Birds Of A Feather: The Physics Of Flocks

September 25, 1998

AIP Physics News Preview: Birds Of A Feather

What Is The News?

--Studying the incredible feat of how hundreds of birds can move as a single unit, physicists have devised a detailed theory of the flocking process.

--Their theory can potentially be extended to herds of wildebeest, schools of fish, and any collection of independently moving animals that rely on each other's cues to move as a group.

---These results may additionally provide insights into the seemingly unrelated topic of auto traffic flow.

---To explain the flocking process, the physicists make intriguing analogies to physics phenomena such as the lining up of magnets, the flow of fluids, and the transfer of heat.

College Park, MD--Watching a flock of dozens or even hundreds of birds can be amazing because the separate birds often move as if they possess a single mind. What's more amazing, bird flocks often move harmoniously without any sort of leader or external cue, especially when they are traveling over short distances. Studying bird flocks may seem to be the exclusive domain of ornithologists. But physicists too have become captivated by the remarkable ability of birds--and many other living creatures--to move flawlessly as an organized group.

In the October issue of the journal Physical Review E, John Toner of the University of Oregon and Yuhai Tu of the IBM Watson Research Center in Yorktown Heights, NY present a detailed theory of how birds manage to move together as a single unit, even if the individual birds make frequent misjudgments and can only see an extremely small fraction of the other birds in the flock.

By making a few simple modifications, the authors say that their theory can also describe movements in herds of wildebeest, schools of fish, swarms of bacteria--in short, any collection of independently moving animals that rely on each other's cues to move as a group. Potentially, their theory can even be applied to the seemingly unrelated topic of automobile traffic flow.

To explain the flocking process, the researchers draw parallels between the motion of flocks and several phenomena in physics: namely, the lining up of magnets, the flow of fluids, and the transfer of heat. But the field of "flocking dynamics" didn't begin in the realm of physics. It didn't really begin in the outdoors. It began in the world of microscopic organisms.

Back in 1993, at his laboratory in Budapest, physicist Tamas Vicsek was watching movies of bacteria colonies made by a group of biologists in Germany and his collaborators in Israel. As the bacteria multiplied, he noticed, parts of the colonies of bacillus circulans and a strain of bacillus subtilis formed circles. Looking more closely at the circles, Vicsek saw that the groups of bacteria moved clockwise in half of the circles and counterclockwise in the others. Whether they chose to move clockwise or counterclockwise was completely random, he realized. Because of the lining up of the bacteria Vicsek immediately thought of permanent magnets--similar to those on your refrigerator.


A permanent magnet consists of many atoms, each of which acts as if it is a tiny bar magnet with a north and south pole. At room temperature, many of these atoms line up their magnets with those of their neighbors, adding together to create in effect a large bar magnet and giving the overall material a strong ability to attract or repel other magnetic materials.

Permanent magnets, also known as "ferromagnets," depend solely upon the interactions between neighboring atoms. It costs a minimum amount of energy for any two atoms to line up their magnets parallel to each other. These cooperative gestures between neighbors quickly lead to a situation in which many of the magnets in the material are lined up--bringing about magnetism on a large scale.

The Curie Point

But despite their name, "permanent" magnets can lose their ability to attract or repel objects. When you raise the temperature of a ferromagnet above a certain level (known as the "Curie point"), the heat can misalign the atoms' respective magnets in random ways. Roughly speaking, there tends to be an equal number of atoms with magnets pointing in opposite directions, so that their magnetic effects cancel out.

But when the magnet dips below the Curie temperature, the material again recovers it ability to attract and repel objects. This is because many of the atoms line up their magnets with each other once again. But this time they may be lined up in a different direction than before. What's amazing is that the magnets can line up in one of many directions, but the atoms have chosen a single direction out of all other possiblities.

This is an example of what physicists call "spontaneously broken symmetry"--the unmagnetized material has properties that are the same in every direction and physicists would therefore say it possesses "symmetry." But chilling the material below its Curie temperature breaks the symmetry, as the atoms spontaneously choose a preferred direction in which to line up their magnets.

An analogous phenomenon, Vicsek realized, happened to each colony of bacteria. At some point while the colony was growing, the members chose to move together in a clockwise or counterclockwise direction. And, Vicsek reasoned, something similar was going on with a flock of birds. If they are confined to fly in a cylindrical cage, they will circle in one of the two possible directions. Even when they are unconstrained in the outdoors, they can fly in any direction, but at some point they decide to follow a single direction. Like magnets, the birds in the flock can line themselves up in many possible directions, but they choose one.

Computer Simulation Of Bird Flocks

With these parallels in mind, Vicsek designed a computer model that simulated how birds line themselves up with their neighbors and move in a single direction as a flock. In the simulation, each bird would try to fly in the same direction as its neighbors, defined as those birds within a certain fixed distance from it. Like an atom in a ferromagnet, each bird would respond to only its closest neighbors. At each step of the simulation, it would look at all its neighbors and move in the average direction.

Vicsek added another real-world factor: he imagined that the birds would not have perfect judgment. At each step, each bird would always have a chance of misjudging the proper direction it should move. And because the birds rely on each other to move in the right direction, this error would spread too, causing the neighbors to fly slightly off course.

Suppose the flock is heading east, but one bird misjudges the situation and moves 10 degrees north of east. If each bird has 4 neighbors, each neighbor will then make a 2.5 degree error in direction. That's because each neighbor averages the movements of the erring bird (10 degrees north of east) and its three neighbors (all moving due east) to move 2.5 degrees north of east. The erring bird's four neighbors all make this 2.5 degree error, so that these errors add up to the amount of the original error. In the next step of the simulation, these 2.5 degree errors get transmitted and shared by neighboring birds in the same way. The question is: Would these errors undermine the ability of flocks to stay organized and move in a single direction?

Vicsek again looked to ferromagnets for the answer. In the absence of heat, two neighboring magnets prefer to line up perfectly with each other, because it costs the smallest amount of energy for them to be aligned. But a little heat can add enough energy to the material to misalign the magnets, thereby introducing a sort of "error" in the material.

Vicsek quickly realized a major problem with the ferromagnet analogy. Because heat is always present in real-world situations, each atom's magnet tends to become slightly misaligned with that of its neighbor. And the slight misalignments between neighbors mean that distant magnets will have a good chance of being very misaligned with each other, greatly dimming the prospects for the material to be an effective magnet. In fact, a rock-solid theory in physics, known as the Mermin-Wagner theorem, says that these interactions make it impossible for a flat, two-dimensional array of atoms to line up enough of their magnets to behave as a permanent magnet.

Carrying the ferromagnetic analogy to flocks, this would mean that misjudgments between birds flying in a flat, two-dimensional formation would never be able to organize into a unified flock. Yet in real life, we see flocks of birds that fly principally in a flat two-dimensional plane. Furthermore, even the simulations betrayed the fact that there was more to the story. The numerical simulations carried out by Vicsek and his young colleague Andras Czirok still showed that the birds lined up in their model. Was there something more to their model than the ferromagnet analogy?

Birds As Fluid Particles

In late 1994, at a symposium held at IBM's Thomas Watson Research Center in Yorktown Heights, NY, two physicists heard the visiting Tamas Vicsek give his talk on bird flocking. John Toner and Yuhai Tu were intrigued by Vicsek's model. Together, they developed a detailed theory that ultimately explains how the flock stays together.

The two physicists explored the idea that the flock of birds moved like a fluid--the term in physics used to describe any gas or liquid. They used the well-known equations for ferromagnetism and added a few terms from the Navier-Stokes equation, the basic equation that describes fluid motion.

In a sense, the approach by Tu and Toner took the opposite approach as the one by Vicsek. In Vicsek's ferromagnetism model the basic ingredient was the interaction between neighboring birds and these short-range interactions added up to explain the large-scale behavior of the whole flock. In contrast, the Navier-Stokes equations describe the overall behavior of the fluid while ignoring the fact that the fluid is made up of individual constituents, such as atoms and molecules, or birds in this case.

Taking these equations, and plugging in typical values of such parameters as how fast real flocks move in the air, Tu and Toner come up with realistic predictions of such things as how densely the birds are packed together in certain situations and how this density fluctuates. Most importantly, their equations bring out some of the birds' properties that were not in the original model. Ultimately, these properties explain how the birds could stay together in the presence of errors.

Tu and Toner show that the fluid-like behavior of birds enables errors to become diluted quickly, even on a two-dimensional plane. To understand how this happens, consider the following analogy. Imagine an inattentive driver on the highway. He starts off by moving 65 miles per hour in his lane. But he starts talking on his cell phone and veers off at a 10 degree angle. When he does this, he will still be traveling 64 miles per hour along his lane, but 11 miles per hour side to side! So he makes a big change in his side-to-side speed (over 11 mph), but the change in his forward speed is 1 mph! In short, his side-to-side motion changes drastically, while his motion in the forward direction stays about the same.

Similarly, if a bird in a flock misjudges the direction it should travel, it will swerve side-to-side rapidly. You might think this error in judgment would overwhelm the other birds, causing the flock to become disoriented and fly apart very quickly. But this process actually helps keep these misjudgments under control, by quickly spreading the error among many birds so that it becomes very diluted. At one instant the erring bird might be on the left side of the flock, and the error affects its neighbors that are on the left side. But the next instant, the bird may be in the middle of the flock, and the error influences its new neighbors over there. A moment later, and the bird is on the right hand side, and it spreads to its neighbors there. As a result, the error is quickly shared among many birds and it is diluted before it has a real chance to affect the direction of a flock.

Convection And Diffusion

As Tu and Toner note, this process is highly analogous to a method of transferring heat known as convection. Convection occurs whenever you heat a pot of soup. The stove first heats the soup on the bottom. Then, the liquid on the bottom rises to transfer its heat directly to the colder stuff on top. Convection is the most efficient way to transfer heat on large scales. In the bird flocking model, the carrier of information (the bird) moves through the flock to distribute its information. In the flocking model, convection is the most efficient way for spreading information and diluting the effects of errors.

In ferromagnets, by contrast, information and errors spread only by a slower, less efficient process known as diffusion: each atom stays fixed and information travels from neighbor to neighbor until it is transmitted through the entire material. But like an ink drop spreading through a cup of water, diffusion is slow and the errors are more burdensome as they are shared by fewer birds. In the bird flocking model, it is truly the convection process that makes the difference.

Future Directions

The theory of bird flock dynamics is still in its infancy. The theory does not yet factor in many external cues that birds may use, such as seasonal temperature and even the Earth's magnetic field (some birds have magnetic material in the brain which help them to navigate using the Earth's magnetic field). But for non-migrating birds travelling over short distances, and for many other groups of animals, it is likely that they depend on information from their neighbors for choosing their direction of motion and staying together. Still, the authors plan to incorporate the effects of internal compasses into future versions of the theory.

In addition, the researchers plan to expand the theory to three dimensions. Future versions of the theory aim to describe and predict the shape of flocks, and how it fluctuates. And perhaps most profoundly, the researchers intend to study how a disordered group of birds transforms into a unified flock. Knowledge about flocks of birds can be applied to collective motion in other groups of animals, and to other problems, including the behavior of humans in their automobiles. Understanding how groups of birds move together might give transportation officials and highway designers new strategies for making the rush-hour commute a bit more smooth.

Journal Article:
John Toner and Yuhai Tu, Flocks, Herds, and Schools: A quantitative theory of flocking, to appear in Physical Review E, October 1998.

--John Toner, University of Oregon, 541-346-0979,
--Yuhai Tu, IBM T.J. Watson Research Center, 914-945-2767,
--Tamas Vicsek, Eotvos University, Budapest, Hungary,

Additional References
1) Tamas Vicsek, Andras Czirok, Eshel Ben-Jacob, and Inon Cohen, Novel type of phase transition in a system of self-driven particles'' Phys. Rev. Lett. 75, 1226 (1995). 2) A. Czirok, H. E. Stanley and T. Vicsek, Spontaneously ordered motion of self-propelled particles, J. Phys A30 , 1375 (1997). 3) A. Czirok, E. Ben-Jacob, I. Cohen, O. Shochet and T. Vicsek, Formation of complex bacterial colonies via self generated vortices'', Phys. Rev. E 54 , 1791 (1996)

Figures and an expanded version of this writeup can be found at

American Institute of Physics

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