Answered Discussion of Adaptive Biasing Dynamics

Definitions and mathematical statement

Let $X_t$ and $Y_t$ denote, respectively, the original dynamics and the adaptive-biasing dynamics named in the supplied record. The record does not specify their state spaces, time domain, evolution equations, initial conditions, or the adaptive-biasing mechanism.

The recorded question is: What is gained by the adaptive-biasing dynamics $Y_t$ compared with the original dynamics $X_t$?

This does not determine a precise mathematical assertion: neither the quantity to be compared nor a criterion for improvement is specified. The supplied statement-verification field describes the question as a rhetorical transition immediately followed in the report by the assertion that $Y_t$ converges essentially as fast as dynamics biased by the exact free energy. This is a supplied description, not an independently verified theorem here. The exact free energy, the benchmark dynamics, the meaning of convergence, the quantitative meaning of "essentially as fast," and the hypotheses for that comparison are absent.

The record retains an open label but marks the statement unrecoverable and says that it is not presented as a separate unsolved problem. Accordingly, no precise open conjecture or quantitative convergence claim can be recovered from the supplied evidence.

Short English statement

The record asks what adaptive biasing improves over the original dynamics, but supplies no mathematical definitions or comparison criterion. Its verification note describes an answered discussion rather than a separate open problem. A precise statement requires the missing dynamics, hypotheses, and convergence criterion.

Sources

• Supplied local registry record: permanent UnsolvedMath ID $30001621$, source problem code catalogue entry, title Answered Discussion of Adaptive Biasing Dynamics, in the Oberwolfach Reports Open Problems collection. No individual proposer is supplied; the proposed year is recorded as 2010.
• Supplied source citation: Large Scale Stochastic Dynamics (2011), Oberwolfach Reports, DOI 10.4171/owr/2010/50, pp. 2877--2975. Source URL: https://doi.org/10.4171/owr/2010/50. The supplied background describes these reports as documenting mathematical workshops and their open-problem discussions.
• Supplied literature source, labeled "original OWR report": https://doi.org/10.4171/owr/2010/50.
• Literature assessment dated exactly 2026-08-21: the supplied curation labels the item open, describes the extracted sentence as an informal methodological question, and reports that no independently checkable resolution of the comparison with the original dynamics was identified. This is supplied dated curation, not independently verified; the background explicitly requests independent verification of time-sensitive status and cited claims.
• The supplied statement-verification field says the report immediately answers the rhetorical question with a comparison to dynamics biased by the exact free energy, does not present it as a separate unsolved problem, and promotes no separate clean statement. It says the original record is retained publicly for provenance. This verification account has not been independently checked here.