Physicists crack unsolvable three-body problem using drunkard's walk

A physics problem that has plagued science since the days of Isaac Newton is closer to being solved, say a pair of Israeli researchers. The duo used "the drunkard's walk" to calculate the outcome of a cosmic dance between three massive objects, or the so-called three-body problem.

For physicists, predicting the motion of two massive objects, like a pair of stars, is a piece of cake. But when a third object enters the picture, the problem becomes unsolvable. That's because when two massive objects get close to each other, their gravitational attraction influences the paths they take in a way that can be described by a simple mathematical formula. But adding a third object isn't so simple: Suddenly, the interactions between the three objects become chaotic. Instead of following a predictable path defined by a mathematical formula, the behavior of the three objects becomes sensitive to what scientists call "initial conditions" — that is, whatever speed and position they were in previously. Any slight difference in those initial conditions changes their future behavior drastically, and because there's always some uncertainty in what we know about those conditions, their behavior is impossible to calculate far out into the future. In one scenario, two of the objects might orbit each other closely while the third is flung into a wide orbit; in another, the third object might be ejected from the other two, never to return, and so on.

In a paper published in the journal Physical Review X, scientists used the frustrating unpredictability of the three-body problem to their advantage. 

"[The three-body problem] depends very, very sensitively on initial conditions, so essentially it means that the outcome is basically random," said Yonadav Barry Ginat, a doctoral student at Technion-Israel Institute of Technology who co-authored the paper with Hagai Perets, a physicist at the same university. "But that doesn't mean that we cannot calculate what probability each outcome has."

To do that, they relied on the theory of random walks — also known as "the drunkard's walk." The idea is that a drunkard walks in random directions, with the same chance of taking a step to the right as taking a step to the left. If you know those chances, you can calculate the probability of the drunkard ending up in any given spot at some later point in time. 

So in the new study, Ginat and Perets looked at systems of three bodies, where the third object approaches a pair of objects in orbit. In their solution, each of the drunkard's "steps" corresponds to the velocity of the third object relative to the other two. 

"One can calculate what the probabilities for each of those possible speeds of the third body is, and then you can compose all those steps and all those probabilities to find the final probability of what's going to happen to the three-body system in a long time from now," meaning whether the third object will be flung out for good, or whether it might come back, for instance, Ginat said. 

But the scientists' solution goes further than that. In most simulations of the three-body problem, the three objects are treated as so-called ideal particles, with no internal properties at play. But stars and planets interact in more complicated ways: Just think about the way the moon's gravity tugs on the Earth to produce the tides. Those tidal forces steal some energy from the interaction between the two bodies, and that changes the way each body moves. 

Because this solution calculates the probability of each "step" of the three-body interaction, it can account for these additional forces to more precisely calculate the outcome.

This is a big step forward for the three-body problem, but Ginat says it's certainly not the end. The researchers now hope to figure out what happens when the three bodies are in special configurations — for example, all three on a flat plane. Another challenge is to see if they can generalize these ideas to four bodies.

Source: Live Science