Hydrodynamics

Hydrodynamics is the effective theory of a many-body system on distances and times long compared with its microscopic relaxation scales. Its starting point is unusually economical: identify the quantities that are conserved, or sufficiently slowly relaxing, and write the most general constitutive relations compatible with symmetry and thermodynamics. Energy, momentum and charge then obey conservation equations, while transport coefficients such as viscosity, conductivity and diffusion constants determine how the system returns towards equilibrium. The derivative expansion makes this statement quantitative by ordering corrections according to the number of spacetime gradients. Nothing in this construction requires a weakly interacting gas or long-lived quasiparticles, which is why hydrodynamics can apply equally well to ordinary fluids, quark-gluon plasma, strongly correlated electronic matter and many other systems. Its power is also its limitation: hydrodynamics is universal because it forgets most microscopic information, so one must understand both the constraints that make the effective theory consistent and the scales at which the hydrodynamic description ceases to be reliable.

Several developments have changed what we now mean by hydrodynamic theory. Gauge-gravity duality made it possible to calculate transport and real-time response in strongly coupled quantum field theories and, through the fluid-gravity correspondence, showed directly how relativistic fluid dynamics emerges from black-hole dynamics in an appropriate long-wavelength limit. Schwinger-Keldysh effective field theory has supplied a complementary advance: dissipation and thermal noise can be placed in a single action, with dynamical KMS symmetry encoding fluctuation-dissipation relations and higher-point constraints. Relativistic hydrodynamics has also been pushed beyond naive first-order formulations by stable and causal theories with additional relaxation scales. More generally, quasihydrodynamics treats approximately conserved quantities and other long-lived modes on the same footing as the ordinary conserved densities. These ideas are now meeting questions about fluctuations, memory, branch cuts, active matter and driven steady states, so the frontier is no longer simply the computation of another transport coefficient. It is the problem of identifying the correct degrees of freedom and the correct analytic structure of response beyond the strict hydrodynamic limit.

Hydrodynamics is the central thread joining much of our work. We have studied anomalous and magnetohydrodynamic transport in strongly correlated systems, including Weyl semimetals and charge-density-wave phases, and developed relaxed and quasihydrodynamic descriptions of electrically driven non-equilibrium steady states. We have worked on hydrodynamics without boost symmetry, both as a general theoretical framework and because it is directly relevant to systems such as active matter. A second strand asks where the hydrodynamic expansion fails: we study pole collisions, the radius of convergence of gradient expansions, the relation between static and dynamical scales, and effective descriptions in which non-hydrodynamic poles or branch cuts are retained explicitly. A third strand concerns fluctuations. Using Schwinger-Keldysh effective theory, we have investigated Maxwell-Cattaneo transport, local descriptions of additional relaxation modes, and the positivity constraints that must be imposed on non-Gaussian noise if the effective action is to admit a genuine stochastic interpretation. The common aim is to turn hydrodynamics from a formal derivative expansion into a controlled effective theory with a clear domain of validity.