After hundreds of years: scientists have found effective solutions to the famous three-body problem - and what does this have to do with "drunken walking"?
The three-body problem is one of the oldest problems in physics: it concerns the motion of systems with three bodies - for example, the system that includes the sun, the earth and the moon. How does the trajectory of the movement of three bodies in such a system change and develop due to their mutual gravitational force? This problem has been the focus of scientific inquiry since Newton.
When one massive body approaches another body, their trajectory is derived from their mutual attraction resulting from their gravitational force; As they continue to move and change their position along their paths, the forces between them also change, and these affect their paths and so on.
When there are two bodies in the system (for example, the Earth moving around the Sun without the influence of other bodies), the Earth's orbit will constantly move in a very specific curve, which can be accurately described mathematically (an ellipse). When a third person is added, much more complex interactions are created; The system becomes chaotic and unpredictable and its development cannot be predicted in a simple way for long periods of time. Although this phenomenon has been known for more than 400 years, since Newton and Kepler, there is still no neat mathematical description of the three-body problem.
Many physicists, including Newton himself, tried to solve the three-body problem. Oscar II King of Sweden even offered in 1889, on the occasion of his 60th birthday, a prize for whoever could find a general solution. In the end it was the French mathematician Henri Poincaré who won the competition. He shattered any hope of a complete solution by proving that such interactions are chaotic, meaning that the end result is fundamentally random. In fact, his findings opened up a new field of scientific research called Chaos Theory.
Usually a three star system is a chaotic system
The lack of a solution to the three-body problem means that scientists cannot predict what happens during a close interaction between a binary system (consisting of two stars that move in orbit relative to each other, like the Earth and the Sun), and a third star, but only by simulating it on a computer and monitoring Its development step by step. These simulations show that when such an interaction occurs, it proceeds in two phases: the chaotic phase in which the three bodies pull each other violently, until one star is ejected away from the other two settling into an elliptical orbit. If the third star is in a bounded orbit, it will eventually return towards the remaining pair and immediately afterwards the first phase will occur again. This triple dance ends when, in the second stage, one of the stars escapes into an unbounded orbit and never returns.
walking drunk
In the article recently published inPhysical Review, the doctoral student Yondav Beri Ginat and Prof. Hagai Peretz from the Faculty of Physics at the Technion used this randomness to give Statistical solution The whole process has two stages. Instead of predicting the actual outcome they calculated the The probability of any given outcome of the entire interaction in step-1. Chaos means that although it is impossible to reach a complete solution, its random nature allows to calculate the probability that a three-way interaction will end in a certain way and not another way. The entire series of approximations can then be modeled by using a certain type of mathematics known as the "random walk theory", sometimes referred to as the "drunk walk". The term got its name from mathematicians who thought about how a drunk walks and in fact saw it as a random process - as if the drunk doesn't understand where he is and therefore takes the next step every time in some random direction. The triple system behaves, basically, in a similar way. After each close encounter, one of the stars is randomly ejected (but the three stars together still maintain the total energy and momentum of the system). You can think of a series of these close encounters as a drunken walk. Similar to a drunkard's step, a star is ejected at random, returns, and another star (or the same star) is apparently ejected in a different random direction (similar to another step taken by the drunkard) and returns, and so on, until a star is ejected completely and never returns (the drunkard falls into the ditch).
Another way to think about it is to notice the similarities to the weather. The weather also exhibits the same phenomenon of chaos that Poincaré discovered, and is therefore difficult to predict. That's why forecasters should use probabilistic forecasts (for example, determine a 70% chance of rain knowing that there may be a wonderful sunny day). Furthermore, to predict the weather for another week from today, forecasters must take into account the probabilities of all possible types of weather on the different days, and only by adding them together do they get an adequate long-term forecast.
Ginat and Prof. Peretz showed in their research how this can be done in the three-body problem: they calculated the probability of each single-binary configuration of phase 2 (the probability of finding different energies, for example) and then connected all the individual phases, using the random walk theory, to Find the final probability of each possible outcome, similar to what you do to get a long-term weather forecast.
"We thought about the random walk model in 2017, when I was an undergraduate student," Ginat says. "I participated in a course taught by Prof. Peretz in which I had to write an article about the three-body problem. We didn't publish it at the time, but when I started my doctoral studies, we decided to expand the article and publish it."
The three-body problem has been studied independently by various research groups in recent years. One of the studies is by Nicholas Stone from the Hebrew University in Jerusalem, Naitan Lee who was at the time at the American Museum of Natural History, andBarak Kol, also from the Hebrew University. Now, thanks to Ginat and Prof. Peretz's current research, the entire multi-step interaction has been resolved statistically.
"This has important implications for the way we understand gravitational systems, and especially in cases where many encounters between three stars occur, such as in dense star clusters," says Prof. Peretz. "In such regions, many exotic systems create three-body encounters and lead to collisions between stars and compact objects such as black holes, neutron stars and white dwarfs that also produce gravitational waves that have been directly detected for the first time only in recent years. The statistical solution may be an important step in modeling and predicting the formation of such systems."
The random walk model may have other uses: studies of the three-body problem have so far treated individual stars as ideal point particles. In reality, of course, they are not like that, and their internal structure may affect their movement, for example, in tides. Earth's tides are caused by the moon and slightly change the shape of the earth. The friction between the water and the rest of our sphere dissipates some of the tidal energy in the form of heat. However, energy is conserved, so this heat must come from the energy of the moon in its motion relative to the earth. Similar to the three-body problem, tides can pull orbital energy out of the three-body motion.
"The random walk model explains this phenomenon naturally," Ginat said. "All that needs to be done is to subtract the tidal heat from the total energy in each phase, and then connect all the phases. We found that we can calculate the outcome probabilities in this case as well." It turns out that walking drunk can sometimes shed light on some of the most fundamental questions in physics.
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One response
Where is the curvature of space, the theory of relativity...?? Because until today there have been some changes.