Source
Chorin, Hald and Kupferman (2002), Physica D 166:239–257, LIT-1253, read in full in NOTE-975 from the published version. The t-damping equation is the paper’s (26)–(27); the result is its Theorem 8.1 with the proof of §8. The derivation below follows it and fills its steps.
What was actually shown
Setting. A Hamiltonian H(q, p) = T(p) + V(q) on ℝ²ⁿ, with initial data drawn from ρ = Z⁻¹e^{−H}, and e^{−T}, e^{−V} decaying fast enough that boundary terms vanish in integration by parts. The first m conjugate pairs (qᵢ, pᵢ), i ≤ m, are resolved, each with its momentum. L = Σⱼ(H_pⱼ∂_qⱼ − H_qⱼ∂_pⱼ). P is the conditional expectation given (q̂, p̂), and Q = I − P. The effective Hamiltonian is ℋ(q̂, p̂) = −log∫e
The t-damping equation. The Mori–Zwanzig memory term ∫₀ᵗ Kⱼ(ϕ̂(t − s), s) ds, with its integrand replaced by the value at s = 0, is tSⱼ(ϕ̂(t)) with Sⱼ = PLQLxⱼ. So, for i ≤ m, ṗᵢ = PLpᵢ + tPLQLpᵢ and q̇ᵢ = PLqᵢ + tPLQLqᵢ.
Step 1, the Markovian part. Lpᵢ = −H_qᵢ and Lqᵢ = H_pᵢ. Differentiating ℋ under the integral, ℋ_qᵢ = P H_qᵢ and ℋ_pᵢ = P H_pᵢ. So PLpᵢ = −ℋ_qᵢ and PLqᵢ = ℋ_pᵢ: without memory, the reduced flow is Hamiltonian in ℋ and conserves it.
Step 2, factorization. Under ρ conditioned on (q̂, p̂), q̃ and p̃ are independent, since e^{−H} = e^{−T(p)}e^{−V(q)}. So P[a(p)b(q)] = P[a]·P[b].
Step 3, PLQLpᵢ. Let gᵢ = H_qᵢ − ℋ_qᵢ = V_qᵢ − ℋ_qᵢ, a function of q alone, with P gᵢ = 0. QLpᵢ = −gᵢ, so LQLpᵢ = −Σⱼ T_pⱼ ∂_qⱼgᵢ (the ∂_p terms vanish). By Step 2, PLQLpᵢ = −Σⱼ P[T_pⱼ]·P[∂_qⱼgᵢ].
- For j > m: P[T_pⱼ] = −∫∂_pⱼ(e^{−T})dp̃ / ∫e^{−T}dp̃ = 0.
- For j ≤ m: P[T_pⱼ] = ℋ_pⱼ (Step 1).
- For j ≤ m, differentiate P gᵢ = ∫gᵢe^{−V}dq̃ / ∫e^{−V}dq̃ = 0 in qⱼ: P[∂_qⱼgᵢ] − P[gᵢV_qⱼ] + P[gᵢ]P[V_qⱼ] = 0. The last term is zero, and P[gᵢℋ_qⱼ] = ℋ_qⱼP[gᵢ] = 0, so P[∂_qⱼgᵢ] = P[gᵢgⱼ].
So PLQLpᵢ = −P[gᵢ Σ_{j≤m} gⱼℋ_pⱼ].
Step 4, PLQLqᵢ. By the same steps with kᵢ = H_pᵢ − ℋ_pᵢ, a function of p alone: LQLqᵢ = −Σⱼ V_qⱼ ∂_pⱼkᵢ, P[V_qⱼ] is 0 for j > m and ℋ_qⱼ for j ≤ m, and P[∂pⱼkᵢ] = P[kᵢkⱼ]. So PLQLqᵢ = −P[kᵢ Σ{j≤m} kⱼℋ_qⱼ].
Step 5, the energy. Along the t-damping flow, by the chain rule and Step 1, the Markovian parts cancel:
dℋ/dt = Σᵢ(ℋ_qᵢq̇ᵢ + ℋ_pᵢṗᵢ) = t Σᵢ(ℋ_qᵢ PLQLqᵢ + ℋ_pᵢ PLQLpᵢ).
Since ℋ_qᵢ and ℋ_pᵢ are functions of (q̂, p̂), they move inside P:
dℋ/dt = −t P[(Σᵢ kᵢℋ_qᵢ)²] − t P[(Σᵢ gᵢℋ_pᵢ)²] ≤ 0 for t ≥ 0,
because a conditional expectation of a square is non-negative.
Check on the paper’s model (two oscillators, H = ½(x₁² + x₂² + x₃² + x₄² + x₁²x₃²), q = (x₁, x₃), p = (x₂, x₄), resolving (x₁, x₂)). Here ℋ = ½x₂² + ½x₁² + ½log(1 + x₁²) + const, k₁ = 0, and g₁ = x₁x₃² − x₁/(1 + x₁²) with x₃ | x₁ ~ N(0, 1/(1 + x₁²)). The formula gives dℋ/dt = −t x₂²·x₁²·Var(x₃² | x₁) = −2t x₁²x₂²/(1 + x₁²)². Computing dℋ/dt directly from the paper’s (28), dΦ₂/dt = −Φ₁(1 + 1/(1 + Φ₁²)) − 2tΦ₁²Φ₂/(1 + Φ₁²)², gives the same.
What could have failed: the leading-order memory term could have been energy-neutral or energy-injecting, so that the cheapest memory correction did not reproduce the decay of the conditional mean at all. Under these conditions it cannot be either.
What this does not say
- Not that the t-model predicts well. It shows the direction of the effect, not its rate. On the paper’s model it over-damps the first peak (0.39 against 0.63 at t ≈ 5.2) and is no better than the Markovian equation to t ≈ 4 (re-check in NOTE-975). No error bound is given.
- Not that the true conditional mean dissipates ℋ. In the re-check, ℋ evaluated on the Monte Carlo mean rises from 0.0091 at t = 15 to 0.0273 at t = 25, and the mean keeps a residual oscillation of about 0.1, while the t-damped solution goes to rest. The t-model’s monotone dissipation is a property of the approximation.
- Not for other projections. The linear and finite-rank projections are not positivity-preserving, so Step 5’s sign fails in general.
- Not without separability or conjugate pairs. Step 2 needs e^{−H} = e^{−T}e^{−V}; Step 1 needs each resolved coordinate resolved with its momentum.
- Not a short-memory result. Evaluating the integrand at lag zero over all of [0, t] treats the kernel as constant over the window. The paper calls the t-model a short-memory approximation; as an expansion it is valid for t short against the memory time.
- Not a Markovian augmentation. The t-model adds no memory coordinates. It is a non-autonomous equation in the resolved variables alone.
Connections
- QUESTION-222: answers in part. For Hamiltonian systems under these conditions, the leading-order memory of the eliminated variables is dissipative for the resolved free energy and grows linearly in time, and it can be computed from the projection at one instant. That gives the form of the memory’s first effect. It leaves the question’s main ask open: a finite-dimensional augmentation with k memory coordinates and a bound on its prediction error.
- THEORY-257. That criterion conditions on the coarse variable’s own history in a stationary process; optimal prediction conditions on the resolved initial value under an initial law. Neither result bears on the other directly.