Dissipation anomaly in gradient-driven nonequilibrium steady states

TL;DR

This work demonstrates that dissipation anomaly–the persistence of finite energy dissipation in the inviscid limit–occurs beyond turbulence in gradient-driven nonequilibrium steady states. By combining fluctuating hydrodynamics simulations with self-consistent mode-coupling theory, it shows that giant, long-range nonequilibrium fluctuations amplified by a constant gradient sustain finite dissipation as $\nu_0, D_0 \to 0$. Linear theory predicts divergence, but nonlinear mode coupling regularizes it, yielding a finite anomalous dissipation tied to the gradients of nonequilibrium fluctuations. The results reveal a non-turbulent arena for dissipation anomaly and highlight the essential role of thermal noise interacting with external driving in producing singular hydrodynamic behavior, with implications for broader gradient-driven systems and potential links to dissipative weak solutions in stochastic fluids.

Abstract

Dissipation anomaly-the persistence of finite energy dissipation in the inviscid limit-is a hallmark of turbulence, sometimes regarded as the "zeroth law" of turbulent flows. Here, we demonstrate that this phenomenon is not exclusive to turbulence. Using fluctuating hydrodynamics, we show that a simple gradient-driven nonequilibrium steady state, in which a fluid is subjected to a constant scalar gradient but remains macroscopically quiescent, also exhibits dissipation anomaly. Direct numerical simulations and self-consistent mode-coupling theory reveal that the anomaly originates from giant, long-range nonequilibrium fluctuations amplified by the imposed gradient. While linear theory predicts a divergent dissipation in the inviscid limit, nonlinear mode coupling regularizes the divergence, yielding a finite anomalous dissipation. Our findings identify a new, non-turbulent arena for dissipation anomaly and establish the interplay between thermal noise and nonequilibrium driving as a fundamental route to singular behavior in hydrodynamics.

Figures (5)

• Figure 1: Numerical results for the anomalous dissipation in our model. The parameters are set to $\rho_0=k_B T=1$, $\chi=0.01$, $L=2048$, $a_{\rm uv}=32$, and $G=0.2/L$. (a) The noise-averaged dissipation rate $\langle \dot{\varepsilon}_{\rm diss} \rangle$ as a function of $D_0$ for a fixed Schmidt number $S_c = 1$. The work rate $\langle \dot{\varepsilon}_{\rm work} \rangle$, calculated independently, is overlaid as the red symbols. (b) The quantity $D_0 \langle |\nabla \delta \psi_{\rm neq}|^2 \rangle$ as a function of $D_0$ for several Schmidt numbers.
• Figure 2: Numerical confirmation of long-range correlations in the linear regime ($D_0=2.0$). Simulation parameters are otherwise the same as in Fig. \ref{['fig1']}. (a) A typical real-space snapshot of the fluctuation field $\delta\psi$. The black arrow is the direction of the imposed gradient $G$. (b) The corresponding spatial correlation of the non-equilibrium fluctuations, $S_{\rm neq}(\bm{k})$, plotted along the $k_x$ axis ($k_y=0$). The numerical data are compared with the prediction from the linearized theory [Eq. (\ref{['eq:stfac_lin']})], shown as a dashed red line.
• Figure 3: Quantitative comparison between the numerical simulation results and the theoretical predictions. Simulation parameters are otherwise the same as in Fig. \ref{['fig1']}. The fluctuation-induced part of the dissipation rate, $\langle\dot{\varepsilon}_{\mathrm{diss}}\rangle - (D_0/\chi)G^2$, is plotted as a function of $D_0$ for several $S_c$. The numerical data are compared with the predictions of the self-consistent MCT (black dashed curves) and the linearized theory (gray dashed curve).
• Figure 4: Relaxation to the non-equilibrium steady state in the numerical simulation. The time evolution of the mean-squared fluctuation, $\langle |\delta \psi|^2 \rangle$, is plotted as a function of time $t$.
• Figure 5: Numerical results for the amplitude of the nonequilibrium fluctuations. Simulation parameters are the same as in Fig. \ref{['fig1']}. (a) The root-mean-square (RMS) amplitude, $\sqrt{\langle |\delta \psi_{\rm neq}|^2\rangle}$, as a function of $D_0$ for several Schmidt numbers $S_c$. (b) The product of the diffusion coefficient and the mean-squared amplitude, $D_0 \langle |\delta \psi_{\rm neq}|^2 \rangle$, as a function of $D_0$.