Universal Spatial Structure of Nonequilibrium Steady States

We describe a large family of nonequilibrium steady states (NESS) corresponding to forced flows over obstacles. The spatial structure at large distances from the obstacle is shown to be universal, and can be quantitatively characterized in terms of certain collective modes of the strongly coupled many body system, which we define in this work. In holography, these modes are spatial analogues of quasinormal modes, which are known to be responsible for universal aspects of relaxation of time dependent systems. These modes can be both hydrodynamical or nonhydrodynamical in origin. The decay lengths of the hydrodynamic modes are set by $\eta /s$, the shear viscosity over entropy density ratio, suggesting a new route to experimentally measuring this ratio. We also point out a new class of nonequilibrium phase transitions, across which the spatial structure of the NESS undergoes a dramatic change, characterized by the properties of the spectrum of these spatial collective modes.

Figure 1

Schematic representation of the NESS considered, showing the imprint of spatial collective modes which describe the return to equilibrium far from an obstacle. The vertical axis $f$ refers to a quantity of interest, such as the expectation value of the stress tensor, with a value $f_L$ on the left and $f_R$ on the right.

Figure 2

The discrete spectrum of spatial collective modes as a function of asymptotic flow velocity $k(v)$ for a ${\mathrm{CFT}}_3$, computed holographically using stationary perturbations of boosted Schwarzschild-${\mathrm{AdS}}_4$. Here we show the case of flow incident angle $\theta =0$ (black). There is a $(v,k)\to (-v,-k)$ symmetry that connects some of the modes shown through $v=0$. Also shown is the conformal relativistic hydrodynamic spectrum (red dashed) valid to first order in small $k$. All modes found have $\mathrm{Re}k=0$. On the downstream side, for some flow velocities $v$ there are no modes of hydrodynamic origin (blue shaded region). In the longitudinal channel there is a phase transition as the velocity is increased through $c_s$ (arrows) giving rise to discontinuities in $k_{*}$.

Figure 3

Asymptotically subsonic-to-subsonic NESS, with finite transverse velocity. We show (with black circles) ${\kappa }_{ϵ}$ for the longitudinal channel (upper panel) and ${\kappa }_{v^y}$ for the transverse channel (lower panel), as defined in Eq. (7). Also shown are the values of $\mathrm{Im}k/{ϵ}^{1/3}$ for the spatial collective modes, computed directly given the left or right moduli of the asymptotic equilibrium. The red solid lines are continuously connected to the hydrodynamic modes labeled, while the blue solid line is a nonhydrodynamic mode.

Figure 4

Demonstration of the new nonequilibrium phase transition on the downstream, right-hand side of a NESS, from $v_R<c_s$ (left column) to $v_R>c_s$ (right column). Top row: locations of the spatial collective modes at these $v_R$ in the complex $k$ plane, displaying one mode of hydrodynamic origin (red x) and one nonhydrodynamic mode (blue circle). Bottom row: Spatial profile of $ϵ$ on the right-hand side of a NESS (black circles) together with an amplitude-fit collective mode from the spectrum above with the longest decay length (solid lines).