Exact transport in an equilibrium Pareto random flight

Definitions and mathematical statement

The original question asks for transport coefficients or related physical quantities for various billiard systems; it supplies no particular table or observable. The aid instead specifies an equilibrium renewal-flight surrogate. Its predictions for a billiard require a separately justified approximation.

Let $\alpha>1$. Take independent identically distributed flight durations $\tau_j$ with survival function \[ \Pr(\tau_j>t)=(1+t)^{-\alpha}\quad(t\geq0),\qquad {\mathbb{E}}\tau_j=\frac1{\alpha-1}. \] Use the stationary equilibrium renewal process on the time line, rather than starting at a renewal at time $0$. During each flight use a fresh independent velocity with a symmetric law, speed $v$, mean zero and covariance $Q={\mathbb{E}}[V V^{\mathsf T}]$. Velocities and durations are independent. Write $V(t)$ for the stationary velocity and $X(T)=\int_0^T V(t)\,{\,\mathrm{d}} t$. This covariance convention absorbs the squared speed into $Q$.

The aid reports the exact velocity covariance and displacement second-moment matrix \[ K(t)={\mathbb{E}}[V(t)V(0)^{\mathsf T}]=Q(1+t)^{1-\alpha},\qquad \Sigma(T)={\mathbb{E}}[X(T)X(T)^{\mathsf T}] =2Q\int_0^T(T-t)(1+t)^{1-\alpha}\,{\,\mathrm{d}} t. \] Define the finite-time Einstein and truncated Green--Kubo matrices by \[ E_T=\frac{\Sigma(T)}{2T},\qquad G_T=\int_0^T K(t)\,{\,\mathrm{d}} t. \] For $\alpha>2$, the reported common finite limit is \[ \lim_{T\to\infty}E_T=\lim_{T\to\infty}G_T=\frac{Q}{\alpha-2}. \] For nonzero $Q$, this finite diffusive transport limit occurs exactly when $\alpha>2$; zero-variance directions are degenerate exceptions. At the threshold and below it, the aid gives \[ \Sigma(T)\sim2QT\log T\quad(\alpha=2),\qquad \Sigma(T)\sim\frac{2Q}{(2-\alpha)(3-\alpha)}T^{3-\alpha} \quad(1<\alpha<2), \] where the scalar asymptotics are interpreted on directions with positive $Q$-variance. In particular, for such a vector $u$, \[ \frac{u^{\mathsf T}E_Tu}{u^{\mathsf T}G_Tu}\longrightarrow\frac1{3-\alpha} \quad(1<\alpha<2). \] At $\alpha=2$, the exact expressions are \[ G_T=Q\log(1+T),\qquad E_T=Q\left[\left(1+\frac1T\right)\log(1+T)-1\right], \qquad G_T-E_T\longrightarrow Q. \] Thus the sign in the matrix difference is fixed by these definitions.

The aid also reports stable weak limits on the $T^{1/\alpha}$ scale for $1<\alpha<2$, a critical Gaussian limit \[ \frac{X(T)}{\sqrt{T\log T}}\ \Longrightarrow\ N(0,Q)\quad(\alpha=2), \] and the classical Gaussian limit $X(T)/\sqrt T\Longrightarrow N(0,2Q/(\alpha-2))$ for $\alpha>2$. The stable law depends on the full directional distribution, which is not specified merely by $Q$. At the critical exponent, \[ \frac{\Sigma(T)}{T\log T}\longrightarrow2Q, \] so the weak Gaussian covariance $Q$ is half the limiting normalized second-moment matrix. Weak convergence alone does not justify convergence of transport moments.

The aid's transfer diagnostic distinguishes two requirements. A coupled billiard displacement $Y(T)$ shares a weak limit of $X(T)/b_T$ if $(Y(T)-X(T))/b_T\to0$ in probability. Transferring second-moment transport asymptotics requires corresponding $L^2$ control, together with control of the reference moments; for example $\|Y(T)-X(T)\|_2=o(c_T)$ and $\|X(T)\|_2=O(c_T)$ suffice for equality of normalized second moments on scale $c_T^2$. No such approximation for a specified billiard is supplied.

Short English statement

An equilibrium Pareto renewal-flight model gives exact finite-time transport matrices and three regimes: finite diffusion for $\alpha>2$, logarithmic enhancement at $2$, and stronger anomalous second-moment growth below $2$. Its critical weak Gaussian covariance differs from the normalized second-moment limit, so billiard approximations must separately justify weak limits and transport coefficients.

Sources

• Local UnsolvedMath registry, permanent ID $20001411$, code catalogue entry; canonical aim-dynamical-systems-notes.json, notes[68].
• AIM, "Stochastic methods for non-equilibrium dynamical systems," Problem 26, https://aimath.org/pastworkshops/noneqdynsysproblems.pdf. The supplied verification says that "billard" is the PDF spelling; Problem 26 is the first item after "Open Problems and Questions: Thursday," between Problem 25 on shrinking strip targets and Problem 27 on martingale decompositions. This is a supplied verification report, not a claim of PDF inspection here.
• Supplied research attempt 1: "partial_result," "partially_solved," rigor label "proved," novelty confidence "low." No additional authoritative literature citation is supplied. The machine-generated aid is not peer reviewed; independently verify before citation.