Mean-Field Limits with Finite-Volume Effects

Definitions and mathematical statement

Source terminology. The supplied record concerns a microscopic particle system with finite-volume effects, described in its background as excluded-volume interactions, and a corresponding macroscopic mean-field system. It does not specify either system mathematically. In particular, it supplies no particle dynamics, interaction law, spatial domain, boundary conditions, scaling regime, initial data, or macroscopic evolution equations.

Problem as supplied. In the presence of finite-volume effects, rigorously justify the limiting passage from the microscopic particle system to the macroscopic mean-field system.

Unresolved formulation. A precise limit theorem cannot be stated from this record alone. The microscopic and macroscopic systems, the parameter tending to a limit, the scaling of particle size and excluded volume, the admissible data, the time interval, and the notion of convergence must first be identified. The record also does not determine whether the intended claim concerns a particular model or a class of models, or what quantifiers and hypotheses such a claim should have. These choices remain unresolved; no specific dynamics or convergence assertion is imposed here.

Short English statement

The problem asks for a rigorous derivation of macroscopic mean-field behavior from a particle system while accounting for finite-volume effects. The supplied record omits the equations and limiting regime needed for a precise theorem. Its open status is supplied dated curation, not an independently verified conclusion.

Sources

• Local UnsolvedMath registry: data/unsolvedmath/problems.json; permanent ID 30003988; exact source problem code catalogue entry; title: Mean-Field Limits with Finite-Volume Effects. The supplied record belongs to the Oberwolfach Reports Open Problems collection. No individual proposer is identified.
• Supplied source citation: Emergence of Structures in Particle Systems: Mechanics, Analysis and Computation (2019), Oberwolfach Reports, pp. 2915--2967, DOI 10.4171/owr/2018/49. Supplied source and literature-reference link, labeled Oberwolfach Report 49/2018: https://doi.org/10.4171/owr/2018/49. The record gives the proposed year as 2018; the citation's year 2019 is retained as supplied.
• Supplied literature reference: Macroscopic Limits of Systems with Excluded-Volume Interactions, https://publications.mfo.de/bitstream/handle/mfo/4281/OWR_2025_23.pdf?isAllowed=y&sequence=4.
• Literature assessment dated exactly 2026-08-21: the supplied curation labels the problem open and states that the general passage from microscopic excluded-volume systems to macroscopic laws remains unresolved, while later work treats selected weak, moderate, or strong scaling regimes. This is supplied dated curation, not independently verified literature assessment. The record expressly requests independent verification of time-sensitive status and cited claims.