From: magnasco@research.nj.nec.com (Marcelo Magnasco)
Subject: Re: Yang-Lee Zeroes
Summary: Y-L zeroes and RG
Originator: magnasco@asterion
Keywords: iteration fractal solution transition
Organization: NEC Research Institute
Date: Tue, 18 Aug 92 23:15:21 GMT

In article <1992Aug18.201038.27173@etek.chalmers.se> cafer@etek.chalmers.se (Tommy Vaske) writes:
>While doing some research for a paper on fractals,
>I hit the paragraph about Yang-Lee zeroes in _the_beauty_of_
>fractals_. Maybe I'm not wizardious enough, but I felt the book
>was very unclear there;
It is. Perhaps because the identifications occur at a very deep 
mathematical level and it is very difficult to explain.

>It was something about using Newton's (or was it N-Raphson's?)
>method to solve a complex polynomial equation. Somehow, this
>would resemble the clusters of atoms in different states, in
>a sheet of some material in some transition state. (Magnetic,
>nonmagnetic, fluid or firm (bad word choice, I know), etc.)
No. It doesn't resemble any physical configuration of anything. The space
you are visualizing is complex TEMPERATURE plane, and things happen
on the fractal (water boils or iron becomes magnetic or any other
`phase transition'). Of course, it will be difficult for you to set
your microwave oven to heat water to a complex temperature...

When a large collection of identical units is studied, you must 
describe it in statistical fashion, since it will be difficult to 
say anything more precise than statistical. When you talk about the
temperature of water, you are doing statistical mechanics, because 
`temperature' is physicist's shorthand for the average kinetic energy 
of all the water molecules. 

One of the basic entities in statistical mechanics is the `partition 
function', which basically looks like a sum of exponentials. In order 
to have a `phase transition', it can be proven that the logarithm 
of the partition function should have a singularity, (the p.f. has 
a zero). Now, the p.f. is a sum of exponentials, so all the terms 
are positive; how could it possibly have a zero?

The answer is that it can have zeros in the complex temperature plane. As you
keep adding particles to your system, you get more and more terms in the p.f.,
and you get more and more zeroes out there in the complex plane. These
zeroes are called Yang-Lee zeroes. They can accumulate and have a 
limit in the real temperature line. 

>As I understood it, the distribution of singularities would 
>somehow be especially important.
It is. The accumulation points of the yang-lee zeroes on the real temp.
line mark the presence of a phase transition, a temperature at which something
drastic happens to the system. As I said, water boils or iron becomes 
nonmagnetic or ... 

There is a technique to deal with phase transitions called `renormalization
group'. It was invented around 1967 by Leo Kadanoff. The only thing I will
say now is that when you are done, you have a recursion relation (an 
iterated map) for the temperature, whose fixed point gives you the phase
transition. The point being mentioned in the book is that 
"the julia set of the r.g. recursion is the closure of the Yang-Lee zeroes"

So you can see its very abstract. But it is particularly interesting because
it is also very very deep. 

>Exactly in what sence it would resemble said material, I felt
>to be very unclear - and then again, I might just lack the
>knowledge. Could anybody fill me in?
I hope I have been a bit clearer. Unfortunately, the math involved is 
way too esoteric. 
>--tommy  

Marcelo Magnasco
marcelo@mjf.rockefeller.edu	magnasco@research.nj.nec.com
Newsgroups: alt.fractals