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Imagine a 20-lane highway with no road signs, speed limits or white lines. The result, needless to say, would be chaos, with drivers constantly swerving, speeding up or slowing down to adjust to the changing flow. But a group at the Weizmann Institute recently looked at a similarly chaotic system and found a surprising underlying order that helps determine the movement on a tiny, busy “road.” Their findings, which recently appeared in Nature Physics, may help reveal hidden patterns in many kinds of complex chaotic systems, as well as providing new insight into the properties of flow for those working in the burgeoning field of microfluidics.
In such systems as traffic, long-range interactions become important – that is, they are the result of all the individual players interacting with one another. Whether they are powered by car engines, liquid flow or the force of gravity, systems with long-range forces tend to be unpredictable and hard to characterize. Several years ago, Prof. Roy Bar-Ziv of the Institute’s Materials and Interfaces Department, together with former graduate student Dr. Tsevi Beatus and theoretical physicist Prof. Tsvi Tlusty, developed a setup for observing what happens to particles in a traffic-like system with many elements.
Their trick was to make their system two-dimensional: They began with a very thin conduit – so thin that microscopic drops of water were flattened into “pancakes” by the top and bottom of the channel. Individual drops were carried along in a stream of oil, but as they were held back by the friction created by the top and bottom of the conduit, the drops moved much more slowly than the friction-resistant oil. Having this simplified, two-dimensional system enabled them to observe and measure things that would be impossibly complex in a three-dimensional arrangement.
In the original system water drops “marched” in single file; in the present study, the passage was widened to enable two-dimensional “traffic” patterns. The channel was half a millimeter in width and a few centimeters long; the drops just fit the height of the channel – around 10 microns (one one-hundredth of a millimeter) – so they could move freely in two dimensions between the channel’s side-walls. Research student Itamar Shani in Bar-Ziv’s lab watched through a microscope and on computer recordings as the tiny drops, pushed by the oil, flowed in streams as chaotic as those of the hypothetical unmarked superhighway on a Monday morning. And yet, as the researchers discovered, there was an underlying pattern – one that revealed a new type of long-range order between particles in a non-equilibrium system. Essentially, their findings show how each of the seemingly independent actors in the scenario is, in fact, influenced by every other actor.
The order is in the velocities of the drops. This became apparent when the team used sophisticated software they had developed to instantaneously capture the velocities of many drops at once. Mapping them out and color-coding them according to relative speed – faster than average or slower than average – revealed an arrangement in which groups of drops were coordinating their velocities in a way that was unusually persistent.
The drops’ collective action begins with the friction differential: If the oil were to flow alone, only the friction with the channel’s top, bottom and sides would affect its velocity. But when moving oil meets the friction-bound drops, some of the forward momentum transfers to the drops. Each time this happens, a bump appears in the velocity field in the channel; each distortion in the field then affects the movement of every other drop, to a greater or lesser extent.
To understand the phenomenon further, Bar-Ziv and Shani, along with Beatus* and Tlusty,* looked at how pairs of drops affect each other’s velocities. Taking large numbers of drop pairs, they looked for correlations between the measurements of their spatial separation and their relative velocities. Indeed, the resulting graph showed neat correlations – both positive and negative – depending on the drops’ relative positions. Positive and negative, in this case, meant how closely the drops moved in tandem; so, for example, positive correlations – pairs showing joint motion – were found between droplets lined up either perpendicular or parallel to the flow. The perpendicular drops were faster, while parallel ones were slower. Negative correlations – pairs with other types of motion, including those moving toward or away from each other, or rotating around each other – tended to be farther off the axes, along the diagonals.
The team then turned to an elegant model based on previous work by the group, in which the water drops create patterns of flow that are similar to the invisible lines of force around everyday magnets: These are dipoles. In other words, the distorted flow diverges and converges from two opposite points on the drops. Like the two sides of a magnet, the poles in the current around each drop have push or pull effects on other drops in the stream. The team again looked at pairs – this time theoretically – and calculated the mutual effects of only two dipole drops positioned at different angles to each other.
This “two-body” solution fit some of the observed correlations, but not all – in particular, not the negative ones. The scientists found that the mutual interaction in their model fell off too quickly with distance. They then realized that the reason the velocities were correlated was because they “felt” similar surroundings – due to their interactions with all the other droplets – rather than being an example of simple causation due to mutual interaction. Rather than try to work out the impact of every drop on every other drop, they added to the two-body model a third body that represented all the other drops – another dipole drop that interfered, from various directions, with the relations between the pairs. Now, with a mere three bodies, the team's theoretical model could describe – using simple mathematics and the positions of three drops – so-called “first principles,” the new long-range order that emerged from the chaotic flow.
These findings may be relevant to any number of chaotic systems with long-range interactions. They may be especially useful to those designing microsystems based on the flow of particles in various fluids.
Bar-Ziv: “This research is unique in the study of many-body systems with long-range forces, in that it presents a neat solution, with mathematical simplicity.” Questions the team intends to ask in future research include: What happens when changes are introduced into the system? And, can the underlying principle they discovered be applied to mapping chaos and turbulence?
*Dr. Tsevi Beatus is currently at Cornell University, Ithaca, New York; Prof. Tsvi Tlusty is at the Institute for Advanced Study, Princeton, New Jersey.
Prof. Roy Bar Ziv’s research is supported by the Yeda-Sela Center for Basic Research.
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