Exact diffusion and chiral area transport in a four-direction collision model

Definitions and mathematical statement

The original problem asks to calculate transport coefficients or related physical quantities for a variety of billiard systems. It specifies neither a billiard table nor a coefficient, equilibrium measure, normalization or limiting regime. The supplied aid treats the following stationary collision surrogate, not a deterministic billiard realization.

Fix a step length $\ell>0$, time per step $\tau>0$, and probabilities $p_0,p_L,p_2,p_R\geq0$ with sum $1$. Let $\theta_0$ be uniform on $\{0,\pi/2,\pi,3\pi/2\}$, and let the independent relative turns have angles $0,\pi/2,\pi,-\pi/2$ with those probabilities, respectively. Set \[ \theta_{j+1}=\theta_j+\Delta_j\pmod{2\pi},\quad V_j=\ell(\cos\theta_j,\sin\theta_j),\quad X_n=\sum_{j=0}^{n-1}V_j,\quad \lambda=p_0-p_2+i(p_L-p_R). \] For $h\geq0$, the aid's explicit lag correlation, with the indicated ordering of the vectors, is \[ C_h={\mathbb{E}}[V_hV_0^{\mathsf T}] =\frac{\ell^2}{2} \begin{pmatrix}\operatorname{Re}\lambda^h&-\operatorname{Im}\lambda^h\\ \operatorname{Im}\lambda^h&\operatorname{Re}\lambda^h\end{pmatrix}. \] Define the signed polygonal area, closing the path by the segment to the origin, and the normalized coefficients by \[ A_n=\frac12\sum_{0\leq j<k<n}(V_j\wedge V_k),\qquad d_n=\frac{{\mathbb{E}}|X_n|^2}{4n\tau},\qquad a_n=\frac{{\mathbb{E}} A_n}{n\tau}, \] where $u\wedge v=u_1v_2-u_2v_1$, so counterclockwise area is positive. The aid reports the joint formula \[ d_n+ia_n=\frac{\ell^2}{4\tau} \left[1+2\sum_{h=1}^{n-1}\left(1-\frac hn\right)\lambda^h\right]. \] For $\lambda\ne1$, it gives \[ d_n+ia_n =\frac{\ell^2}{4\tau}\frac{1+\lambda}{1-\lambda} -\frac{\ell^2\lambda(1-\lambda^n)}{2\tau n(1-\lambda)^2}, \qquad d+ia=\lim_{n\to\infty}(d_n+ia_n) =\frac{\ell^2}{4\tau}\frac{1+\lambda}{1-\lambda}. \] Thus the exact finite-window bias is the second term and is $O(n^{-1})$ for each fixed $\lambda\ne1$. The diffusion component is \[ d=\frac{\ell^2}{4\tau}\frac{1-|\lambda|^2}{|1-\lambda|^2}, \] and with $\zeta=(4\tau/\ell^2)(d+ia)$, the aid reports first-mode identification \[ \lambda=\frac{\zeta-1}{\zeta+1}. \] This identifies the first Fourier mode, not all four turn probabilities.

The aid states that both the full correlation series and its symmetrized version converge absolutely exactly when scattering is nondeterministic, equivalently $|\lambda|<1$ in this four-direction model. If one turn has probability $1$, then $\lambda\in\{1,-1,i,-i\}$, and ordinary correlation summation fails. For $\lambda=-1,i,-i$, the finite-window transport limit above nevertheless exists; it must not be identified with a convergent ordinary Green--Kubo series. Straight deterministic motion, $\lambda=1$, is excluded from the rational formula and is ballistic, with $d_n=n\ell^2/(4\tau)$ and $a_n=0$. Applying the surrogate coefficients to a genuine billiard would require a justified reduction to these stationary turn statistics, which is not supplied.

Short English statement

The broad billiard transport problem is represented by a precisely defined four-direction random collision model. Its supplied exact formulas combine diffusion and signed area drift, include the finite-time bias, and distinguish ordinary Green--Kubo summability from finite-window limits at deterministic periodic endpoints.

Sources

• Local UnsolvedMath registry, permanent ID $20001405$, code catalogue entry; canonical aim-dynamical-systems-notes.json, notes[62].
• AIM workshop list "Stochastic methods for non-equilibrium dynamical systems," Problem 20, https://aimath.org/pastworkshops/noneqdynsysproblems.pdf. The registry quotes the PDF question; this draft does not claim to have read the PDF. No additional authoritative literature citation or proposer is supplied.
• Supplied research attempt 1: result "partial_result," status "partially_solved," rigor label "proved," novelty confidence "low." The research aid is machine generated, not peer reviewed; independently verify before citation.