Tsuki
Terence TaoThe behavior of a high-dimensional dynamical system can, very roughly speaking, be divided into two regimes. The first is what one might call the "effective dynamics" regime, in which the complex, high-dimensional dynamics can be well approximated (in the observables that one particularly cares about, at least) by lower-dimensional effective equations or models that emerge from the more fundamental laws of motion, and are easier to understand and analyze. A classical example is the laws of thermodynamics, which can effectively govern (some of the) macroscopic behavior of a large number of interacting particles, due to mixing effects that greatly simplify the impact of most of the degrees of freedom. Another example from physics is Hooke's law, that asserts that an elastic object, such as a spring, exerts a linear restoring force to push it in the direction of its equilibrium. Similar linear restoring force phenomena can be seen across the sciences (such as climate science, biology, economics, or even political science): not as fundamental laws of nature, but as empirically observable laws that emerge from more fundamental ones. Such effective laws can provide a valuable amount of long-term stability, predictability, and simplification to the dynamical understanding of many real-world complex systems. (1/4)
In many cases, the effective dynamics essentially decouple many of the degrees of freedom, allowing one to study simpler subsystems almost in isolation, with only a few bulk variables remaining to represent all the exogenous factors. (This is for instance the case in modern economics, in which a complex economy of a large number of independent agents can be decoupled, as an initial approximation, into independent microeconomic systems, interacting with some background macroeconomic variables such as inflation, interest rates, or unemployment.)
However, there is also complementary regime of "no effective dynamics", where the hypotheses on the system state that permit simpler approximations to the dynamics to be effective break down (particularly over longer time scales), because some key variables become non-perturbative, or so correlated with other variables that statistical laws such as the law of large numbers are no longer accurate. For instance, if one pulls a spring too far from its equilibrium, then the internal structure of the spring can be impacted, and the restoring force can become nonlinear, or cease to exist entirely (or in more plain language, the spring can break if pulled too hard). (2/4)
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