Logarithmic Sobolev Inequalities for Nonconvex Spin Systems

Definitions and mathematical statement

The supplied record asks for sufficient conditions for logarithmic Sobolev inequalities for spin systems with a nonconvex Hamiltonian, together with associated hydrodynamic-limit estimates. It does not specify a configuration space, equilibrium ensemble, dynamics, or limiting equation. The following notation describes the inequality without selecting those missing data.

For a probability space $(\Omega,\mu)$ and a nonnegative integrable function $g$ with $0<\int_\Omega g\,\mathrm d\mu<\infty$, define \[ \operatorname{Ent}_{\mu}(g) =\int_\Omega g\log\!\left(\frac{g}{\int_\Omega g\,\mathrm d\mu}\right)\,\mathrm d\mu, \] with $0\log 0=0$, whenever the integral is defined. Set $\operatorname{Ent}_{\mu}(0)=0$. Given a Dirichlet form $\mathcal E$ on $L^2(\mu)$, a logarithmic Sobolev inequality with constant $\rho>0$ means \[ \operatorname{Ent}_{\mu}(f^2) \leq \frac{2}{\rho}\mathcal E(f,f) \qquad\text{for every }f\in\operatorname{Dom}(\mathcal E). \] This fixes a convention for the constant only; the source does not specify $\mu$, $\mathcal E$, or their dependence on system size and other parameters.

Question. What explicit conditions on a spin system with a nonconvex Hamiltonian guarantee a logarithmic Sobolev inequality for its equilibrium measure and the relevant Dirichlet form, and what conditions yield the associated hydrodynamic-limit estimates?

Here nonconvexity refers to failure of convexity in the relevant spin coordinates, but the source does not identify those coordinates or their domain. The admissible spin systems, interactions, boundary conditions, constraints, dependence of $\rho$ on parameters, and meaning of the associated hydrodynamic-limit estimates remain unspecified. In particular, the source does not state whether constants must be uniform in system size, which stochastic evolution is considered, or which scaling, limiting equation, topology, or error bounds are intended. Consequently, this is a framework for the supplied research question; these missing qualifications prevent its conversion into a uniquely specified quantified theorem.

Short English statement

Find sufficient hypotheses under which a spin system with a nonconvex Hamiltonian satisfies a logarithmic Sobolev inequality and admits the intended hydrodynamic-limit estimates. The supplied record leaves the model, dynamics, uniformity requirements, and target estimates unspecified, so a precise theorem requires further source clarification.

Sources

• Supplied local UnsolvedMath registry record, permanent ID $30001484$, source problem code catalogue entry, title: Logarithmic Sobolev Inequalities for Nonconvex Spin Systems. The record associates the problem with 2010 and names no individual proposer.
• Phase Transitions (2010), Oberwolfach Reports, pp. 1431--1484, DOI 10.4171/owr/2010/24: https://doi.org/10.4171/owr/2010/24. This is the supplied authoritative report citation. The background describes Oberwolfach Reports as documenting mathematical workshops and their open-problem discussions; no problem-specific page or passage is supplied.
• Supplied literature reference labeled Menz--Otto: https://arxiv.org/abs/1307.2338. The record supplies no further bibliographic details or precise theorem supporting its assessment.
• Literature assessment dated exactly 2026-08-21: the supplied curation labels the problem open and states that partial sufficient-condition results exist for nonconvex logarithmic Sobolev inequalities while the broad nonconvex question remains unresolved. This is supplied dated curation, not an independently verified literature assessment; its status and cited claims require independent verification.
• The registry labels the statement exact and reports that its mathematical content was checked against the official Oberwolfach report, with notation, context, and question form normalized without changing the source-level claim. This verification claim is supplied provenance and has not been independently checked for this draft.