From: Brian Atkins (brian@posthuman.com)
Date: Thu Dec 04 2003 - 10:46:52 MST
Glorious noise
New Scientist vol 161 issue 2168 - 09 January 1999, page 36
Random noise can be used to drive chemical reactions or stop a
spacecraft spinning out of control—you just have to know how to tweak
it. Peter McClintock and Dmitri Luchinsky tune in
THERE'S no doubt it's had a bad press. For most of us, noise is what
goes on next door when you're trying to sleep. It's that horrible hiss
and crackle when an old recording ends or when you try to tune your
short-wave radio into the BBC World Service when you're in the Brazilian
jungle.
But within that seemingly meaningless barrage of uncontrolled and
usually disruptive rapid-fire shocks lies something far more
interesting. If those shocks arrive in just the right sequence,
interesting things happen. Without noise, ice would never grow on a lake
in winter, nor would chemical reactions happen; noise drives our muscles
to contract when they should and it helps our cells to pump crucial
materials though their membranes. Life itself wouldn't exist without
that messy stuff called noise.
But how does noise do its work? And if we understand it better, can we
harness its power? After studying the problem for some years, we think
we've found some answers. Unlike most forces, when noise makes
significant things happen, it doesn't do so through a gradual
accumulation of effects. Instead, noise does its work all at once in
dramatic, exceptional events. To understand the workings of a noisy
system, it seems, you can ignore most of what goes on, so long as you
keep track of the rare events that really count.
Physicists' present fascination with noise stems from a discovery made
in the 1980s. Suppose you send a weak periodic signal through a noisy
black box, and look at the ratio of signal and noise strengths coming
out—a measure of quality called the signal-to-noise ratio (SNR). For
some black boxes, it turns out that adding noise at the input actually
increases both the signal and the SNR at the output. Adding noise can,
paradoxically, boost a signal's ability to get through the system
("Noises on", New Scientist, 1 June 1996, p 28).
Stochastic resonance, as this effect is called, has now been seen in
black boxes ranging from the sensory neurons in crayfish tails to the
microscopic ion channels that carry messages across cell membranes. In
the case of crayfish, Frank Moss and his colleagues at the University of
Missouri, St Louis, have shown that adding noisy, irregular currents in
the water near the crayfish's tail increases its ability to detect
regular, periodic fluid motions which might betray the presence of a
predator.
How does this strange effect work? Imagine that the black box is a light
switch, and the incoming signal is your finger, which tries to flick the
switch on and off with a steady rhythm. The output signal is the light
intensity of a lamp to which the switch is attached. If the signal of
your finger is strong, then the output signal is also strong—the light
flickers on and off in the same rhythm. But if you have a broken finger
and can flick the switch only weakly, you might not be able to move it,
in which case no signal will get through. Here is where noise can help.
Suppose that the switch is slightly noisy, and has a tendency to vibrate
although not so vigorously that it would actually flip between on and
off by itself. When your broken finger tries to flick this noisy switch,
it will occasionally be reinforced by a fortuitous vibration acting in
the same direction, and so push the switch over the hump (see Diagram, p
38). The noise increases the ability of your finger to make things
happen, so at the lamp, there now appears a signal, not a perfectly
regular signal, to be sure, but a signal nonetheless, which goes on and
off in rough synchrony with your finger.
Triggering a switch using random noise
Breaking waves
Physicists now know that this two-state switching scenario is just one
of many ways in which stochastic resonance can occur. In 1990, physicist
Mark Dykman, then at the Institute for Semiconductors in Kiev, Ukraine,
realised that stochastic resonance could be brought within the fold of
traditional theoretical physics by using something called linear
response theory. This provides a simple and general way of describing
how a fluctuating system responds to a weak periodic driving force.
Using Dykman's ideas, we showed that the conditions under which
stochastic resonance arises should occur in just about every noisy
nonlinear system you can imagine.
But stochastic resonance is just one of noise's odd effects. Another
occurs in a device known as a stochastic ratchet. Suppose you have some
particles trapped in a ratchet-shaped base—some gravel at rest in a
strip of corrugated steel, for example, where the corrugations all lean
in one direction like breaking waves. Here, it turns out that purely
random jiggling of the base can cause the particles all to drift in just
one direction. This seems to defy the second law of thermodynamics,
since you can extract useful work from seemingly random noise. But it works.
The trick, as physicist Marcelo Magnasco of Rockefeller University in
New York pointed out in 1993, is that the noise has to be "coloured".
The archetype of noise—the kind considered originally by Einstein—is
"white" noise, in which each little noisy shock is independent of its
predecessors. White noise has no memory. In coloured noise, however,
there is a kind of memory at work—a shock that pushes a bit of gravel to
the right is more likely to be followed by another similar shock, rather
than by one which pushes back to the left. Coloured noise is still
random, and ultimately the bits of gravel get little kicks to the left
and right in equal proportion. Yet the sequence can conspire with the
ratchet to move the particles. The direction of the steady flow can even
be changed without altering the ratchet merely by altering the colour of
the noise—that is, by changing the pattern in which successive shocks
tend to follow one another.
This mechanism probably lies at the root of how living cells move
molecules around. They may also help engineers to build nanoscale
motors, able to function in the micro-world where noise rules and
conventional engineering techniques fail.
So physicists have come to understand several ways in which noise can be
useful. But is there any way to get inside stochastic resonance, or the
workings of a stochastic ratchet, and understand them in the same way
that we understand, say, how a clock works, or a car? Think of the noisy
switch. Starting from the off position, there are infinitely many
possible paths that it could follow in executing a noisy transition from
off to on. It could make the change quickly, or vibrate about in the off
position for several minutes, and then suddenly hop over to on. Then
again, it might just take a slow, gradual noisy walk over the hill
between those states. The same goes for a piece of gravel moving from
one dip in the corrugated steel to another.
Wait long enough, and you'll see all these possibilities play out. There
is no "one way" that noise can make things happen. But we have found
that in practice, almost all of those infinitely many possible motions
turns out to be irrelevant to significant events—only a tiny subset
really matters. We have also found that by focusing on these special
paths, we can understand the workings of noise with just a few simple
equations. The important thing, then, is to identify the few pathways
that matter.
The ideas and theoretical results leading to this view of noise have
accumulated over decades, stretching back to Einstein and Boltzmann. But
in the intervening years, nobody could see how to test things
experimentally. The breakthrough came in 1992, when we worked out with
Dykman, now at Michigan State University in East Lansing, and Vladim
Smelyanskiy, who works at NASA, how to measure and interpret a new
physical quantity, the forbiddingly named prehistory probability density.
The idea is to take some noisy system and monitor its fluctuations,
waiting for something rare and dramatic to happen—a flip of the switch,
for example. When it does, you examine the immediate prehistory of the
system and record exactly how it happened. It's rather like recording,
every time you spill the milk, a detailed history of what led up to the
event. Doing this over and over, thousands of times, you accumulate
information on the histories that are most likely to lead to it. This is
the prehistory probability density.
Large fluctuations
Our first experiment in 1992 used electronic circuits from which it is
easy to record detailed histories and get good data. We didn't drive our
circuits—they were in simple equilibrium with their noisy surroundings.
One circuit we studied was rather like a switch in that it could be in
one of either two stable states. Thermal noise made it vibrate, leading
to occasional "large fluctuation" events in which it moved a very long
way from a stable state. The experiments verified our ideas: on almost
every occasion, the events took place by way of a special pathway,
predicted by theory, that mirrored the path by which the system relaxed
back to its closest stable state.
Getting theoretical results for non-equilibrium systems is fundamentally
more difficult because that time symmetry. But these systems are among
the most important for applications because they are so common,
especially in biology. Theory suggested that they must display very
complicated behaviour. To study this, we turned again to our switch-like
circuit. The state of this circuit is given by a variable q, and its two
stable states are q = -1 and q = 1. (You might think of these states as
"off" and "on" for a switch.) We started with the circuit in the state
at q = -1, and then subjected it to random noise, as well as a regular,
oscillating force. This force pushed the circuit out of equilibrium. We
then waited for an unusual state to occur.
Once in a great while, we found it to be in the state q = -0.63, having
wandered well away from the stable point q = -1. Every time we found it
there, we immediately looked in detail at the events that brought it
there. To make things simple, we always took that moment when the system
arrived at q = -0.63 as the time t = 0. This was so we could compare a
large number of case histories of the same special event, and not get
confused by the different times when they occurred (see Diagram, below
right).
The probability density of Noise
In the diagram, the mountainous heap towards the bottom shows the
resulting prehistory probability density. Where many histories pass
through a certain point, the mountain is high. Where few histories pass,
the mountain is low. The upper plane shows in red dots the location of
the mountain's ridges, which first separate and then come together
again, giving the mountain the appearance of volcano. The two ridges
correspond to the two special paths that the circuit is most likely to
follow if it begins near q = -1 and later arrives at q = -0.63. In
almost all cases, it is by these two pathways, and these two pathways
alone, that the system reaches this particular, unusual state. The upper
plane also shows the pattern of special pathways (grey lines) predicted
by mathematical theory, which seem to fit our data .
That's great for us, but will this new understanding of the work habits
of noise be of use? We think so. Instead of making our "special event" q
= -0.63, for example, we could just as well have made it q = 1, in which
case the special pathways would show the circuit's preferred ways of
switching from one stable state to the other. Since events like these
typically underlie stochastic resonance or the workings of a stochastic
ratchet, this theory should make it possible to understand these things
in detail, even in complicated problems in the real world.
Making noise work
There may be other uses too. Suppose there is some event that you want
to happen—a chemical reaction, for example, which is the key step in
manufacturing an expensive drug. Random thermal noise will drive the
reaction at a certain rate. But you might do better. If you want to
speed things up, and follow the most energy-efficient approach, you
might try to apply well-designed forces to push the molecules involved
along one of the special paths most likely to lead to the reaction. By
carefully measuring the chemical system, you should be able to work out
the required forces directly. This is exactly what chemist Herschel
Rabitz and his collaborators at Princeton University are doing, using
lasers to do the "pushing" (see "No toil no trouble", New Scientist, 18
July 1998, p 33).
The new view on noise could also help in the world of far bigger things.
Suppose, say, there is some event you don't want to happen. You might be
a physicist who doesn't want the power in your laser to exceed some
limit which would destroy it, or guiding a spacecraft or oil tanker, and
want to keep it from being driven by random and uncontrollable forces
into some dangerous predicament. In principle, you ought to be able to
calculate the special paths along which such hazardous fluctuations
would be bound to develop. When something ominous does start to happen,
you can be ready to apply small corrective forces to steer away from
trouble. This approach would be much more efficient and cheaper than
applying correction forces all the time, or applying huge forces at the
last moment.
So noise works in a relatively simple way, which we can understand, and
even control. In future, we should be able to command noisy systems with
the same ease as ordinary machinery. Far from being a disruptive
nuisance, it looks as if noise could turn out to be a valuable ally.
-- Brian Atkins Singularity Institute for Artificial Intelligence http://www.intelligence.org/
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