Long-range influence of a pump on a critical fluid

A pump coupled to a conserved density generates long-range modulations, resulting from the non-equilibrium nature of the dynamics. We study how these modulations are modified at the critical point where the system exhibits intrinsic long-range correlations. To do so, we consider a pump in a diffusive fluid, which is known to generate a density profile in the form of an electric dipole potential and a current in the form of a dipolar field above the critical point. We demonstrate that while the current retains its form at the critical point, the density profile changes drastically. At criticality, in $d<4$ dimensions, the deviation of the density from the average is given by ${\rm sgn}(\cos(\theta))|\cos({\theta})/r^{(d-1)}|^{1/\delta}$ at large distance $r$ from the pump and angle $\theta$ with respect to the pump's orientation. At short distances, there is a crossover to a $\cos({\theta})/r^{d-3+\eta}$ profile. Here $\delta$ and $\eta$ are Ising critical exponents. The effect of the local pump on the domain wall structure below the critical point is also considered.