This short article explains how to efficiently simulate the evolution of agglomerative processes, and visualize their behavior with data animations. I use a generic, simple model for illustration purposes: atoms, initially consisting of one electron, collide and merge over time, with a pre-specified maximum number of electrons per atom: the maximum limit. Given enough time, small atoms eventually disappear, leading to a universe with heavy atoms only.
The focus is on the distribution of atom sizes over time and the number of collisions, becoming rarer and rarer as atoms get bigger and the density of atoms per unit volume decreases. You can use the method and Python implementation in various contexts by updating the algorithm accordingly, and changing the terminology: replacing atoms by particles, or small celestial bodies in the birth of a start system with planets, or molecules, or even the soap bubbles merging together. A potential simple improvement is to allow the atoms to not only merge and grow in size, but also to split or lose electrons.
Video: Atom size distribution (max = 50) evolving over time
Perhaps the most interesting feature is that you can simulate the evolution and interactions of the 1080 atoms in our universe without working with individual atoms. Indeed, I use very fast and efficiently simulations based on arrays with fewer than 100 cells, to study the macro behavior. The implementation allows you to simulate either one or hundreds of evolution paths in parallel. The second option leads to the theoretical evolving distribution of atom sizes over time. It can be customized to a variety of agglomerative processes. The context may be very different from statistical physics and could even include mergers and acquisitions in the business world, celestial mechanics, or applications in chemistry.
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The technical article, entitled Generating and Videolizing Agglomerative Processes, is accessible in the “Free Books and Articles” section, here. It contains links to my GitHub files, to easily copy and paste the code. The text highlighted in orange in this PDF document are keywords that will be incorporated in the index, when I aggregate all my related articles into books about machine learning, visualization and Python. The text highlighted in blue corresponds to external clickable links, mostly references. And red is used for internal links, pointing to a section, bibliography entry, equation, and so on. To not miss future articles, sign-up to our newsletter, here.
About the Author
Vincent Granville is a pioneering data scientist and machine learning expert, co-founder of Data Science Central (acquired by TechTarget in 2020), founder of MLTechniques.com, former VC-funded executive, author and patent owner. Vincent’s past corporate experience includes Visa, Wells Fargo, eBay, NBC, Microsoft, and CNET. Vincent is also a former post-doc at Cambridge University, and the National Institute of Statistical Sciences (NISS). He published in Journal of Number Theory, Journal of the Royal Statistical Society (Series B), and IEEE Transactions on Pattern Analysis and Machine Intelligence. He is also the author of multiple books, including “Intuitive Machine Learning and Explainable AI”, available here. Vincent lives in Washington state, and enjoys doing research on spatial stochastic processes, chaotic dynamical systems, experimental math and probabilistic number theory.