Entropy production rate in thermodynamically consistent flocks

TL;DR

This work analyzes the entropy production rate (EPR) in a thermodynamically consistent lattice model of aligning self-propelled particles undergoing a flocking transition. Using an exact coarse-graining to fluctuating hydrodynamics, the authors prove a direct correspondence between the microscopic EPR and the hydrodynamic IEPR, and reveal that EPR is maximal in homogeneous states while traveling-band states exhibit reduced dissipation due to strong spatial modulations and edge effects. They decompose the EPR into bulk, interfacial, and reversible contributions, showing that TB states realize a thermodynamic cycle in density–magnetization space and that edge-driven energy exchanges govern local dissipation. In the weak self-propulsion regime, EPR shows singular scaling and non-analytic traveling-band profiles, providing general insights into the energetics of nonequilibrium pattern formation in thermodynamically consistent active matter.

Abstract

We study the entropy production rate (EPR) of aligning self-propelled particles which undergo a flocking transition towards a polarized collective motion. In our thermodynamically consistent lattice model, individual self-propulsion is the exclusive source of irreversibility. We derive the fluctuating hydrodynamics for large system sizes using a controlled coarse-graining: our procedure entails an exact correspondence between the EPR evaluated at the hydrodynamic and particle-based levels. We reveal that EPR is maximal when the system adopts a homogeneous configuration, either apolar or polar, and reduced in the non-homogeneous state where a polar band travels in a apolar background due to strong spatial EPR modulations. By analyzing the latter we also show that asymmetric energetic exchanges occur at the trailing and leading edges, which we map into a thermodynamic cycle in density-polarization space. Finally, we demonstrate that the regime of weak self-propulsion features a singular scaling of EPR, and a non-analyticity of the travelling band profiles.

Figures (9)

• Figure 1: Summary of results from agranov_thermodynamically_2024 for the lattice model of thermodynamically consistent flocking. (a) Each particle can be in either one of two states ($+$ and $-$) which determine the direction of biased diffusion (respectively to the right and left). The aligning Hamiltonian $H$ [Eq. \ref{['h']}] constrains both the change of particle states and their diffusive hops. (b) Phase diagram at finite activity ($\text{Pe}=0.5$). The colored dashed and solid lines represent the spinodals and binodals, respectively. (c) Profiles of density and magnetization (respectively, $\rho$ and $m$) corresponding to a travelling band, obtained by simulation of the deterministic hydrodynamics [Eq. \ref{['d2']}] with $\text{Pe}=0.5$, $\rho_0\simeq3.62$, and $\beta^{-1}\simeq0.71$. The arrow indicates the direction of band propagation.
• Figure 2: Phase diagrams (a) at equilibrium ($\text{Pe}=0$) and (b) close to equilibrium ($\text{Pe}=0.08$). The faded blue and red curves in panel (b) represent the binodals of the equilibrium limit in panel (a), terminating at the tricritical point ($\beta^{-1}=\beta^{-1}_{\text{tri}}$). In panel (b), the binodal gap extends above the tricritical temperature ($\beta^{-1}>\beta^{-1}_{\text{tri}}$). (c) The binodal gap $\rho_l-\rho_g$ and the liquid magnetization $m_{l}$, evaluated from the numerical solution of the deterministic hydrodynamics [Eq. \ref{['d2']}] at varying $\text{Pe}$ and for $\beta^{-1}=0.8$. The blue and orange lines have slopes $1/2$ and $1$, respectively, following the analytical predictions [Eq. \ref{['expand0']}].
• Figure 3: (a) Profiles of density and magnetization with the leading and trailing interfaces in solid and dashed lines, respectively. The green dot marks the point where the interfacial profile crosses the spinodal values $(\rho = \varphi_g, m\simeq 0)$. (b) Plot of the scaled profiles [Eq. \ref{['equ:travel-scale-main']}] displaying the expected scaling at low $\text{Pe}$. The value of $x_{\text{lead}}$ here corresponds to intersection with the gas spinodal. Parameters: $\text{Pe}=0.02$, $\rho_0\simeq5.1$, $\beta^{-1}=0.8$. (c) Parametric representation of the same profile $\left[\rho(x,t),m(x,t)\right]$ with $x\in[0,\ell_s)$.
• Figure 4: The reversible EPR density $s_{\rm rev}$ [purple line, Eq. \ref{['eq:rev']}] displays opposite peaks that are localized at interfaces of the density profile [black line]. The area beneath these localized modulations is given by the difference in $\Phi$ [green line, Eq. \ref{['equ:SP']}]. Parameters: $\text{Pe}=1$, $\rho_0=5.25$.
• Figure 5: The interfacial EPR density $s_{\rm interface}$ [top row, orange line, Eq. \ref{['equ:sint']}] displays opposite peaks that are localized at interfaces of the density profile [black line]. It has a behavior similar qualitative to the reversible EPR density $s_{\rm rev}$ [bottom row, purple line, Eq. \ref{['eq:rev']}]. Here $\text{Pe}=1$ and the mean densities are $\rho_0=(5.25, 3.28, 2.62)$ in (a,b,c), respectively.