Entropy Production in Field Theories without Time-Reversal Symmetry: Quantifying the Non-Equilibrium Character of Active Matter

Active-matter systems operate far from equilibrium because of the continuous energy injection at the scale of constituent particles. At larger scales, described by coarse-grained models, the global entropy production rate $\mathcal{S}$ quantifies the probability ratio of forward and reversed dynamics and hence the importance of irreversibility at such scales: It vanishes whenever the coarse-grained dynamics of the active system reduces to that of an effective equilibrium model. We evaluate $\mathcal{S}$ for a class of scalar stochastic field theories describing the coarse-grained density of self-propelled particles without alignment interactions, capturing such key phenomena as motility-induced phase separation. We show how the entropy production can be decomposed locally (in real space) or spectrally (in Fourier space), allowing detailed examination of the spatial structure and correlations that underly departures from equilibrium. For phase-separated systems, the local entropy production is concentrated mainly on interfaces, with a bulk contribution that tends to zero in the weak-noise limit. In homogeneous states, we find a generalized Harada-Sasa relation that directly expresses the entropy production in terms of the wave-vector-dependent deviation from the fluctuation-dissipation relation between response functions and correlators. We discuss extensions to the case where the particle density is coupled to a momentum-conserving solvent and to situations where the particle current, rather than the density, should be chosen as the dynamical field. We expect the new conceptual tools developed here to be broadly useful in the context of active matter, allowing one to distinguish when and where activity plays an essential role in the dynamics.

Popular Summary

Flocks of birds, swarms of bacteria, and millimeter-sized robots are all examples of active matter, where individual units in an ensemble self-propel by extracting energy from the environment. The constant churn of energy drives such systems far from equilibrium: A movie of the dynamics played forward looks different than one played backward, a feature known as a breakdown of time-reversal symmetry. It is often difficult, however, to pinpoint nonequilibrium signatures by looking only at macroscopic properties. Obvious markers, such as the presence of steady-state currents, are often absent. Theoretical approaches have relied on or implied a restoration of time-reversal symmetry at a coarse-grained level. So far, there has been no clear methodology for quantifying the degree to which time reversibility is broken at the macroscopic level. We introduce a quantitative diagnosis of time-reversal-symmetry breakdown and apply it to continuous descriptions of active matter.

We consider the entropy production rate, which quantifies how strongly time reversibility is broken, and evaluate it for systems that are homogeneous, as well as separated between dense and dilute phases. In the latter case, we show that the main contribution to entropy production is due to dynamics at the boundaries that separate phases. In homogeneous systems, we provide a quantitative connection between entropy production and a violation of the fluctuation-dissipation theorem, which, when time-reversal symmetry is respected, connects the response of the system to an external perturbation to correlations.

Hopefully, our work will reduce false-positive claims on the breakdown of time-reversal symmetry in active matter and make the question accessible from experimental and numerical viewpoints. We expect this work to be broadly useful, allowing one to distinguish when and where activity plays an essential role in the dynamics.

Figure 1

Left panel: Density map of a fluctuating phase-separated droplet in Active Model B in 2D. Center panel: Local contribution to the entropy production $\sigma (\mathbf{r})$ showing a strong contribution at the interfaces. Right panel: Density and entropy production for a 1D system comprising a single domain wall, for various noise levels $D\ll a_2^2/4a_4$. The entropy production is strongly inhomogeneous, attaining a finite value as $D\to 0$ at the interface between dense and dilute regions and converging to zero in the bulk in this limit. Values of the parameters used are $a_2=-0.125$, $a_4=0.125$, $\kappa =8$, and $\lambda =2$.

Figure 2

Total entropy production ${\mathcal{S}}_{\varphi }$ for Active Model B in one dimension for the case where the system consists of a single uniform phase ($a_2>0$, $a_4>0$, red line) and for the case where it shows coexistence between a high-density and a low-density phase ($a_2<0$, $a_4>0$, green line). [Parameter values are $a_2=\pm 0.25$ for the uniform $(+)$ and phase-separated $(-)$ states, with $a_4=0.25$, $\kappa =4$, $\lambda =1$.] In the two-phase system, any putative subextensive (interfacial) term scaling as $\sqrt{D}$ would be swamped by the extensive $D^1$ contribution from bulk phases.

Figure 3

Scaling of the total entropy production as a function of $\lambda$, showing that ${\mathcal{S}}_{\varphi }\propto {\lambda }^2$, as predicted analytically. Here, $\xi =\sqrt{-(\kappa /2a_2)}$ is the passive interface width. Values of the parameters used are $(a_2=-0.5,a_4=0.5,\kappa =2,D=0.001)$, $(-0.25,0.25,4,0.001)$, and $(-0.125,0.125,8,0.001)$ for the red, green, and blue points, respectively.

Figure 4

(a) Correlation $\omega C(\mathbf{k},\omega )$ and (b) response $2DR(\mathbf{k},\omega )$ for Active Model B in one dimension, found by numerical simulations as detailed in the main text. Parameter values are $-a_2=a_4=0.25$, $k=4$, $D=0.05$, and $\lambda =8$.

Figure 5

(a) Correlator-response mismatch $\omega C-2D\tilde{R}\propto {\sigma }_{\varphi }(\mathbf{k},\omega )k^2/\omega$ and (b) normalized effective temperature $\mathcal{D}(\mathbf{k},\omega )/D$ as a function of $\mathbf{k}$ and $\omega$ for Active Model B in one dimension. Parameter values as in Fig. 4. (Control simulations for the passive limit $\lambda =0$ recover a statistically insignificant signal for both plots, not shown, confirming that our numerical algorithm respects TRS when present.)