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Mail-from: From lynch@theory.lcs.mit.edu Tue Apr 18 11:26:06 2000
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Date: Tue, 18 Apr 2000 11:22:24 -0400 (EDT)
From: Nancy Lynch <lynch@theory.lcs.mit.edu>
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To: shishir@theory.lcs.mit.edu
Subject: pset and tutorial
cc: karger@theory.lcs.mit.edu, lynch@theory.lcs.mit.edu

*** EOOH ***
Date: Tue, 18 Apr 2000 11:22:24 -0400 (EDT)
From: Nancy Lynch <lynch@theory.lcs.mit.edu>
To: shishir@theory.lcs.mit.edu
Subject: pset and tutorial
cc: karger@theory.lcs.mit.edu, lynch@theory.lcs.mit.edu


David sent these notes on physics.
Could we make a HW problem out of this?
They could set up the four expressions, and possible calculate and
compare the bounds.
If it's too late for the HW problem, than maybe a tutorial problem
would be fun.

\section{Physics}

{\em Statistical Mechanics} studies the behavior of large systems of
particles (e.g., boiling a kettle of water).  One of the most basic
physical systems is a {\em gas:} a collection of particles (electrons,
atoms, photons) floating around in a large container. 

One way to describe this system is to divide the container into lots
of tiny boxes.  We can then describe the system by saying, for each
particle, which tiny box it is located in.  We call this the ``state''
of the system.  The system is equally likely to be in any one of its
states (assuming they all have the same energy, but discussing this
would take us too far afield).  Many facts about the aggregate
behavior of the system arise from averaging over all the possible
states in the system.

A single particle is equally likely to be in any one of the tiny
boxes.  The obvious thing to do is to describe states by where
particle $1$ is, where particle $2$ is, and so on.  This means that if
there are $r$ particles and $n$ positions, there are $n^r$ possible
states.  This gives rise to the {\em Maxwell-Boltzman statistics} for
the system.  Nature follows these rules when all the particles are
distinct (e.g, different atoms).

You would think it doesn't make a difference if the particles are all
the same---after all, we could label them.  Surprisingly, it matters!
In a cloud of identical photons, nature actually obeys our balls in
bins model.  Thus there are ${n+r-1 \choose n-1}$ possible states of
the system, each equally likely.  Gases made up of photons are said to
obey {\em Bose-Einstein} statistics; photons are a kind of {\em Boson}
and so is Helium 2 (the superconducting stuff).  This has impacts on
the physical behavior of the system.  For example, suppose $r/n =
\lambda$ is the average number of particles per cell.  Under
Maxwell-Boltzman Statistics, the fraction of empty cells is about
\[
\lambda e^{-\lambda}
\]
while under Bose--Einstein Statistics, the fraction of empty cells is about
\[
\frac{1}{1+\lambda}
\]
When $\lambda$ is large, the first quantity is a lot less than the
second.  In other words, Maxwell-Boltzman predicts a more spread-out
gas than you would actually find.

A different story happens with electrons, protons, and other ``solid''
particles called {\em Fermions}.  At most one fermion will be present
in a given cell (thus we'd better have $n>r$).  If the particles were
distinguishable, there would be $P(n,r)=n(n-1)\cdots(n-r+1)$ different
states.  But when the particles are all the same, nature acts as if
there are ${n \choose r}$ different states.  This means that gases
made up of just one atom behave differently than gasses made up of
more than one atom.  In particular, the {\em entropy} of a gas, a
physically measurable quantity, depends on the number of states.
Physical experiments have shown that the entropy agrees with the
indistinguishable particles model, and not the other.
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