Local entropy production rate of run-and-tumble particles

TL;DR

This work develops a local entropy production rate (EPR) framework for non-interacting run-and-tumble particles in a thermal bath, deriving time-dependent and stationary expressions that depend solely on the stationary distribution $P(x)$ and its gradients. In the stationary regime, pi(x) equals phi(x), enabling closed-form representations of the local EPR in terms of $P$ and $Q$ (and their derivatives) without requiring currents. The authors solve exact or perturbative problems for space-dependent velocity (piecewise constant and sinusoidal) and for external force fields (harmonic and piecewise-linear potentials), validating the results against numerical simulations. They reveal simple inverse relations between EPR and density in the absence of external forces, and more intricate, multi-peak dissipation patterns under confinement, underscoring how local dissipation encodes the competition between self-propulsion and external fields. The results offer a practical pathway to spatially resolve dissipation in active matter experiments, with potential extensions to higher dimensions and interacting or density-dependent velocity schemes.

Abstract

We study the local entropy production rate and the local entropy flow in active systems composed of non-interacting run-and-tumble particles in a thermal bath. After providing generic time-dependend expressions, we focus on the stationary regime. Remarkably, in this regime the two entropies are equal and depend only on the distribution function and its spatial derivatives. We discuss in details two case studies, relevant to real situations. First, we analyze the case of space dependent speed,describing photokinetic bacteria, cosidering two different shapes of the speed, piecewise constant and sinusoidal. Finally, we investigate the case of external force fields, focusing on harmonic and linear potentials, which allow analytical treatment. In all investigated cases, we compare exact and approximated analytical results with numerical simulations.

Figures (6)

• Figure 1: Run-and-tumble particles with piecewise constant velocity field. (a) Local entropy production rate (blue) and stationary distribution (red) of RT immersed in a piecewise constant speed with $v_1=1/2$ (for $x<\lambda=1/2$) and $v_2=1$ (for $1/2<x<L=1$). Dashed lines refer to theoretical predictions, solid lines are numerical simulations ($\alpha=1$ and $D=0.1$). (b) Analytical expression of $P(x)$ as $D$ increases (increasing values from violet to yellow, see legend) (c) The corresponding local entropy production rate $\pi(x) D$.
• Figure 2: Run-and-tumble particles with sinusoidal velocity field. (a) Local entropy production rate $\pi(x)$ from numerical simulations ($\alpha=1$, $v_0=1$, $L=1$) for increasing values of $D$ (from violet to yellow, see legend). (b) Stationary distribution computed from numerical simulations (different colors indicate different $\alpha$ values, see the legend, for $D=10^{-2},v=1$), the dashed red lines are $P(x)$ computed using perturbation theory ($\alpha=0.01,10$). The solid blue line is the velocity profile. (c) Local entropy production rate $\pi(x)$ from numerical simulations (same color code as in (b)) and using perturbation theory (dashed red lines as in (b)).
• Figure 3: Run-and-tumble particle in a harmonic trap. (a) Local entropy production rate $\pi(x)$ computed through (\ref{['pi_1']}) from numerical simulations, with $\pi_1(x) \equiv \frac{\Delta^2}{4 D}(\frac{1}{R} + \frac{1}{L})$ and $\pi_2(x) \equiv \frac{\alpha Q}{2} \log \frac{R}{L}$. (b) $\phi(x)=\pi(x)$ computed from numerical data through (\ref{['pi_2']}) ($\alpha=0.08$, $\mu=v=k=1$, and $D=0.1$).
• Figure 4: Run-and-tumble particle in a harmonic trap. (a) Stationary distribution: comparison between numerical simulations (solid lines) and (\ref{['P_harm']}) ($\alpha \in [0.08,2.20]$ (see legend in (b)), $\mu=v=k=1$, and $D=0.1$). (b) $\pi(x)$ computed in numerical simulations (solid line) compared with (\ref{['eprharmexpl']}). Different colors refer to different $\alpha$ values (see legend). We consider $\mu=v=k=1$, and $D=0.1$.
• Figure 5: Run-and-tumble particle in a linear potential. (a) Stationary distribution $P(x)$ obtained from (\ref{['Pst_lin']}) ($\alpha \in [0.01,1.0]$, increasing values of $\alpha$ from violet to yellow, as reported in legend of panel (b)), $\mu=v=1$, $f_0=0.5$, and $D=0.1$). (b) The corresponding local EPR $\pi(x)$ from (\ref{['epr_lin']}).