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We study heat transport in two systems without momentum conservation: ahydrodynamic system, and a holographic system with spatially dependent,massless scalar fields. When momentum dissipates slowly, there is awell-defined, coherent collective excitation in the AC heat conductivity, and acrossover between sound-like and diffusive transport at small and largedistance scales. When momentum dissipates quickly, there is no such excitationin the incoherent AC heat conductivity, and diffusion dominates at all distancescales. For a critical value of the momentum dissipation rate, we compute exactexpressions for the Green's functions of our holographic system due to anemergent gravitational self-duality, similar to electric/magnetic duality, andSL(2,R) symmetries. We extend the coherent/incoherent classification toexamples of charge transport in other holographic systems: probe brane theoriesand neutral theories with non-Maxwell actions.

Momentum dissipation and effective theories of coherent and incoherent transport