When the sun is completely obscured by the moon, the originally dazzling sun is obscured by the moon's shadow, and the earth falls into brief total darkness. Such a rare astronomical phenomenon is a total solar eclipse, which occurs in one place at an average interval of about 400 years. Scientists have long predicted that the next total solar eclipse will occur on December 4, 2021, and the best place to observe it is (76°47.0′S, 46°9.7′W) about 110 kilometers northeast of the coast of Antarctica's Berkner Island.
Behind this key weapon in mastering the laws of heaven and earth is mathematics, which has been used by mathematical models to predict the universe for centuries.
▲ Antikythera machine, a bronze machine designed in ancient Greece to calculate the position of celestial bodies in the sky, belongs to an analog computer (from Wiki, author Therese Clutario)
Around the end of the 19th century, scientists largely held the Cartesian view of the cosmology of matter. That is to say, except for God, the operation of all things can be explained by mechanical principles. In this way, if the position and velocity of all the particles in the universe can be measured, then in theory, human beings can predict everything.
But if that's the case, even in this era of supercomputers and big data, why are there still some unpredictable events in the world? If we can predict eclipses a century in advance, why can we only predict the weather a week or two in advance?
We know that weather systems also follow the laws of physics, but it also seems to be random. Chaos theory is widely used to study seemingly random systems, and one of the pioneers of the theory was Edward Norton Lorenz, a meteorologist at the Massachusetts Institute of Technology.
Lorenz received his master's degree from Harvard University in 1940, when World War II changed his career— serving in the Air Force after graduation. At that time, the Air Force was in urgent need of weather forecasters, so he went to MIT for special training, studied meteorology, and received a doctorate in meteorology, staying in the academy to teach, working with noam Philips and Jule Chaney, pioneers in the field of weather forecasting. At the time, Lorenz was trying to build a mathematical model with variables that represent information such as temperature and wind speed, like real weather, so that he could use the model to test the accuracy of different forecasting methods.
In World War II and the decades that followed, computers enabled scientists to take a crucial step forward in weather forecasting. Lorenz was approved to use the "Royal McBee LGP-30" vacuum tube computer for modeling, with the assistance of experts from the Department of Mathematics to improve the program. Margaret Hamilton was one of the experts, and the code for the later Apollo lunar mission was also written by him.
▲ Margaret Heatherfield Hamilton, August 17, 1936 – present. Her immediate boss at the time was Lorenz himself.
In 1961, Lorenz was working on a set of mathematical model equations used to simulate atmospheric flow, in which 12 variables changed over time. The computer then calculates the value of the variable at the next moment by bringing the value of the variable from the previous moment into the equation, and Lorenz observes this process. He needed to make sure that the process did effectively simulate weather conditions and not just pointless iterations.
At that time, the Lorenz team needed to reprint some of the simulation results, they would restart the program from an earlier point in time, and the calculation program would continue. Specifically, the earlier value is substituted to re-use as the starting value. Since the values and equations are consistent, they are convinced that the results of the two programs will be consistent, but this is not the case, and there are often cases where the results are far from each other.
At first, Lorenz thought the computer was malfunctioning, but he eventually found the source of the error — the accuracy of the data. Although the number stored on the computer's memory has six decimal places, the print result format includes only three decimal places, so the input values actually have a small error. From this, Lorenz realized that in some systems, small differences in initial values can lead to dramatic changes in future behavior.
Scientists call this situation Sensitive Dependence on Initial Conditions, which is not a completely new concept. As early as the 1780s, the great French mathematician Henri Poincaré observed a similar situation when studying the three-body problem in space, which was the beginning of chaotic thought.
Back in 1963, Lorenz devoted himself to finding the simplest set of equations to express the chaotic situation of the system. Eventually, he found a simplified set of equations that could simulate atmospheric convection processes in which only three variables change over time. Let's try to draw the orbit of the Lorentz equation solution in three-dimensional space on a modern computer. The position of the first point is arbitrary, and for each subsequent point position, the computer can calculate it by substituting the previous point value into the equation. It looks as if the track is changing direction randomly and doesn't reach the same point twice — otherwise, the track will obviously go into a repeat iteration. This type of system is now known as the "Singular Attractor".
For example, the following figure randomly gives the location of 100 points and brings them into the Lorenz system of equations. At first, their solution trajectories looked rather random, but soon bizarre scenes appeared.
The trajectories formed by the outer points first travel in a random direction, but as long as they are within the confinement domain, these trajectories will eventually reach the center of attraction from the outside. This is what the word "attraction" means in the term "singular attractor," which emphasizes that the trajectory does not really fall into the center of attraction. Each trajectory remains independent and never intersects, which means that the solution trajectory in the system is still determined by the initial value.
When Lorenz reported on his research at an academic conference that year, he ended with a meteorological note: "If a seagull flaps its wings, will the weather change because of it?" What he really wants to express is that the nonlinear equations that describe the process of weather change are so unstable. This is also the meaning of the famous saying "a thousand miles of loss of a millimeter" that is said on weekdays.
Let's look at the pendulum. This is a cyclical, nonlinear system with repetitive motion. If we make two releases from roughly the same position, then the two swings we get will also be roughly the same.
If you connect the two pendulum ends, you still get a nonlinear system, only this time it is non-periodic. Similarly, we release twice from roughly the same position, and this time we get two swings that are very different. It is what Lorenz calls "deterministic chaos."
"Determinism" means that the evolution of a system is still determined by the laws of physics it follows, even though it appears to be random. If the weather system is a deterministic chaotic system, it is easy to understand why it is so difficult to predict. But what if we could scan the entire planet with extreme precision? Even if we drastically reduce observational errors, will we still not be able to predict the weather in advance?
In a paper published in 1969, Lorenz reviewed the problem of "seagull flapping its wings," in which he took a closer look at how tiny movements in the air expanded over time to affect entire atmospheric systems, and how small-scale measurement errors eventually led to severe deviations in predictions. To this end, Lorenz proposed a mathematical model and came to a surprising conclusion: the system in the model can only predict the future for a short period of time, beyond which reducing the initial value error will no longer improve the prediction accuracy.
Lorenz is not sure whether the true evolution of the atmosphere fits this point, and we are still unable to make a conclusion today. But his model still has significant implications because it suggests that, in terms of predictability, a world determined by the laws of physics may end up with a completely random world. This is undoubtedly a major challenge to Descartes' cosmology of matter.
In 1972, when Lorenz was presenting his latest work at an academic conference, because he did not provide the title of his report in time, the moderator of the conference drafted a proposal for him, called "Will The Flapping Wings of Brazilian Butterflies Trigger a Hurricane in Texas?" So this little butterfly, instead of the original seagull, became the "image endorsement" of chaos theory.
Chaos theory has had an increasingly profound impact on many disciplines. Lorenz's 1963 paper was cited only a few dozen times in the first 10 years, but 10 years later it was cited more than 9,000 times in papers in almost every scientific field.
In 1987, James Gleick's best-selling book Chaos: Pioneering New Science introduced the work of Lorenz and other chaos theory pioneers to the world, and the "butterfly effect" became part of popular culture. The butterfly effect is a theory according to which even the most inconspicuous situations can create great disasters after a series of developments, such as a butterfly flapping its wings and causing a tornado.
▲ The weather is unpredictable, which is the mystery of nature
Mathematicians are also continuing to build more mathematical models of nonlinear systems, and if these chaos models are correct, then deterministic chaos theory shines in a number of important areas, including financial markets, eye tracking for patients with brain diseases, the pathogenesis of sickle cell disease, and human and animal population fluctuations.
In 1992, scientists at the Massachusetts Institute of Technology used a supercomputer to simulate the future operation of the entire solar system for nearly 100 million years, and finally found that the really valid prediction range was only about 12 million years. After that, planets may collide or may have left the solar system.
So even the planetary motions that humans have been predicting for centuries may not be as simple as we previously thought.
In the eyes of his friends, Lorenz was quiet, humble and unpretentious. It is known that late in his career, the world began to worship him as a pioneer in the new realm of chaos. In 1991, he received the Basic Science Prize, the prestigious Kyoto Prize. Lorenz passed away on 16 April 2008 at the age of 90. He will pass on his name as a great scientist to future generations, because he fundamentally changed man's understanding of the universe and the way he views himself and all things.
Author: Sheng Yan (Meet the core members of the math creation team)