Mori-Zwanzig Formalims

24 Jul 2023 22:09

Yet Another Inadequate Placeholder, or perhaps a confession of incompetence

This is an idea from statistical mechanics, which I understand in broad outline as follows. We have a system obeying some sort of high-dimensional, microscopic dynamics, with state (say) \( X \) that evolves deterministically. (I'm most interested in the classical case, but of course quantum time evolution is deterministic too [between observations]), say with evolution semi-group \( \rho_t \), so \( X(t) = \rho_t X(0) \). We also have a favored macroscopic observable, say \( M(t) = m(X(t)) = m \circ \rho_t X(0) \). This is in general not a deterministic system at the level of the macroscopic variable \( M \). Mori-Zwanzig is a formalism getting an autonomous stochastic dynamical system for the \( M \) level, as a deterministic function of the history of \( M \) plus noise from the "unresolved" microscopic degrees of freedom. The classic examples lead to first-order stochastic differential equations for \( M \), i.e., \( \frac{dM}{dt} = a M(t) + b \xi(t) \) where \( \xi(t) \) is white noise. (Or, really, \( dM = a M(t) dt + b dW \) to appears the stochastic-calculus gods....)

I can follow derivations about Mori-Zwanzig when I read them, but something's missing in my understanding, because I can only follow the derivations. My hope in writing this notebook, and collecting these things to read, is that if I immerse myself in it enough, it will eventually click for me. At that point, presumably, I'll get how much of it really involves physics, and how much would work for any dynamical system, or even any stochastic processes.