Medaille College News

Dr. Vincenzo Isaia, assistant professor of mathematics and sciences, will present “Intermediate Asymptotic Behavior, Self-Similarity and the Renormalization Group” on Friday, December 7 at 3 p.m. in the Main Building, Room 200.

Renormalization Group (RG) is a tool used by scientists in theoretical physics to analyze physical changes at different scales. According to Dr. Isaia, “RG started as a tool in quantum mechanics, and with help from Murray Gell-Man and Richard Feynman moved to field theory and critical phenomena. Kenneth Wilson won the 1983 Nobel prize in Physics for using RG to connect critical phenomena and phase transitions, showing that the RG may be more general than its original job of curing divergences in field theory. Mitchell Feigenbaum used RG for studying bifurcations in dynamical systems, a story made popular in James Gleick’s book Chaos (a topic which was further popularized by Michael Crichton’s book Jurassic Park). Here, RG has jumped to the world of partial differential equations (PDEs), and finds its home in intermediate asymptotic behavior.”

Dr. Isaia explains: “If the state of a physical system can be described by a single value (or more generally a vector) at every point in time, then such a system is called an evolution system, and upon prescribing an initial value, along with a law that governs the evolution (i.e. a law which describes how the state may change), then one can predict the state of the system at any time if one can solve the resulting equation.

“If the system also has a spatial dependence, then the resulting law may usually be described by a partial differential equation (PDE), and the resulting PDE with the initial value is called an initial value problem (IVP). An example would be how temperature distributes itself in a partially heated bar as time increases. Physically, we expect that if one waits long enough, the system’s behavior may ‘forget’ its initial value as well as become accustomed to any spatially imposed boundary conditions.

“The behavior during the range of time where we have waited ‘long enough’ is called the intermediate asymptotic behavior of the IVP, and it is reasonable to expect this behavior to be simpler than the original ‘short time’ behavior coming from a complicated initial condition. The resulting simplification that is usually observed in physical problems is called self-similarity: the state at time A may be reproduced by a proper time scaling of the state at any different time B. This property of self-similarity represents a stabilization of the problem at hand. Mathematically justifying the above discussion is the content of the talk. The tool which allows for all of this is based upon the extension of dimensional analysis through the use of the physics topic of Renormalization Group (RG).”

Related Link:

Wikipedia’s entry on Renormalization Group

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