General conditions ensuring the Fisher-metric monotonicity needed for TURs

Establish, under assumptions more general than cosh-type dissipation potentials, that for any gradient flow system on a hypergraph (X, Ψ*, E, H) the monotonicity condition ∥j∥_{g_{J_{x, τ j}}} ≥ ∥j∥_{g_{J_{x, j}}} holds for all τ ∈ [0,1] and all flux vectors j, where g_{J_{x, ⋅}} denotes the Fisher Riemannian metric on the flux space J induced by the Hessian of the dual dissipation potential Ψ at (x, ⋅).

Background

In Section 5.1, the authors derive a thermodynamic uncertainty relation (TUR) by representing the dissipation rate as an integral over the Fisher metric on the flux space. The derivation hinges on a monotonicity condition along the ray τ ↦ τ j in flux space: the norm of j measured at τ j must be no smaller than the norm at j for all τ ∈ [0,1].

They verify this condition explicitly for chemical reaction networks and Markov jump processes (Appendix A) and observe that the same argument extends to cosh-type gradient flow systems. However, they do not have a proof that the condition holds in greater generality. The open problem is to establish this inequality under more general assumptions on the dissipation potential or system structure, thereby extending the TUR result beyond these specific cases.