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LIT-1245: Optimal prediction and the Mori–Zwanzig representation of irreversible processes

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StatusActive — read 2026-10-20 (NOTE-989) in full, from the authors' PostScript copy on Chorin's Berkeley page; worth reading for the generalized Langevin identity of the resolved variables, with conditional expectation under an invariant measure as the projection (Markovian term R(A) = E[LA|A], memory K(s, A) = E[LS⊥ˢF|A], noise S⊥ᵗF with zero conditional mean), for first-order optimal prediction as its Markovian part, and for the renormalized Hamiltonian −log∫e^{−H} in the canonical case. Read it knowing that its claim that no m-equation Markovian model beats first-order optimal prediction in mean square holds only for the instantaneous rate, that it states no fluctuation–dissipation relation and no criterion for when memory may be dropped, and that its one demonstration (m = 0, perturbative, kernels cut off at a time chosen to give a reasonable decay) shows the decay of the mean but not a faithful mean after t ≈ 2.
Version2
Tagsmathematics, natural-sciences, probabilistic-modeling
SupportsCLAIM-389 — In a stable linear Gaussian VAR(1), exact dynamical independence of a linear coarse-graining implies that its Mori–Zwanzig memory vanishes, but not conversely: memory needs coupling out of the coarse variable and back, and dependence needs only the way back, so a coarse variable can be Markov in its own law with zero memory and still be dynamically dependent
Cited as “grounds” byCLAIM-480 — Composing interactions can turn an interior into a constraint, and sometimes into a constraint of a new kind: in the chain E = ½k₁(x₁ − x₂)² + ½k₂(x₂ − x₃)², with x₂ internal and equilibrated, the boundary sees E_eff = ½·k₁k₂/(k₁ + k₂)·(x₁ − x₃)², a coupling between x₁ and x₃ that neither part has; if the interior relaxes slowly, eliminating it leaves a history-dependent constraint, a memory kernel, so interconnection fixes an effective constraint on the boundary's trajectories and separation of time scales decides only whether that constraint is approximately memoryless; and for cellular sheaves the reduced constraint can relate three boundary values where every relation among the parts is pairwise
Cited as “paper” byNOTE-989 — Chorin, Hald and Kupferman — Optimal prediction and the Mori–Zwanzig representation

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Chorin, A. J., Hald, O. H. & Kupferman, R. (2000). Optimal prediction and the Mori–Zwanzig representation of irreversible processes. Proceedings of the National Academy of Sciences 97(7): 2968–2973. doi:10.1073/pnas.97.7.2968. Open copy: PMC16175.

Read in full 2026-10-20 (NOTE-989), from the authors’ copy at https://math.berkeley.edu/~chorin/CHK.ps.

Why this is here

A3 and A5 put Mori–Zwanzig first among the physics readings: it says what becomes of eliminated degrees of freedom, which come back as memory and noise. Bears on QUESTION-222 and THEORY-257.

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