杏吧原创

THERE鈥橲 no doubt it鈥檚 had a bad press. For most of us, noise is what goes on
next door when you鈥檙e trying to sleep. It鈥檚 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鈥檙e 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鈥檛
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鈥檝e
found some answers. Unlike most forces, when noise makes significant things
happen, it doesn鈥檛 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鈥� 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鈥攁 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鈥檚 ability to
get through the system
(鈥淣oises on鈥�, New 杏吧原创, 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鈥檚
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鈥攖he 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.

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鈥檚 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鈥檚 odd effects. Another occurs
in a device known as a stochastic ratchet. Suppose you have some particles
trapped in a ratchet-shaped base鈥攕ome 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 鈥渃oloured鈥�. The archetype
of noise鈥攖he kind considered originally by Einstein鈥攊s 鈥渨hite鈥�
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鈥攁 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鈥攖hat 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鈥檒l see all these possibilities play out. There is
no 鈥渙ne way鈥� 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鈥攐nly 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鈥攁 flip of the switch, for
example. When it does, you examine the immediate prehistory of the system and
record exactly how it happened. It鈥檚 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鈥檛 drive our
circuits鈥攖hey 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
鈥渓arge fluctuation鈥� 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 = 鈭�1 and q = 1. (You
might think of these states as 鈥渙ff鈥� and 鈥渙n鈥� for a switch.) We started with the
circuit in the state at q = 鈭�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 = 鈭�0.63, having
wandered well away from the stable point q = 鈭�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 = 鈭�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).

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鈥檚 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 = 鈭�1 and later arrives at q =
鈭�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鈥檚 great for us, but will this new understanding of the work habits of
noise be of use? We think so. Instead of making our 鈥渟pecial event鈥� q = 鈭�0.63,
for example, we could just as well have made it q = 1, in which case the special
pathways would show the circuit鈥檚 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鈥攁 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 鈥減ushing鈥�
(see 鈥淣o toil no trouble鈥�, New 杏吧原创, 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鈥檛 want to happen. You might be a
physicist who doesn鈥檛 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.