For example, in chemistry, we may find a lot of these investigations 10. Renormalization Group theory, as a typical method of analyzing increasing complexity, was originated by K. Wilson and has been developed gradually over past decades 12.

Our paper 1 deals with more general problems, mainly constructing the fractal networks from Scale Transformation theory in statistical mechanics, especially Renormalization Group theory 12, and is not based on any specific example. It is our logical conclusion that the present method is very general. By the factors group or parameters group transformations at different hierarchies or scales:
$Kn=RTGKn−1$
(2)
This method is a kind of typical geometric analysis in statistical perspective.
According to Renormalization Group theory 12, there are two directions for analyzing complex systems: zoom-in and zoom-out. Considering the growth processes, the present analysis 1 was based on zoom-out, that is, from small scale to large scale. We can derive the fractal structure in following way for two-dimensional problems,
$ΩnΩn−1≈Junα1−Junα2Jun−1α1−Jun−1α2=hnln2hn−1ln−12=lnln−12−p$
(3a)
Or for three-dimensional problems,
$ΩnΩn−1≈Junα1−Junα2Jun−1α1−Jun−1α2=hnln3hn−1ln−13=lnln−13−p$
(3b)
Depending on actual conditions, p, as a specific parameter ranging from 0 to 1, reflects the effects of the system’s internal configurations and environment on dynamics fractal structure of evolutional complex system. System adaptation to environment is highly exhibited.

## Other Related Problems

Open systems must optimize to live, which is generally common sense 18. Modern sciences are trying to repair the gap between living science and non-living science 6. Indeed, the present analysis 1 shows that same principle could be applied for any open dissipative systems, whether they are living or non-living.

Time direction arouses many interests, though it is still a puzzling problem 6. As for present authors’ viewpoints, though affected by literature 15, often other researchers state it similarly too 6,10,18. In some cases, they even express it as common sense.

Compared to available theories and to our best knowledge, in our present paper 1, we indeed derived detailed dissipative structure from a general viewpoint, which is always our objective in studying dissipative structure theory of Nicolis and Prigogine 7. In addition, our paper 1 derived detailed fractal structure from a general viewpoint. From the beginning of knowing fractal theory, we have realized that available fractal theory can only describe fractal structures, but not explain the physics of the formation of fractal structures.

From 1970s, urban evolution was a typical dissipative example even. We can find it in many books on self-organization; especially the self-organization of social problems. Professor Bejan’s theory 15 is a typical description on urban evolution.

In our paper, the term “Equilibrate means death” 1 means that it is impossible for energy or flow to be equally distributed in open systems. The present theory stresses the feedback effect 1, and it is by feedback effect that the ordered network can be constructed. The dominant subsystems must satisfy two conditions. Firstly, they need to get enough flux from the environment to destabilize them. In this case, new modes will be formed and more flux will be consumed. Secondly, the maintaining of this new mode, i.e., the maintaining of the ordered structure, will need more flux from environment. Actually, according to the above theory, tree-like network means that by competition and natural selection this new mode will indeed get most of the flux from the environment.

As for Figs. 4–8 in paper 1, we can only say that it is possible that we received some hints from Bejan’s book 15. The senior author has developed the idea of describing dynamics processes by using this type of figure over a period of time, and it cannot be attributed to a single source. It seems very natural to keep in mind the processes of Renormalization Group Transformation (RGT). More importantly, our figures differ greatly from Professor Bejan’s figures 15: For instance, in Professor Bejan’s figures 15, there are only two or three types of lattices dealing with only two or three types of sub-systems or factors. However, in our figures, we deal with real complex systems, which include amounts of sub-systems. So, in our figures, we draw many lattices indicated by many specific symbols dealing with enormous factors. Even though at first glance our figures are similar to Professor Bejan’s figures 15, they reflect completely different physical means. Professor Bejan’s figures 15 deal with solid geometry, whereas our figures stand for geometry in a statistical view, which were indicated by lots of symbols in our figures. Moreover, these symbols have specific statistical representing means. In conclusion, our figures differ from Professor Bejan’s figures 15 both in much-detailed configurations and in physical means.

## Conclusions

In conclusion, our paper 1 is making a renewed effort to develop the theories of complex systems from statistical mechanics. By paying more attention to understanding the processes of non-equilibrium phase change, the present investigations analyzed the physics of fractal structure and the mechanism of dissipative structure from statistical mechanics and renormalization perspectives. Of course, many results in our paper 1 agree well with the results derived by using Professor Bejan’s Constructal Theory 15, which demonstrated the unifications of natural laws. According to the history of any scientific developments, whether our paper lives in the large umbrella of Constructal Theory, or whether it is only a supplement to Constructal Theory will have to wait the test of time.

## Acknowledgment

The Project is Currently Sponsored by the Scientific Research Foundation for the Returned Overseas Chinese Scholars, State Education Ministry.

## Nomenclature

• h

= generalized flux transfer rate

•
• l

= scale, m

School of Environmental Science and Engineering, Tianjin University, Tianjin 300072, China; e-mail: shoji@photon.t.u-tokyo.ac.jp

Department of Mechanical Engineering, The University of Tokyo, Hongo 7-3-1, Bunkyo-ku, Tokyo 113-8656, Japan; e-mail: shoji@photon.t.u-tokyo.ac.jp

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