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This article comes from the official account of Wechat: back to Park (ID:fanpu2019), author: Jordan Ellenberg (Professor of Mathematics at the University of Wisconsin-Madison), compiled by Xu Zhaoqing

The complexity of everything in the world, many of which start with some simple rules, are produced through the spontaneous interaction of the system, which is the magical self-organization. Self-organization has been found in many fields of natural science and even social science. This paper mainly introduces the earliest mathematical conceptual model: Abel sand pile model.

Have you ever heard of domino theory (Domino theory)? This is the geopolitical theory put forward by the United States to contain communism during the Cold War, which means that socialist countries will radiate and influence neighboring countries to carry out socialist changes. This theory greatly influenced the foreign policy of the United States in the middle of the 20th century and was used to justify its hegemonic behavior. But political theory aside, there are similar dominoes in nature. From a physical point of view, it should be called "sand pile theory (sandpile theory)".

Real-world regime changes tend not to happen methodically, but in sudden co-ordination, such as the Arab Spring and upheaval in Eastern Europe (eventually the collapse of the Soviet Union). In these historical events, there is a crisis hidden in a period of calm, and then suddenly collapsed at some point. Like a sand pile, if you put some more gravel on top of a sand pile, the sand pile may not change significantly in a short period of time. However, in an instant, similar to an avalanche, the gravel at the top suddenly rushes down in an irregular way and is likely to cause small secondary quicksand in the process.

This metaphor does not necessarily bring us anything. After all, real sand is hard to analyze, just like real-world politics. But there are also miracles. Physicists Per Bak, Chao Tang and Kurt Wiesenfeld proposed an abstract "Abel Sand pile Model (Abelian sandpile model)" in 1987. While keeping it simple enough to facilitate the study of applied mathematics, this model seems to be able to depict some interesting but disordered characteristics of the real sand pile, and is suitable for other complex systems derived from biology, physics and social sciences [1].

The process of the Abel sand heap model goes like this: we can imagine an infinite grid with a small pile of sand on each grid, and the number of gravel is represented by numbers in each grid.

However, there are certain restrictions on the height of the sand pile in the vertical direction. So it is assumed that every time the number of gravel in the grid reaches four, the four gravel will flow to the four surrounding grids. So if you start with four grains of gravel in two grids:

After the sand pile is dispersed, the grid on the left becomes:

At this point, the right grid has more than four grains of sand, so it will continue to disperse one grain of sand toward each of the four surrounding grids:

Now, because the number of gravel in all locations is no more than four, and each grid point is in a stable state, the process of sand dispersion has stopped.

In the above analysis, we first carried out the dispersion on the left side of the original two grids, followed by the one on the right side. How do we know which grid should flow around first? The good news is that the order of choice doesn't matter. Because we can get from the symmetry of the steady-state grid, the final state of this "Abel sand pile" does not depend on the order in which we choose to simulate the quicksand grid. This is why it is named Abel, which means that the order in which we choose does not affect the final result. Translator's note: Abel in mathematical nomenclature is usually in memory of the Norwegian mathematician Nils Abel (Niels Henrik Abel,1802-1829). He opened the research in many fields and is famous for proving that the root general solution of quintic equation does not exist and the study of elliptic function. Despite his great achievements in mathematics, he encountered many difficulties in his life and finally died of tuberculosis under the age of 27.]

For example, the operation of addition is Abel's, and we mean that the elements in addition are interchangeable: adding 2 and then 3 is equivalent to adding 3 and then 2. But most of the operations or operations are not Abel's. For example, first unlock the car, then open the door, then the door opens; but first pull the door, and then unlock the car, the result is completely different-the door is still closed. So the Abel nature of the sand pile can be regarded as a surprise.

So you might ask, what happens if we put a lot of gravel, say 1 million grains, on a grid? What will it look like when the gravel continues to flow around and finally stabilizes? You might imagine that it would end up with a huge, flat pile of sand, in which a large area near the center would have many grids of three gravel.

But that's not the case. The following picture shows the grid after it is finally stabilized:

Millions of grains of sand: about a large amount of gravel (to be exact, to be exact, to the 20th power of 2) accumulates on the central point to simulate an Abel sand pile. (the color indicates the heap height. Blue means there is no sand. Purple means one, yellow means two, and maroon means three.) Photo Source: Wes Pegden well, is it possible that 1 million is not enough to make the number of sand piles smooth? If we use a billion grains of gravel, will we get a large, flat pile of sand? The final image looks like this:

Simulation picture of a billion grains of gravel? photo source: Wes Pegden, the flat situation we expected did not happen. On the contrary, those strange fractal patterns persist. Near the center, the complex pattern is like a dome, with many squares embedded inside, which look like some kind of geometric pattern but seem random; at the boundary of the sand pile, there are many consistent triangles, tightly connected in a regular pattern.

The images were drawn by Wes Pegden, a professor of mathematics at Carnegie Mellon University, and collaborators Lionel Levine and Charlie Smart of Cornell University in cutting-edge research on the sand pile [2]. On Professor Pegden's website, we can even see interactive pictures of a billion grains of sand, which we can zoom in or move to any location, such as looking directly at the center of the sand pile:

The central image of a billion grains of sand? photo source: Wes Pegden or see sharp and strange details on the outer edge:

Image of the edge of a billion grains of sand. Source: Wes Pegden, if you look closely, you can also observe a more detailed local structure. Like many math articles, we give interested readers an assignment here: after the sand pile is stable, please explain why the two adjacent grids cannot be empty at the same time. (there is an answer! In fact, some experiments suggest that we may have a stronger conclusion: not only are empty grids not adjacent to each other, they even tend not to be close to each other, just like particles with the same charge, they repel each other.

Reference answer: suppose there are adjacent grids 1 and 2. And the two grids are inside the sand pile rather than at the edge, so we might as well assume that grid 1 is the last one of the two grids to flow around before stabilizing. Then when the gravel in grid 1 is dispersed to the adjacent grid, grid 2 will contain a grain of sand from grid 1. But because grid 1 is the last one to disperse, the gravel of grid 2 is not lost, so the two grids will not be empty at the same time.

The simple law of complexity before you actually look at the sand pile with a microscope, I have to remind you that real sand pile does not produce such a spontaneous structure [3]. The Abel sand pile model here does not even have the property of simulating real physical materials. Instead, all the complexity we see comes from an abstract, simple, deterministic algorithm that can even be written in just five lines of code. This is reminiscent of John John Conway's Conway's Game of Life, https://playgameoflife.com/. The game also produces a wealth of complexity from very simple rules. Like the Game of Life, the Abel Sand pile is a cellular automaton (cellular automaton): it is a miniature universe in which the laws of operation can be fully described in discrete languages acceptable to the computer. In the sand pile, each grid has a number from 0 to 4, and the value of the adjacent grid is determined by a simple rule. In the Game of Life, the state of the grid is simpler, and each state is either born (with a value of 1) or dead (with a value of 0).

But there are differences between the two: "Life Game" as a typical cellular automata, complex behavior can appear, but more inclined to a simple pattern [4]; but for the sand pile model, we do not seem to need to set special initial conditions, it will automatically tend to complex patterns.

In fact, the occurrence of complex behavior in the sand pile depends on a so-called critical threshold, near which complex behavior often occurs. We are familiar with the concept of critical threshold in nature: water is a disordered liquid at higher temperatures, but when the temperature drops to a certain critical value, water changes dramatically-crystallization into ice. For the sand pile, its density is similar to the temperature of water. (the density here refers to the average amount of gravel in each grid. If there is too much gravel, the sand pile will be unstable and an "avalanche" will occur; if there is too little gravel, the sand pile will soon stabilize. So how much are we talking too much? In fact, the answer is surprisingly simple. The dividing line between great and small changes is an average of 2.125 grains of gravel in each grid.

It is worth noting that when the grid is limited and assuming that the gravel will disappear when it reaches the edge grid, the average number of gravel per grid will be 2.125. At the beginning, all the grids were empty, and we put the gravel into the central grid one by one. After a while, the sand begins to disperse around, slowly forming an image similar to the one generated by Professor Pegden, which we showed earlier (this image assumes that there are an infinite number of grids in all directions. (we drop a grain of sand, and when the sand pile is stable, we drop another grain, so that there will be more and more gravel. But if the gravel reaches the edge, the scattered gravel disappears. After that, the sand pile will approach a balance: the rate at which the gravel falls at the boundary is equal to the rate at which we increase the gravel, and the density will stabilize at a certain critical value. Of course, there will be local fluctuations in the system, and there will be some alternating changes between low-density and high-density places over time, but for the whole, the average number of gravel per grid will be about 2.125.

What would happen if at first we filled each grid with gravel as much as possible, that is, three grains per grid? This initial layout is stable, but it is fragile and stable. We put a grain of gravel in any grid, and then a huge avalanche begins, and it doesn't stop until the density drops to 2.125.

So what happens when the gravel density reaches a critical value? At this time, the sand pile will be in the most interesting state. The process of dispersion occurs all the time, but it is not a continuous state of widespread chaos; on the contrary, it is similar to wave after wave, with rare avalanche disasters across all grids from time to time. However, the distribution of dispersion activity under the threshold density seems to follow the power law, and the frequency of dispersion activity is inversely proportional to its scale. There are also continuous diaspora activities, but they have some structure and regularity. Not only that, in order to show its complex behavior, the sand pile does not need to make fine adjustments, it has the ability to adjust itself. No matter where the system starts, as long as the new gravel increases at a constant rate, the system will lead to the critical threshold state.

Seeing is believing. R.M. Dimeo of the National Institute of Standards and Technology (NIST) has produced a series of boring movies in which the sand pile is in a critical state.

To me, this process looks like vitality, not a coincidence. Thinking about how rich life structures emerge from simple systems, they automatically find critical thresholds, and it is a popular way to use the concept of self-organized criticality. Some biologists believe that self-organized criticality is a potential unified theory of complex biological behavior. This theory dominates the pattern of simultaneous flight of a flock of birds, just as genetic information dominates the development of individual birds [5]. Stuart Kaufman (Stuart Kauffman), a theoretical biologist, wrote, "Life systems exist in a stable state near the edge of chaos, and it is natural selection that achieves and maintains this equilibrium." It's like the sand we're talking about. It's not a life, of course, but it's alive, isn't it?

Sand pile is the first and most frequently studied example of self-organized criticality. There are many other examples. (there are also some on Professor Pegden's website. However, we do not know what the scattering rule of sand pile is, why it makes the system inevitably develop towards a complex critical state, and it is not clear which cellular automata may show this self-organized criticality.

Some profound understanding may arise from the surprising connection between sand pile theory and other mathematical theories. For geologists like me, sand pile theory is related to the recently emerging theory of tropical geometry (tropical geometry), the goal of which is to use similar discrete geometric phenomena to simulate continuous geometric phenomena. [translator's note: the theory of tropical geometry was first developed by Brazilian mathematician and computer scientist Imre Imre Simon in the 1980s, and the word "tropical" stems from the stereotype of Brazil by some French mathematicians. Tropical geometry can be regarded as piecewise linearized algebraic geometry and has important applications in counting algebraic geometry.]

For probabilists, sand piles are closely related to the so-called spanning tree. Spanning tree (on a square grid) is a branching path that touches every point on the grid but does not form a closed loop. No matter where these understandings come from, sand pile theory reminds us that very interesting phenomena in mathematics, like many interesting phenomena in physics, often occur in phase transitions. It is here that we are between two different mathematical theories, which not only have their characteristics, but also transmit information and problems across boundaries. Of course, generally speaking, there are more questions than answers.

references

[1] Bak, P.A., Tang, C., & Wiesenfeld, K. Self-organized criticality: An explanation of the 1 An explanation of the f noise. Physical Review Letters 59,381-384 (1987).

[2] Levine, L., Pegden, W., & Smart, C.K. Apollonian structure in the Abelian sandpile. Preprint arXiv.:1208.4839 (2014).

[3] Mehta, A. & Barker, G.C. Disorder, memory and avalanches in sandpiles. Europhysics Letters 27,501506 (1994).

[4] Aron, J. First replicating creature spawned in life simulator. New Scientist 2765, 6-7 (2010).

[5] Mora, T. & Bialek, W. Are biological systems at criticality? Journal of Statistical Physics 144,268-302 (2011).

This article is translated from the April 2015 issue of Nautilus Dominoes.

Https://nautil.us/issue/107/the-edge/the-math-of-the-amazing-sandpile

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