Validity of Murray’s law in out-of-equilibrium adaptive Hagen–Poiseuille flows

Determine whether the generalized Murray’s law—namely, at every branching node the sum of r_ij^3 over channels with positive flux equals the sum of r_ij^3 over channels with negative flux—remains valid during out-of-equilibrium (transient) regimes of adaptive Hagen–Poiseuille flows on connected graphs with elastic, time-varying conductivities governed by the Almeida–Dilão adaptation law (g(Q)=Q^{2/3}).

Background

In this work, adaptive Hagen–Poiseuille flows are studied on channel networks with elastic, time-varying conductivities that evolve according to an adaptation law derived from energy minimization. At steady state, the model yields a generalized Murray’s law at branching nodes, equating the sum of cubes of radii for inflowing and outflowing channels.

While the generalized Murray’s law is a direct consequence of the steady-state conditions in the model, the authors explicitly raise the question of whether this relationship holds during transient, out-of-equilibrium states. They subsequently present simulations indicating that Murray’s law is only validated at steady state, but they do not establish a general proof for all out-of-equilibrium regimes, leaving the broader question open.