Strongly interacting electrons can move in a neatly coordinated way, reminiscent of the movement of viscous fluids. Here, we show that in viscous flows, interactions facilitate transport, allowing conductance to exceed the fundamental Landauer's ballistic limit G ball . The effect is particularly striking for the flow through a viscous point contact, a constriction exhibiting the quantum mechanical ballistic transport at T = 0 but governed by electron hydrodynamics at elevated temperatures. We develop a theory of the ballistic-to-viscous crossover using an approach based on quasi-hydrodynamic variables. Conductance is found to obey an additive relation G = G ball + G vis , where the viscous contribution G vis dominates over G ball in the hydrodynamic limit. The superballistic, low-dissipation transport is a generic feature of viscous electronics.electron hydrodynamics | graphene | strongly correlated systems F ree electron flow through constrictions in metals is often regarded as an ultimate high-conduction charge transfer mechanism (1-5). Can conductance ever exceed the ballistic limit value? Here we show that superballistic conduction is possible for strongly interacting systems in which electron movement resembles that of viscous fluids. Electron fluids are predicted to occur in quantum-critical systems and in high-mobility conductors, so long as momentum-conserving electron-electron (ee) scattering dominates over other scattering processes (6-9). Viscous electron flows feature a host of novel transport behaviors (10)(11)(12)(13)(14)(15)(16)(17)(18)(19)(20)(21)(22). Signatures of such flows have been observed in ultraclean GaAs, graphene, and ultrapure PdCoO2 (23-26).We will see that electrons in a viscous flow can achieve through cooperation what they cannot accomplish individually. As a result, resistance and dissipation of a viscous flow can be markedly smaller than that for the free-fermion transport. As a simplest realization, we discuss viscous point contact (VPC), where correlations act as a "lubricant" facilitating the flow. The reduction in resistance arises due to the streaming effect illustrated in Fig. 1, wherein electron currents bundle up to form streams that bypass the boundaries, where momentum loss occurs. This surprising behavior is in a clear departure from the common view that regards electron interactions as an impediment for transport.A simplest VPC is a 2D constriction pictured in Fig. 1A. The interaction effects dominate in constrictions of width w exceeding the carrier collision mean free path lee (and much greater than the Fermi wavelength λF ). The VPC conductance, evaluated in the absence of impurity scattering, scales as a square of the width w and inversely with the electron viscosity η,where n and e are the carrier density and charge. In the opposite limit, lee w , the ballistic free-fermion model (1, 5) predicts the conductance G ball = 2e 2 /h N , where N ≈ 2w /λF is the number of Landauer's open transmission channels. The conductance G vis grows with width faster than G ball ...

Interactions in electron systems can lead to viscous flows in which correlations allow electrons to avoid disorder scattering, reducing momentum loss and dissipation. We illustrate this behavior in a viscous pinball model, describing electrons moving in the presence of dilute point-like defects. Conductivity is found to obey an additive relation σ = σ0 + ∆σ, with a non-interacting Drude contribution σ0 and a contribution ∆σ > 0 describing conductivity enhancement due to interactions. The quantity ∆σ is enhanced by a logarithmically large factor originating from the Stokes paradox at the hydrodynamic lengthscales and, in addition, from an effect of repeated returns to the same scatterer due to backreflection in the carrier-carrier collisions occurring at the ballistic lengthscales. The interplay between these effects is essential at the ballistic-to-viscous crossover.

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In the Bona-Masso formulation, Einstein equations are written as a set of flux conservative first order hyperbolic equations that resemble fluid dynamics equations. Based on this formulation, we construct a lattice Boltzmann model for Numerical Relativity. Our model is validated with wellestablished tests, showing good agreement with analytical solutions. Furthermore, we show that by increasing the relaxation time, we gain stability at the cost of losing accuracy, and by decreasing the lattice spacings while keeping a constant numerical diffusivity, the accuracy and stability of our simulations improves. Finally, in order to show the potential of our approach a linear scaling law for parallelisation with respect to number of CPU cores is demonstrated. Our model represents the first step in using lattice kinetic theory to solve gravitational problems.

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