Green--Kubo reduction and an exact persistent-flight transport diagnostic

Definitions and mathematical statement

Problem 12 asks to calculate transport coefficients or related physical quantities for a variety of billiard systems. It does not specify tables, scatterers, horizon, boundary conditions, driving, invariant measure, observable or transport normalization. The aid gives a general correlation reduction and an exactly solvable persistent-flight surrogate; it does not identify that surrogate with an arbitrary billiard.

For a stationary centered square-integrable collision displacement $\xi_j\in{\mathbb{R}}^p$, let $S_n=\sum_{j=0}^{n-1}\xi_j$ and $C_k={\mathbb{E}}[\xi_0\xi_k^{\mathsf T}]$. If $\sum_{k\ge1}\|C_k\|<\infty$, the aid's symmetrized Green--Kubo covariance statement is \[ \frac1n\operatorname{Cov}(S_n)=C_0+\sum_{k=1}^{n-1}(1-k/n)(C_k+C_k^{\mathsf T})\longrightarrow \Sigma=C_0+\sum_{k\ge1}(C_k+C_k^{\mathsf T}). \] This summability statement alone is not a functional central limit theorem. To pass to physical time one additionally needs the stated functional-limit, clock/time-change, remainder and uniform-integrability hypotheses. In a precise conditional framework, if the rescaled collision process converges to Brownian motion of covariance $\Sigma$, the collision clock has asymptotic rate $1/\bar\tau$, the unfinished-flight remainder is negligible on the square-root-time scale, and squared normalized physical displacements are uniformly integrable, then physical covariance per unit time is $\Sigma/\bar\tau$. With the convention $\operatorname{Cov}(Y(t))\sim2tD$, the physical diffusion tensor is \[ D=\frac{\Sigma}{2\bar\tau}. \] Thus the collision diffusion tensor $\Sigma/2$ is divided by mean flight time. This explicit factor-two convention avoids conflating covariance tensors and diffusion tensors; applicability of the required hypotheses to a billiard must be proved separately.

The aid's scalar surrogate has stationary signs $\epsilon_j\in\{-1,1\}$ with equal stationary probabilities and $\Pr(\epsilon_{j+1}=\epsilon_j)=q\in(0,1)$. Let displacement per step be $\ell\epsilon_j$, fixed flight length $\ell>0$, and step time $\tau>0$. With $r=2q-1\in(-1,1)$, \[ C_k=\ell^2r^k,\qquad D=\frac{\ell^2q}{2\tau(1-q)}. \] For $D_n=\operatorname{Var}(S_n)/(2n\tau)$, the exact finite-time bias is \[ D_n-D=-\frac{\ell^2r(1-r^n)}{\tau n(1-r)^2}. \] It is negative for persistent signs $r>0$, positive for $r<0$, and zero for $r=0$. The one-lag ratio $C_1/C_0=r$, together with $\ell,\tau$, predicts the complete geometric correlation sequence and finite-time bias in this surrogate, rather than merely its limiting coefficient.

The aid further gives a Perron path-tilt response identity. Normalize the tilt by physical step displacement, so the transition-weight matrix is $M_h(i,j)=P(i,j)e^{h\ell j}$, $i,j\in\{-1,1\}$. If $\lambda(h)$ is its Perron eigenvalue and the associated normalized tilted chain is used, its stationary velocity is $v(h)=\tau^{-1}(\log\lambda)'(h)$. Then the reported identity is \[ v'(0)=2D. \] Only if a physical forcing convention supplies $h=\beta F/2$ does this imply the Einstein relation $({\,\mathrm{d}} v/{\,\mathrm{d}} F)(0)=\beta D$. The matching of field $F$ to the tilt is an extra hypothesis, not an automatic property of a billiard model.

Finally, for centered scalar currents with positive correlation tail $C_k\sim c/k$, $c>0$, the same finite-time covariance sum gives \[ D_n\sim(c/\tau)\log n. \] Thus the usual finite diffusion coefficient fails in that diagnostic regime; summable-correlations formulas for a finite limit cannot be applied there.

The supplied verification says "billard" is printed in the source itself, almost certainly a misspelling of billiard, rather than a corpus extraction error. The registry labels the entry partially solved and reconstructed unverified. Attempt 2 is a partial result, problem partially solved at the run, internal rigor proved and novelty confidence low. The aid is machine generated, not peer reviewed, and requires independent verification before citation.

Short English statement

The broad billiard question leaves the physical system unspecified. The aid provides a Green--Kubo covariance reduction with additional conditions for physical time, and exact diffusion, finite-time bias and tilt response for a two-state persistent-flight surrogate. A positive $1/k$ correlation tail instead gives logarithmic growth of the finite-time diffusion coefficient.

Sources

• Local UnsolvedMath registry, permanent ID $20001397$, source code catalogue entry; aim-dynamical-systems-notes.json, notes[54].
• AIM, Stochastic methods for non-equilibrium dynamical systems, Problem 12: https://aimath.org/pastworkshops/noneqdynsysproblems.pdf. The source spelling "billard" is noted by the supplied verification; this draft does not claim independent PDF inspection.
• Supplied research attempt 2 and registry update August 14, 2026. The original literature field is empty, and no authoritative external citations for the Green--Kubo, time-change, Perron-tilt or surrogate formulas are supplied.