Emergence of Irreversible Macroscopic Dynamics

Definitions and mathematical statement

The source asks what additional nonmechanical probabilistic information or assumption is needed to obtain irreversible thermodynamic, macroscopic or kinetic behavior from time-reversal-invariant microscopic Newtonian or Schrödinger dynamics. It is a conceptual derivation question, not a specified universal convergence theorem.

For a classical formulation, let $\Phi_t$ be an invertible microscopic flow on phase space and let $\mathcal R$ be a time-reversal involution satisfying \[ \mathcal R\Phi_t\mathcal R=\Phi_{-t}. \] A quantum version uses unitary Schrödinger evolution and the relevant antiunitary reversal symmetry, when that symmetry is present. The source's hypothesis does not claim that every Hamiltonian has such a symmetry.

A probabilistic macroscopic derivation would additionally specify initial microscopic laws $\mu_N$, macroscopic observables $A_N$, a scaling/limit regime, and a convergence notion for distributions such as \[ (A_N\circ\Phi_t)_*\mu_N \] toward an effective kinetic or thermodynamic evolution. Initial-state restrictions, ensemble/typicality choices, decorrelation or molecular-chaos assumptions, coarse graining and limiting procedures are distinct possible inputs. The question asks which are justified and sufficient in the intended setting; none is asserted here to be a uniquely necessary or universally sufficient answer.

The target "irreversible" must also be fixed: noninvertibility of an effective semigroup, entropy production, a dissipative Lyapunov law or a time-asymmetric typicality statement are different claims. Effective coarse-grained entropy change must be distinguished from fine-grained entropy invariance under reversible measure-preserving or unitary dynamics where those quantities are defined. Changing the microscopic model itself to a stochastic irreversible law would require explaining how that addresses, rather than assumes, the requested emergence.

No microscopic interaction, initial ensemble, macroscopic variables/equation, size/time scaling, entropy notion, error estimate or time interval is supplied. Thus the record cannot identify one quantitative theorem or a unique extra assumption without further specification.

The literature triage checked 21 August 2026 explicitly calls this broad and conceptual, reports no definitive universal answer and retains it as open. This is a dated supplied assessment, not a denial of rigorous results in particular microscopic/macroscopic regimes or an independent current literature survey.

Short English statement

Which justified probabilistic assumptions connect reversible microscopic dynamics to an irreversible effective description? A precise derivation needs an initial ensemble, observables, scaling limit and meaning of irreversibility. The source is a broad conceptual question, not a claim that one universal assumption or theorem has been specified.

Sources

• Local UnsolvedMath registry, permanent ID 30000595, exact source code catalogue entry; Emergence of Irreversible Macroscopic Dynamics. Proposed year 2006, status open; corrected_verified extraction changes only standard spelling/typography and retains the conceptual scope.
• Supplied report citation: PDE and Materials (2007), Oberwolfach Reports, DOI 10.4171/owr/2006/44, pp. 2631--2698, https://doi.org/10.4171/owr/2006/44. Title year differs from DOI/proposed year.
• Supplied literature triage checked 2026-08-21, citing only "original OWR report", https://doi.org/10.4171/owr/2006/44. It classifies the question as conceptual without a precise theorem or definitive universal answer. The dated OWR-derived curation requests independent verification of time-sensitive status and cited claims.