## Mori-Zwanzig Formalims

*01 May 2021 10:52*

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.

- Specific questions:
- How does this relate to the old Volterra / Wiener theory of nonlinear systems, where we extract successively higher-order kernels to represent memory effects?
- How does this relate to the De Roeck / Maes / Netocny results linking autonomous evolution of macroscopic variables to Boltzmann style H-theorems for those variables? (See under nonequilibrium statistical mechanics.)

See also: Emergent Properties; Large Deviations; Macroscopic Consequences of Microscopic Interactions; Nonequilibrium Statistical Mechanics;

- Recommended:
- Alexandre J. Chorin and Ole H. Hald, Stochastic Tools in Mathematics and Science

- To read:
- Yen Ting Lin, Yifeng Tian, Daniel Livescu, Marian Anghel, "Data-driven learning for the Mori--Zwanzig formalism: a generalization of the Koopman learning framework", arxiv:2101.05873
- E. A. J. F. Peters, "Projection-operator formalism and coarse-graining", arxiv:0810.2894
- Jeroen Wouters, Valerio Lucarini, "Multi-level Dynamical Systems: Connecting the Ruelle Response Theory and the Mori-Zwanzig Approach",
Journal of Statistical Physics
**151**(2013): 850--860, arxiv:1208.3080 - Yuanran Zhu, Huan Lei, "Effective Mori-Zwanzig equation for the reduced-order modeling of stochastic systems", arxiv:2102.01377
- Robert Zwanzig, Nonequilibrium Statistical Mechanics