Gradient-flow formulation for nonexponential response functions

Establish whether a nonexponential response function in the spontaneous aggregation Fokker–Planck equation can admit a gradient-flow formulation based on a metric different from the generalized Wasserstein structure used for the exponential response, or on additional state variables.

Background

The paper proves that, within a natural class of free energies consisting of a convex local entropy and a symmetric quadratic interaction energy, the spontaneous aggregation model has a generalized Wasserstein gradient-flow structure only when the response function is exponential. The authors explicitly qualify this result as non-exhaustive: it does not rule out formulations using a different metric or additional variables. Such a formulation would extend the variational framework beyond the class analyzed in the paper and could clarify whether nonexponential response laws possess hidden gradient-flow structures.

References

This does not constitute an impossibility result for every conceivable variational formulation: we do not exclude the possibility that a nonexponential response function might admit a gradient structure based on a different metric or on additional state variables.

— Gradient Flow Structure of the Spontaneous Aggregation Model  (2608.19005 - Haskovec, 19 Aug 2026) in Section 1, Introduction; Section 2, Remark following the discussion of the natural Onsager factorization