Some Thoughts on Dynamical Systems

I’ve resurrected this post from my draft graveyard after chatting with Chas Leichner about the lightly related notion of domain theory, and the interaction between computation and topos theory.

What is fiber bundle dynamics?

A fiber bundle expresses global phenomenon in terms of the output of local data.
Analogously, a geometric multi-scale modeling technique, aptly named fiber bundle dynamics, expresses macroscale data in terms of the output of microscale models.

For example: if the microscale model is molecular dynamics and the macroscale model is continuum hydrodynamics, then this formula is the Irving-Kirkwood formula that expresses stress in terms of the atomistic data from molecular dynamics. If the microscale model is replaced by Brownian dynamics, then this link is replaced by Kramer’s expression, etc.​

“…in concurrent coupling methods, one does not compute the constitutive relation within the full range of these variables – only the values that actually occur in the simulation are needed, and these might be a very small subset of the entire range.”
Ren, Seamless Multiscale Modeling of Complex Fluids via Fiber Bundle Dynamics

It’s interesting to think of (1) dependent types as fibrations and (2) a multi-scale model as a generalized transition system. Continue reading Some Thoughts on Dynamical Systems