On 13-Aug-07 12:12:50, Paul Spicker wrote:
> There is a remarkable claim in the "news" section of New Scientist
> this week, in a piece entitled "Predicting change, not a moment
> too soon". 
> A piece in Physical Review E (which I don't have access to: ref DOI:
> 10.1103/PhysRevE.75.051125) claims that changes in the behaviour of
> swarms, crowds, traffic jams or similar systems can be predicted by
> monitoring a very small number of particles.  "Using a mathematical
> model of a phase transition, they attempted to detect an oncoming
> change by monitoring only a small fraction of the elements in the
> system. They found that they could do so by focusing on the 'mutual
> information' shared by those elements.
> ...In a disordered state, looking at a particle gives no information
> about what others are doing.  As the system approaches a phase
> transformation, the mutual information betwen particles increases,
> so that one particle's behaviour does provide information about
> the speed and trajectory of other particles. ... The researcher's
> simulations suggest that in a crowd of, say, 1000 people, observations
> on as few as five people might be sufficient."  
>  
> I think it was George Gallup who suggested that it might be ultimately
> be possible to predict election results using five people. I'm not,
> sure, though, that I believe it.  If anyone's in a position to
> appraise the technique it would be fascinating to know about it.
>  
> Paul Spicker

I've not managed to read the News Sceintist on-line (the site crashes
all my browsers), so will comment without knowing what the NS said.

There are two key issues above:
1. Mutual information
2. Phase transition

Mutual information is, essentially, a measure of the agreement
between one probability distribution and another. The more of
this there is, the closer one is to the other.

Quite how a situation of high mutual information between two parts
of a system may arise is qnother question, and will depend on the
type of system being studied. Probably there is some discussion of
this in the NS article.

Phase transition refers to the passage from one "state of matter"
to another, e.g. the transition from the liquid phase to the
solid phase ("freezing") or from the liquid phase to the gasious
phase ("boiling").

You could tell whether a cup of water was about to freeze by
monitoring the movement of a few molecules, since the temperature
of the liquid is, physically speaking, equivalent to the mean
squared velocity of its molecules.

Provided, of course, you were confident that the velocities of
a few molecules iwere "representative" of the distribution of
the velocities in the whole (i.e. good mutual information).
If not, then (in plain speech) the temperature of the molecules
being monitored would be different from the temperature of other
collections of molecules, and you would probably not be able to
redict much.

How would you ensure representativeness?

If you could stick labels on a few hundred molecules scattered
around the water in the cup, and track these, then you'd be looking
at a random sample of the whole set of velocities, and could closely
infer those of the whole. But that's technically pretty demanding
(i.e. close to impossible).

Alternatively, you could ensure that the water was well mixed,
(i.e. stir it well). Then the temperature at one point would be
close to the temperature at any other -- the distribution of
the velocities would be similar everywhere (high mutual information).

In that case you could readily monitor the mean squared velocities
at one point, and be confident that it was representative of
the whole. How best to do this in practice? Well, just stick a
thermometer in. This will register the temperature -- i.e. the mean
squared velocities -- of the tiny fraction of the molecules that
are impinging on the glass of the thermometer. Then you will know
whether the water is about to freeze.

Similar considerations apply to predicting whether it is about to boil.

Of course the above may be totally naive. In particular, it does
not refer to the quote:

  "...In a disordered state, looking at a particle gives no
   information about what others are doing. As the system
   approaches a phase transformation, the mutual information
   between particles increases, so that one particle's behaviour
   does provide information about the speed and trajectory of
   other particles."

The implication there, in the boiling/freezing context, might
be that as the phase transition is approached the similarities
between velocity distributions at different points increase.
I can see this for boiling, perhaps; but I'm not so sure about
the freezing. This is where I wish I could read that NS article!

To come back to Gallup and the Holy Grail: Perhaps the outcome
of an election could be predicted from a few voters, if the
voters as a whole were sufficiently stirred?

Best wishes to all,
Ted.

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Date: 13-Aug-07                                       Time: 14:04:02
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