Divergent Conductivity in Low-Dimensional Stochastic Lattices

Definitions and mathematical statement

For the energy--momentum-conserving stochastic lattice model of the source, in a nonequilibrium stationary state, determine the finite-system conductivity's divergence order in low-dimensional unpinned regimes. Unpinned ordinarily means absence of an on-site pinning potential, but the Hamiltonian, stochastic generator, noise, boundary reservoirs, and stationary-state construction are not supplied.

Writing $\kappa_N$ for the intended conductivity, the report gives the predictions \[ \kappa_N\text{ of order }\sqrt N\quad(d=1),\qquad \kappa_N\text{ of order }\log N\quad(d=2). \] These are attributed Green--Kubo predictions, not established universal asymptotics. The source does not say whether $N$ is linear size or total site count, how conductivity is normalized, or whether order means two-sided bounds, an asymptotic equivalent, or another comparison. No constants or temperature dependence are added.

The source also asks:

1. How do random masses and random interactions change conductivity and its divergence order?
2. What happens for nonlinear interactions?
3. How does the noise strength $\gamma$ affect the behavior, including the original observation that increasing $\gamma$ appears to increase the divergence order?

The last item asks to explain an apparent behavior in the report's model; it does not assert a general monotonicity theorem. The original clause is retained even though the restored statement summarizes it more broadly as dependence on $\gamma$.

A complete statement requires the mass/interaction distributions, coupling and noise law, fixed-versus-size-dependent $\gamma$, observation direction, temperature regime, and quenched or averaged disorder convention. Green--Kubo conductivity defined through equilibrium current correlations and conductivity inferred from a driven stationary state also require a comparison hypothesis or definition; their equality is not assumed merely from the prediction. The supplied assessment reports rigorous distinctions for particular pinned, higher-dimensional, or low-dimensional unpinned models, but these do not supply a universal nonlinear/disordered theorem.

Short English statement

The question asks for the predicted square-root or logarithmic conductivity growth, its dependence on disorder and nonlinearity, and the reported effect of stronger noise. The exact model, size normalization, conductivity definition, and probabilistic regime remain unspecified.

Sources

• Local UnsolvedMath registry, permanent ID 30000834, code catalogue entry, "Divergent Conductivity in Low-Dimensional Stochastic Lattices." Proposed year 2007; proposer unspecified.
• Large Scale Stochastic Dynamics (2008), Oberwolfach Reports, pp. 2443--2530, DOI 10.4171/owr/2007/42; https://doi.org/10.4171/owr/2007/42. The displayed report year 2008 differs from the DOI and proposed year 2007.
• Supplied "Basile--Bernardin--Olla": https://doi.org/10.1103/PhysRevLett.96.204303.
• Supplied "Bernardin review and open questions": https://arxiv.org/abs/0906.3596.
• Statement status corrected_verified; the supplied note says all conductivity questions and both predicted dimensional orders were preserved while notation was restored. The original's specific apparent increase with noise is retained explicitly here.
• Literature triage checked 2026-08-21: partially solved. It reports rigorous model-specific Green--Kubo results distinguishing finite pinned/higher-dimensional conductivity from low-dimensional unpinned divergence, while general nonlinear and disorder dependence remains incomplete. Exact hypotheses and current status require independent verification. Created 2026-08-25; updated 2026-09-16.
• Registry curation by Ulam AI / UnsolvedMath Contributors, CC BY 4.0; underlying report and literature provenance remain as supplied. Unreviewed statement-only draft.