Considering statistical mechanics problems mathematically, as problems in probability theory

The Sakagawa Group in Keio University's Department of Mathematics studies problems in probability theory, motivated by topics in statistical mechanics.
In probability theory, a major goal is to discover regularities in phenomena that involve randomness, such as repeatedly tossing a coin or rolling a dice, with the aim of providing mathematical explanations.
In statistical mechanics, on the other hand, the aim is to explain phenomena at the macroscopic level, as observed by the human eye, in terms of the microscopic level, which involves vast numbers of randomly moving molecules and atoms.
In terms of this approach, probability theory and statistical mechanics are closely related. Various problems originating in statistical mechanics have come to be considered mathematically, as problems in probability theory.

Q."In other words, we're talking about mathematical physics. In general, there are lots of physical phenomena, and we want to consider mathematical models for them, to explain physical phenomena mathematically. What we actually do is, we extract the aspects of a physical phenomenon that capture its essential nature, and we create a simple model of the phenomenon. Then, we analyze the phenomenon in various ways, to see what we can say about it."

One important problem in statistical mechanics concerns phase transitions, where two different phases coexist; for example, water and ice at zero temperature. The Sakagawa Group creates various mathematical models related to phase transitions. The researchers' aim is to understand the essential nature of phase-change phenomena through aspects of probability theory, such as the limit theorem.

Q."The limit theorem expresses the overall properties of large numbers of random objects, in terms of limits on various quantities. Broadly speaking, there are three main principles involved. One is called the law of large numbers. As I said about dice, this asserts that, when lots of individually random events are averaged out, the result approaches what's called the expectation value. The second principle is called the central limit theorem. This tells us broadly how deviations from average look. It's about the probability of missing the average. It gives us a limit expressing how deviations from the average are distributed. The third principle is called the large deviation theorem. This principle is easy to understand intuitively. The law of large numbers tells us that results will approach the expectation value on average. In that case, the question is, how unlikely are the unlikely results? In other words, if I repeatedly roll a dice, what's the probability that I'll deviate from the average of 3.5? The large deviation theorem is used to investigate that kind of question."

In the physical world, there are many complex phenomena that appear to be random. The Sakagawa Group will keep investigating such elusive phenomena mathematically, using probability theory and statistical mechanics.

Видео Considering statistical mechanics problems mathematically, as problems in probability theory канала 慶應義塾 Keio University