Mean Field Theory

Mean Field Theory

Mean field theory comes from the domain of physics and in particular statistical mechanics where researchers are dealing with many interacting variables, such as gas molecules in a chamber. The main idea of mean field theory is to replace all interactions to any one body with an average interaction. This reduces any multi-body problem into an effective one-body problem. The ease of solving MFT problems means that some insight into the behavior of a complicated system can be obtained at a relatively low cost. This model has been adopted within game theory, giving us mean field game theory which is the study of strategic decision making in very large populations of small interacting individuals. This approach will work in many social scenarios: Whenever we have random or negative correlations, differences nicely cancel one another out and we can use some form of the mean field theory. For example, consider tracking the behavior of a swarm of bees. If you observe any one bee in the swarm its behavior is pretty erratic, making an exact prediction of that bee’s next location nearly impossible; however, keep your eye on the center of the swarm—the average—and you can detect a fairly predictable pattern. In such worlds, assuming behavior embodied by a single representative bee who averages out the flight paths of all of the bees within the swarm both simplifies and improves our ability to predict the future. And this is the nature of simple linear systems where reductionism will often work well as an approximation and we can reduce things down to a single homogenous state variable. This does however not work in nonlinear systems.

#NonlinearSystems #Physics

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