Moreover, he calculated the probabilities that the particles differ from this expected behavior, and show that these probabilities become smaller and smaller. In both cases he was able to show that if the number of particles approaches infinity, the density of the particles more and more satisfies a particular mathematical equation. My question is: how can I bring these models together? How can we derive a mathematical equation that fits both realities and is also rigorous?"įor this, Hoeksema looked at two scenarios: one where particles attract and repel each other (like metal cracks), and one where particles are being added and removed (like changes in population, ecology and tumors). "As an alternative, engineers have come up with alternative models that, although less rigorous, can be used for experimental observations. But these calculations very quickly become very costly. But what if you want to explain how a metal beam bends? You can of course try to aggregate the insights you gained at the microscopic level, and deduce from that how things work at a macroscopic level. It's relatively easy to describe the way these fissures attract and repel each other as a random (stochastic) process. "Take the behavior of minute cracks in metal structures. While microscopic models track the state of all particles in a system individually, macroscopic models look at the world from a more abstract level, describing qualities like movement, density, velocity and temperature." "It serves to bridge two ways of looking at reality that are often worlds apart. "Mathematically, you can even talk about what happens if you take the number of particles to infinity, which is precisely what I did."Īnswering this question does more than merely satisfy a mathematical curiosity, says Hoeksema. Just like a painting in a museum that only makes sense when you look at it from a distance," explains Hoeksema. The more particles there are, and the further you step back, how sharper this picture becomes. These roughly correspond to how many particles there are at any given spot. "Instead of individual particles, you suddenly start to see vague colors or shades of gray. In his thesis, mathematician Jasper Hoeksema has investigated the following question: what happens if you put many particles together, take a few steps back, and squint your eyes a bit? Particles can be molecules, bacteria, pixels, plants, planets, or sometimes even people. If you look at the world with a microscope, or at the universe with a telescope, it turns out that many things can be described in terms of particles: particles that move, wiggle, hop or even pop in and out of existence.
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