Terence Tao on Nostr: In many cases, the effective dynamics essentially decouple many of the degrees of ...

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, 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)