What are phase transition and critical behavior?

You may have heard the word “Rinkai (Critical)” after the unfortunate disasters following the Great East Japan Earthquake. However, you do not expect those phenomena to be ubiquitous, do you? Actually, there are various kinds of phase transition and critical behavior around us, many of which assist us in our daily lives and impact our daily activities. For example, water, after heated, boils around 100 degrees C, keeping the temperature constant until all the water evaporates away (which is why we can cook!). Liquid water and water vapor, both having the same composition (H₂O) at a microscopic level, belong to completely different phases (liquid and gas phases) at a macroscopic scale, which is one example of phase transition. Another example is the progress of an infectious disease, such as influenza. Replace “infectious disease” with “opinion,” and, in the same way, there is a transition from the majority phase, in which the opinion is widely supported, to the minority phase in which it will disappear without advocates. The figure on the left shows samples of percolation, which is a mathematical model of simulating the society with population density p (cited from Reference 1). Assuming the top and bottom walls are the sources of an infectious disease, you can see how the clusters of infected individuals, which are indicated in red, get drastically larger as p increases. Suppose that the society is infinitely large.  It is known that, as the figure on the next page shows, the probability θ(p) of the cluster from one specific point, denoted o, being infinitely large exhibits a phase transition at p_c≒0.593. It is also known that the transition shows unusual behavior and occurs “continuously but non-analytically.” Such abnormal behavior around the critical point is collectively termed critical behavior.

 

What do you mean by mathematically rigorous understanding?

As explained in the previous paragraph, phase transition and critical behavior cover a wide range of phenomena not only in mathematical physics, but also in many other areas, such as chemistry, biology and sociology. Therefore, what would you do if you were to investigate universally and objectively such phenomena as a result of the collaboration of an infinite number of components?

We do so by using mathematics. The phenomena derived from collaboration of an infinite number of components could not be understood using simple addition or integration. Here, the advanced mathematics you can learn at university, such as precise definition of convergence, permutation of multiple limits and various kinds of analytical methods, works effectively. For example, you may not be able to imagine exactly how we can “calculate” the probability of the existence of an infinitely large cluster somewhere in the society.  However, if the society is homogeneous, then there is a limit theorem of mathematics such that the probability is either zero (0) or one (1).

One of the mathematically rigorous approaches to understanding phase transition and critical behavior is the lace expansion, a powerful and beautiful method to which I have a great deal of contribution (refer to Reference 2). Using the lace expansion, we can show that critical behavior in dimensions higher than the so-called upper-critical dimension turns out to be very simple. Although the details cannot be described here due to space limitation, in the case of percolation, the lace expansion of the probability G_p(x) of the origin o and the point x being in the same cluster is shown to satisfy the following recursion equation:

Here, pD(v –u) is the probability that the neighboring two points u and v are directly connected and Π_p(x) is the so-called lace-expansion coefficient. Roughly speaking, the lace-expansion coefficient can be evaluated as the sum of the probabilities that o and x are connected in such complex ways as are depicted in the above diagrams. If the sum converges, then the recursion equation can be solved due to its linear structure and results in the aforementioned simple critical behavior. The lace expansion has been successfully applied not only to percolation, but also to other models, such as the Ising model which serves as a model for magnets and self-avoiding walk (SAW) which models linear polymers, revealing the critical behavior in higher dimensions.

We will encounter more and more problems on phase transition and critical behavior. I look forward to seeing how these problems are solved with mathematical rigor. During the course, we may have to develop new analytical methods. It would be challenging and worth it if I could stay actively involved in these research topics.

 

References

1. http://moreisdifferent.wordpress.com/2013/11/16/percolation-theory-the-deep-subject-with-the-not-so-deep-sounding-name/
2. Sakai, A. A chapter in Analysis and Stochastics of Growth Processes and Interface Models. OxfordUniversity Press, 2008.