Transport and equipartition in a two-mass toral billiard

Definitions and mathematical statement

The source asks, for a finite-horizon toral billiard with two masses $m_1,m_2$, how to compute viscosity/momentum transfer, define heat, check an Einstein relation, and decide whether unequal masses acquire equal kinetic energy in a limit. The printed "Saini billard" is naturally read as Sinai billiard, but the geometry, particle interactions, scatterers, ensemble, conserved quantities, current, and limiting operation are not specified. Finite horizon conventionally bounds free-flight lengths; it does not by itself specify those missing data.

The aid separates different models. For two two-dimensional particles with $m_i>0$, write \[ K_i=|p_i|^2/(2m_i),\qquad H=K_1+K_2=E>0. \] On the full microcanonical Liouville energy shell, with positions independent of momenta in the admissibility constraint and no fixed total-momentum condition, it reports \[ K_1/E\sim\operatorname{Uniform}[0,1],\qquad {\mathbb{E}} K_1={\mathbb{E}} K_2=E/2. \] If the chosen billiard flow is ergodic for that full-shell measure, this gives almost-everywhere Cesàro time-average equipartition, \[ \lim_{T\to\infty}\frac1T\int_0^TK_i(t)\,{\,\mathrm{d}} t=E/2. \] It does not give pointwise convergence of instantaneous energies, nor establish ergodicity for the unspecified source system.

For two elastically colliding hard disks on a bare torus without fixed scatterers, total momentum $P=p_1+p_2$ is conserved. On the invariant shell $P=0$, the aid reports \[ K_1/K_2=m_2/m_1, \] so unequal masses do not equipartition there. Fixed scatterers can destroy this momentum constraint, and the bare-torus model is not automatically a finite-horizon Sinai model. This is a constraint diagnostic, not an unqualified counterexample to every interpretation of the source.

For a specified scalar stationary centered current $J(t)$ with finite variance and absolutely integrable autocorrelation $C(s)={\mathbb{E}}[J(0)J(s)]$, put $A(t)=\int_0^tJ(s)\,{\,\mathrm{d}} s$. The aid reports the Green--Kubo/Einstein--Helfand identity \[ \lim_{t\to\infty}\frac{{\mathbb{E}}[A(t)^2]}{2t} =\int_0^\infty C(s)\,{\,\mathrm{d}} s. \] Physical viscosity or conductivity requires the appropriate current and its temperature/volume normalization. If the intended Einstein relation instead concerns diffusion versus mobility/linear response, that additional response model is not supplied by this identity.

The aid's coboundary obstruction is that raw net inter-label energy transfer satisfies \[ Q_{12}(t)=K_2(t)-K_2(0) \] when particle exchange is the only source of kinetic-energy change. Since $0\le K_2\le E$, its Einstein--Helfand growth coefficient is zero. Similarly a net momentum increment defined as $p_2(t)-p_2(0)$ is bounded on a fixed-energy shell, but purely interparticle momentum transfer need not equal that increment in a system with external scatterer impulses. These bounded observables do not imply zero physical spatial viscosity or conductivity. Nonzero transport must be formulated using spatial, periodic-boundary-corrected currents or reservoirs, rather than a bounded label-energy difference. The record supplies neither those currents nor the estimates needed to justify integrable correlations for its billiard. All reported formulas are conditional model results, leaving the source's under-specified transport and limiting questions only partially addressed.

Short English statement

The source asks about transport and equal kinetic energies in a two-mass billiard. The supplied aid shows that the answer depends on the invariant ensemble and observable: full-shell averages equipartition, a fixed-zero-momentum shell generally does not, and bounded label-energy transfer cannot serve as a nonzero spatial transport coefficient.

Sources

• Local UnsolvedMath registry, permanent ID $20001387$, source code catalogue entry; aim-dynamical-systems-notes.json, notes[44], tag problem.
• AIM, Open Problems and Questions: Stochastic Methods for Non-Equilibrium Dynamical Systems, notes by Ben Webb, Problem 2: https://aimath.org/pastworkshops/noneqdynsysproblems.pdf. Statement status exact; source spellings and the extracted $m_1\ne m_2$ notation require interpretation. No PDF was accessed for this draft.
• Green--Kubo and Einstein--Helfand formulas are named in the aid without full authoritative references. No original remarks, literature field, or proof artifact for the constraint-sensitive diagnostic is supplied.
• Research attempt 1: partial_result, status partially_solved, classification PARTIAL-PROGRESS, rigor label proved, novelty confidence low. Created and updated 14 August 2026. This machine-generated aid is not peer reviewed; independently verify before citation.
• Ulam AI / UnsolvedMath Contributors; CC BY 4.0.