Irreversibility in scalar active turbulence: The role of topological defects

TL;DR

Understanding irreversibility in inertia-free active turbulence is addressed by a minimal hydrodynamic model (Active Model H) that couples swimmer density to the surrounding fluid via active stress and defines an informatic entropy production rate (IEPR) as a local measure of time-reversal symmetry breaking. In the noiseless limit, the dominant IEPR density is linked to enstrophy through regions of high vorticity, tying irreversibility to the flow structure generated by activity. Embedding the dynamics in a linear irreversible thermodynamics framework shows IEPR is a lower bound on the total steady-state dissipation, while defect geometry in the density field organizes the spatiotemporal irreversibility. Collectively, the results reveal that the statistics and orientation of topological defects dictate where irreversibility concentrates, offering a practical route to estimate dissipation from local flow measurements and to steer active flows by defect engineering.

Abstract

In many active systems, swimmers collectively stir the surrounding fluid to stabilize some self-sustained vortices. The resulting nonequilibrium state is often referred to as active turbulence, by analogy with the turbulence of passive fluids under external stirring. Although active turbulence clearly operates far from equilibrium, it can be challenging to pinpoint which emergent features primarily control the deviation from an equilibrium reversible dynamics. Here, we reveal that dynamical irreversibility essentially stems from singularities in the active stress. Specifically, considering the coupled dynamics of the swimmer density and the stream function, we demonstrate that the symmetries of vortical flows around defects determine the overall irreversibility. Our detailed analysis leads to identifying specific configurations of defect pairs as the dominant contribution to irreversibility.

Figures (8)

• Figure 1: For Active Model H [Eqs. \ref{['eq:phi_dynamics']} and \ref{['eq:stream_dynamics']}], the swimmer density $\phi$ (top) and the scaled stream function $\tilde{\psi} = \frac{\eta}{\zeta+\kappa} \psi$ (bottom) feature three possible steady states for different levels of activity $\zeta$: (a) laminar flow, (b) vortex, and (c) spatio-temporally chaos (referred to as the turbulent state). The black lines with arrows show the velocity field lines. Parameters $-a=b=0.1$, $\kappa=0.1$, and $\eta=1.67$. (d) Velocity spectrum $S(q)$ [Eq. \ref{['eq:S']}] as a function of wavenumber $q$ (scaled by the largest system size) for $\zeta =-1.1$.
• Figure 2: (a) Density field $\phi$ in the turbulent state. The markers indicate the locations of $+\frac{1}{2}$ (green circles) and $-\frac{1}{2}$ (red triangles) topological defects. Panels (b,c): zoomed--in view of the four squares marked in panel (a): comparison between the location of the defects, the nematic director field (black lines) along with (b) the density gradient $|\nabla \phi|$ and (c) the local curvature $H$. (d) Density of defects $\rho_{\rm D}=\langle N_{\rm D} \rangle/ L^{2}$, where $\langle N_{D} \rangle$ is the defect number averaged over time and realizations, and (e) defect lifetime $\tau_{\rm D}$, defined as the average time between pair creation and annihilation, as functions of the activity parameter $\zeta$. Same parameters as in Fig. \ref{['fig:steady_states']}.
• Figure 3: Informatic entropy production rate (IEPR) [Eq. \ref{['loc_epr']}] in steady state as a function of activity $\zeta$, when changing (a) the cost at density gradients $\kappa$, (b) the nonlinear free-energy coefficient $b$, and (c) the viscosity $\eta$. Same parameters as in Fig. \ref{['fig:steady_states']}.
• Figure 4: (a) Spatial distribution of the local IEPR from numerical simulation. The figure refers to the same turbulent state as in Figs. \ref{['fig:steady_states']}(a) and \ref{['fig:gradphi_defect']}(a). The white dashed line represent the phase boundary $\phi =0$. Panels (b,d,f,h): zoomed--in views of isolated defect with charge (b) $+\frac{1}{2}$ and (d) $-\frac{1}{2}$ and defect pairs (f,h) marked resp. with orange, pink, green and red circles in panel (a). Panels (b-i): comparison between simulations and analytical predictions. We numerically evaluate the relative angle $\Phi$ and the distance $d$ between defects [Sec. \ref{['sec:pair']}] for a proper comparison of the local IEPR with analytics. Same parameters as in Fig. \ref{['fig:steady_states']}.
• Figure 5: Analytical prediction for the local IEPR $\dot{\sigma}$ [Eq. \ref{['loc_epr']}]: (a) a pair of $+\frac{1}{2}$ defects, (b) a pair of $-\frac{1}{2}$ defects, and (c) a pair of $+ \frac{1}{2}$ and $- \frac{1}{2}$ defects. The defects are at a fixed distance $d$ within a disc of radius $R$, and the angle of orientation changes from $\Phi=0$ to $\Phi=\pi/2$. The first panel in each row shows the schematic of the configuration. For all panels, the color bars are normalized by the maximum value $\dot{\sigma}_{max}$ attained for a pair of $+\frac{1}{2}$ defects within the disc.