Hydrodynamics of Ergodic Kob–Andersen Models

Definitions and mathematical statement

A Kob--Andersen model is a conservative kinetically constrained particle system: configurations $\eta\in\{0,1\}^{{\mathbb{Z}}^d}$ record occupancy, and particles exchange with neighboring vacancies only when the prescribed local vacancy constraints are satisfied. Write $\nu_\rho$ for the Bernoulli product equilibrium measure of particle density $0\le\rho<1$. Ergodicity here is naturally equilibrium ergodicity with respect to $\nu_\rho$; it does not imply that every individual configuration can move or belongs to the same communicating class.

The verified source question is: for every Kob--Andersen model that is ergodic at density $\rho<1$, does hydrodynamic behavior emerge on sufficiently large spatial and temporal scales? A standard meaning of such a limit is that, for suitable initial laws with a macroscopic profile and some appropriate diverging time scale $\tau(\varepsilon)$, the empirical density \[ \pi_t^\varepsilon=\varepsilon^d\sum_{x\in{\mathbb{Z}}^d} \eta_{\tau(\varepsilon)t}(x)\,\delta_{\varepsilon x} \] converges to a deterministic density $u(t,r)\,{\,\mathrm{d}} r$, tested against compactly supported spatial functions. The governing macroscopic equation, mode of convergence and scale are part of the desired hydrodynamic theory. The source says "sufficiently large" scales; it does not specify $\tau(\varepsilon)=\varepsilon^{-2}$ for every ergodic model.

The precise dimension/constraint parameters, allowed initial distributions, regularity and density range of profiles, and boundary geometry are not supplied. These cannot be omitted from a rigorous limit theorem. In particular, blocked configurations remain possible even when the equilibrium process is ergodic, so the statement must not be interpreted as hydrodynamic convergence from every microscopic initial state.

The literature assessment checked on 21 August 2026 reports ergodicity and diffusive behavior in broad finite-dimensional regimes and a hydrodynamic limit for a softly constrained perturbation. It distinguishes those results from the original hard-constraint hydrodynamic conjecture, whose general initial-data scope remains unresolved in that assessment because of blocked configurations. The partial-progress label is dated triage requiring independent verification.

Short English statement

Does every equilibrium-ergodic Kob--Andersen model below full density admit a macroscopic hydrodynamic limit on suitable large scales? A rigorous version must specify admissible initial laws and must distinguish hard constraints from the softly constrained models covered by some known results.

Sources

• Large Scale Stochastic Dynamics (2005), Oberwolfach Reports, DOI 10.4171/owr/2004/43, pp. 2229--2310, https://doi.org/10.4171/owr/2004/43; proposed year supplied as 2004.
• Literature assessment checked 21 August 2026: Shapira, "hydrodynamic limit for the Kob--Andersen model," https://arxiv.org/abs/2003.08495; Martinelli--Shapira--Toninelli, "diffusive scaling," https://doi.org/10.1214/19-AIHP1035; original OWR report, https://doi.org/10.4171/owr/2004/43.
• Local UnsolvedMath registry, permanent ID $30000125$, exact source code catalogue entry, created 25 August 2026 and updated 16 September 2026. The clean statement is labelled corrected and verified against the official report.