Entropy production in non-reciprocal polar active mixtures

TL;DR

This work addresses how entropy production, as a measure of time-reversal symmetry breaking, reflects collective transitions in a two-species polar active mixture with non-reciprocal couplings. It combines particle-scale Brownian Dynamics simulations with a coarse-grained continuum and a fluctuating-hydrodynamics field theory to show that the informatic entropy production rate not only grows with non-reciprocity in the chiral regime but also exhibits pronounced peaks at exceptional points predicted by the field theory. A key finding is the strong, qualitative link between entropy production and polarization-susceptibility, with the long-wavelength field theory showing that the entropy production rate scales with susceptibilities of polarization perturbations, especially near EPs where Goldstone modes are activated. The results bridge particle- and field-theoretic descriptions, suggesting polarization-susceptibility measurements as practical proxies for dissipation, and deepen understanding of irreversibility in active, non-reciprocal matter with potential implications for control strategies in living and synthetic active systems.

Abstract

The out-of-equilibrium character of active systems is often twofold, arising from both the activity itself and from non-reciprocal couplings between constituents. A well-established measure to quantify the system's distance from equilibrium is the informatic entropy production rate. Here, we ask the question whether and how the informatic entropy production rate reflects collective behaviors and transitions in an active mixture with non-reciprocal polar couplings. In such systems, non-reciprocal orientational couplings can induce chiral motion of particles. At the field-theoretical level, transitions to these time-dependent chiral states are marked by so-called critical exceptional points. Here, we show that at a particle level, the entropy production rate within the chiral states increases with the degree of non-reciprocity, provided it is sufficiently strong. Moreover, even at small degrees of non-reciprocity, the transitions via exceptional points leave clear signatures in the entropy production rate, which exhibits pronounced peaks at coupling strengths corresponding to the field-theoretical exceptional points. Overall, the increase and peaks of the entropy production rate mirror the susceptibility of the polarization vector at the particle level. This correspondence is supported by a field-theoretical analysis, which reveals that, in the long-wavelength limit, the entropy production rate scales with the susceptibilities of the polarization fields.

Figures (12)

• Figure 1: Stability diagram of the homogeneous flocking and antiflocking states against infinite-wavelength perturbations and particle simulation snapshots. The stability diagram in (a) is obtained from linear stability analyses of the continuum Eqs. \ref{['eq:continuum_eq_density']} and \ref{['eq:continuum_eq_polarization_density']} for different interspecies coupling strengths $g_{AB}$ and $g_{BA}$. The intraspecies coupling strength is set to $g_{AA}=g_{BB}=9$. Exceptional points are indicated as black lines. The reciprocal system is marked with a white line. Green, cyan, and pink triangles denote data points corresponding to the particle simulations discussed in Figs. \ref{['fig:BD_pol_sus_entropy_results_reciprocal']}, \ref{['fig:BD_pol_sus_entropy_results_anti-symmetric']}, and \ref{['fig:BD_pol_sus_entropy_results_fixed_dif-5_species']}, respectively. The Brownian Dynamics simulation snapshots correspond to (b) $g_{AB}=g_{BA}=9$, (c) $g_{AB}=g_{BA}=-9$, (d) $g_{AB}=-g_{BA}=-2.5$, (e) $g_{AB}=-g_{BA}=9$, and (f) $g_{AB}=g_{BA}-5=-1.9$. The color code for all snapshots is provided in (f): Particles are colored according to their species ($A$ and $B$ in yellowish-red and bluish colors, respectively), and the color gradient indicates particle orientation. Other parameters are specified in text. The stability diagram and snapshots are consistent with Ref. kreienkamp_klapp_2024_synchronization_exceptional_points.
• Figure 2: Particle-averaged entropy production rates from particle simulations of a reciprocal system with $g_{AB} = g_{BA}=\kappa$. In (a), the entropy production rate is shown as a function of $g_{AB}$ separately for each particle species, while in (b) the contributions from translational motion and alignment couplings are plotted separately. Other parameters are as specified in Sec. \ref{['sec:model']}.
• Figure 3: Particle simulation results for a non-reciprocal, anti-symmetric system with $g_{AB} = -g_{BA} = \delta$. The (a) susceptibility, (b) spontaneous chirality, and (c),(d) particle-averaged entropy production rates are shown as functions of $g_{AB}$. The parameter values corresponding to the snapshots in Fig. \ref{['fig:stability_diagram_snapshots']}(d) and (e) are indicated by cyan markers in (b). In (a), the susceptibility of the global polarization vector is calculated in a co-rotating frame, which is determined by time averaging over $0.03\,\tau$. In (c), the entropy production rate is shown separately for both particle species, while in (d) the contributions from translational motion and alignment couplings are plotted separately. Other parameters are as specified in Sec. \ref{['sec:model']}. The spontaneous chirality is consistent with Ref. kreienkamp_klapp_2024_synchronization_exceptional_points.
• Figure 4: Particle simulation results for a non-reciprocal system with $g_{AB} = g_{BA}-d$ with $d=5$. The (a) susceptibility of the polarization vector, (b) spontaneous chirality, and (c),(d) particle-averaged entropy production rates are shown as functions of $g_{AB}$. The green vertical lines indicate the positions of critical exceptional points, respectively ($g_{AB}^{{\rm EP}>}(d=5)\approx -1.83$ and $g_{AB}^{{\rm EP}<}(d=5)\approx -3.17$). The parameter values corresponding to the snapshots in Fig. \ref{['fig:stability_diagram_snapshots']}(d) and (e) are indicated by pink markers in (b). In (a), the susceptibility of the global polarization vector is calculated in a co-rotating frame, which is determined by time averaging over $0.03\,\tau$. In (c), the entropy production rate is shown separately for both particle species, while in (d) the contributions from translational motion and alignment couplings are plotted separately. Other parameters are as specified in Sec. \ref{['sec:model']}. The spontaneous chirality is consistent with Ref. kreienkamp_klapp_2024_synchronization_exceptional_points.
• Figure 5: Brownian Dynamics simulation snapshots of the Langevin Eqs. \ref{['eq:Langevin_r']} and \ref{['eq:Langevin_theta']}. The upper row shows particle species, and the lower row shows particle orientations, using separate color coding for clarity. The snapshots correspond to the parameter combinations (a) $g_{AB}=g_{BA}=9$, (b) $g_{AB}=g_{BA}=-9$, (c) $g_{AB}=-g_{BA}=-2.5$, (d) $g_{AB}=-g_{BA}=9$, and (e) $g_{AB}=g_{BA}-5=-1.9$. The intraspecies coupling strength is set to $g_{AA}=g_{BB}=9$. Other parameters are specified in text.