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In modern condensed matter physics, electron hydrodynamics emerges as a powerful framework for understanding transport when particle collisions conserve momentum over extended distances. In ultra-clean materials, such as high-m purity graphene or certain Weyl semimetals, electrons can behave like a viscous fluid rather than a simple gas of independent particles. This hydrodynamic regime arises when electron-electron interactions dominate over impurity scattering and phonon processes, enabling collective flow that reshapes how electric current, heat, and momentum propagate. The resulting nonlocal transport features challenge conventional semiclassical pictures, inviting a fluid-like description that connects microscopic interactions to observable macroscopic patterns. As researchers refine experimental probes, the hydrodynamic picture becomes a unifying language for diverse materials.

Central to this perspective is the recognition that nonlocal signals encode information about viscous and inertial forces within the electron fluid. When an electrical current is injected at one location, the surrounding electrons respond through correlated motion that extends beyond the immediate contact, producing voltage patterns and temperature distributions far away from the drive point. The interplay between boundary conditions, device geometry, and intrinsic relaxation times determines how these signals decay or persist. In clean systems, momentum-conserving collisions can transport momentum across sizable distances, thereby linking distant regions of a device. Such nonlocal responses offer a window into the fundamental transport coefficients that characterize the electron liquid.

Experiments designed to probe nonlocal resistance, current spreading, and heat flow in clean materials consistently point toward hydrodynamic behavior. By comparing measured voltages in remote probes with predictions from Navier–Stokes-like equations adapted for electron fluids, researchers extract effective viscosity and conductivity parameters. These studies demonstrate that the viscosity, akin to a measure of internal friction, governs how swiftly perturbations propagate away from disturbances. The resulting transport characteristics depart significantly from simple diffusion models, instead displaying gradients shaped by fluid momentum, boundary slip, and nonlinear effects at higher current densities. The synthesis of theory and measurement strengthens the case for a universal hydrodynamic description in suitable regimes.

Beyond qualitative agreement, quantitative modeling of electron hydrodynamics emphasizes the role of conservation laws and symmetry. In clean materials, momentum, energy, and charge conservation constrain the evolution of flow fields, while Galilean or Lorentzian invariances shape the permissible modes of motion. The effective hydrodynamic equations incorporate terms representing external forces, viscosity, and thermal diffusion, linking microscopic scattering processes to macroscopic observables. Researchers also explore how temperature affects the regime, since phonons ultimately introduce momentum-relaxing channels. By tuning experimental parameters such as carrier density and temperature, one can traverse between ballistic, hydrodynamic, and diffusive regimes, mapping a rich phase space of transport behaviors.

The geometry of a device plays a decisive role in shaping nonlocal transport in the hydrodynamic regime. Narrow channels, constrictions, and patterned edges create shear flows, vortices, and boundary layers that alter current distribution in surprising ways. Slip conditions at boundaries, whether partial or complete, influence how momentum transfers between the electron fluid and the host lattice. In some configurations, backflow or secondary current loops can emerge, producing counterintuitive voltage signals near corners or across abruptly changing cross-sections. Understanding these geometric influences is essential for interpreting measurements and designing devices that exploit or suppress hydrodynamic effects for applications in metrology and sensing.

Refinements in experimental technique have allowed more precise mapping of nonlocal signatures. Scanning probe methods, local thermometry, and multiterminal measurements enable spatially resolved views of current density and heat flow. By correlating these observations with detailed simulations, researchers disentangle hydrodynamic contributions from ballistic or diffusive backgrounds. The emerging consensus emphasizes that clean, well-controlled materials provide a unique platform where electron-electron interactions dominate enough to sustain collective motion, yet impurities remain sufficiently scarce to prevent rapid momentum relaxation. This delicate balance is the cornerstone for observing robust, nonlocal hydrodynamic transport.

At the microscopic level, electrons interact via Coulomb forces and exchange interactions that conserve total momentum in the absence of strong scattering. These interactions generate correlated motion among many electrons, akin to the coordinated flow of a classical fluid. When this collective behavior emerges, local disturbances influence distant regions, producing a spread of current and heat that cannot be captured by single-particle pictures. The hydrodynamic approach translates complex many-body dynamics into tractable equations for velocity fields, pressure, and temperature, enabling predictions for how perturbations dissipate over space and time.

The macroscopic consequences of such a fluid description are particularly vivid in clean two-dimensional materials. In graphene, for example, the relativistic-like dispersion of Dirac fermions leads to distinctive viscous effects and peculiar scaling of resistivity with temperature and density. Nonlocal measurements reveal signatures of shear viscosity and even potential quantum critical scaling in certain regimes. These results underscore how electron hydrodynamics provides a bridge between fundamental interactions and emergent transport properties, highlighting universal aspects that transcend material specifics.

The recognition of hydrodynamic transport in clean materials carries practical implications for technologies relying on precise signal transmission. Devices designed to harness viscous electron flow could exhibit reduced Joule heating in specific configurations, improving efficiency for high-speed electronics or nanoscale interconnects. Conversely, uncontrolled hydrodynamic effects might complicate conventional circuit behavior, necessitating careful engineering of contacts and layouts to manage nonlocal responses. As measurement capabilities advance, engineers can tailor geometries to either amplify desirable fluid-like transport or suppress it where traditional drift-diffusion models suffice.

In tandem with device considerations, electron hydrodynamics informs metrological pursuits by offering alternative routes to probe fundamental constants and material properties. Nonlocal responses provide sensitive gauges of viscosity, entropy production, and momentum relaxation times, enabling high-precision characterization of samples. Such techniques complement conventional conductivity or Hall measurements, delivering spatially resolved fingerprints of how electrons interact and dissipate energy. The convergence of theory, experiment, and application positions hydrodynamic transport as a meaningful tool in material science and metrology alike.

The field is moving toward a cohesive framework that unifies hydrodynamic theory with microscopic models across diverse materials. Researchers seek to identify universal scaling laws that capture how nonlocal signals depend on temperature, density, and geometry, while also accommodating material-specific features such as band structure and interaction strength. By combining numerical simulations with analytic insights, the community builds predictive capabilities that guide experiments and inform device design. The overarching aim is to transform nonlocal transport from a counterintuitive curiosity into a standard diagnostic criterion for electron correlation strength and fluid-like behavior in clean conductors.

Looking ahead, advances in fabrication, control of disorder, and high-resolution imaging promise deeper insights into electron hydrodynamics. As experimental platforms broaden—from graphene to topological materials and moiré superlattices—the range of observable hydrodynamic phenomena expands correspondingly. The ongoing dialogue between theory and experiment sharpens our understanding of how momentum-conserving collisions sculpt nonlocal transport, and it highlights the enduring relevance of a fluid-based view for next-generation electronic materials. In this evolving landscape, nonlocal transport remains a robust probe of collective electron behavior in pristine systems.