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This work investigates the out-of-equilibrium dynamics of dipole andhigher-moment conserving systems with long-range interactions, drawinginspiration from trapped ion experiments in strongly tilted potentials. Weintroduce a hierarchical sequence of multipole-conserving models characterizedby power-law decaying couplings. Although the moments are always globallyconserved, adjusting the power-law exponents of the couplings induces variousregimes in which only a subset of multipole moments are effectively locallyconserved. We examine the late-time hydrodynamics analytically and numericallyusing an effective classical framework, uncovering a rich dynamical phasediagram that includes subdiffusion, conventional diffusion, and L\'evy flights.Our results are unified in an analytic reciprocal relationship that capturesthe nested hierarchy of hydrodynamics in multipole conserving systems whereonly a subset of the moments are locally conserved. Moreover, we extend ourfindings to higher dimensions and explore the emergence of long-time scales,reminiscent of pre-thermal regimes, in systems with low charge density. Lastly,we corroborate our results through state-of-the-art numerical simulations of afully quantum long-range dipole-conserving system and discuss their relevanceto trapped-ion experimental setups.

Hierarchical hydrodynamics in long-range multipole-conserving systems