Effective Entropy Production and Thermodynamic Uncertainty Relation of Active Brownian Particles

TL;DR

This paper develops a coarse-grained thermodynamic framework for active Brownian particles, introducing effective entropy production and thermodynamic uncertainty relations to better understand their irreversibility and phase separation phenomena.

Contribution

It introduces a new coarse-grained Fokker-Planck approach to define effective entropy production in active matter systems, linking it to phase separation and thermodynamic bounds.

Findings

Effective entropy production quantifies irreversibility in active particles.

The framework captures interface and defect phenomena in motility-induced phase separation.

Derived entropic bounds and TUR for generalized currents in active matter.

Abstract

Understanding stochastic thermodynamics of active Brownian particles (ABPs) system has been an important topic in very recent years. In this article we study a general model of active Brownian particle systems by introducing a coarse-grained Fokker-Planck equation, which allows us to identify an effective entropy production along a stochastic trajectory, wherein an activity and configuration dependent diffusion coefficient comes into play with an important role. Although the hidden component between the true entropy production and the effective one is dominant, the effective entropy production still act as a reliable measure to quantify the dynamical irreversibility, capturing important phenomenon such as the interface and defects of motility induced phase separation (MIPS). Furthermore, in this framework, we are able to obtain the entropic bound as well as TUR associated with any generalized currents in the systems. We expect the new conceptual quantities proposed here to be broadly used in the context of active matter.

Equations

$$\dot{\mathbf{x}}(t)=\mu \mathbf{F}(\mathbf{x})+v_0\mathbf{n}(t)+\mathbf{\xi }(t),$$

$${\dot{\theta }}_i(t)={\zeta }_i(t)$$

$$\dot{\mathbf{x}}(t)=\mu \mathbf{F}(\mathbf{x})+\mathbf{\xi }(t)+{\mathbf{\eta }}^A(t).$$

$$⟨{\eta }_i^A(t){\eta }_j^A(s)⟩=\frac{v_0^2}{3}{e}^{-|t-s|/{\tau }_p}{\delta }_{ij},$$

$${\partial }_{t}P(\mathbf{x},t)\approx -\sum _{i=1}^N{\partial }_{x_i}{J}_i(\mathbf{x},t).$$

$$J_i(\mathbf{x},t)=D_i(\mathbf{x})\beta F_i^{eff}(\mathbf{x})P(\mathbf{x},t)-D_i(\mathbf{x}){\partial }_{x_i}P(\mathbf{x},t),$$

$$D_i(\mathbf{x})=D_t+D_a{\left[1-\beta \tau {\partial }_{x_i}{F}_i\left(\mathbf{x}\right)\right]}^{-1}$$

$${\Sigma }_m[\chi ]={\int }_0^{t_f}{\mathbf{F}}^{eff}(\mathbf{x}{)}^{\text{T}}\circ \dot{\mathbf{x}}dt,$$

$${\dot{S}}_{sys}\left(t\right)$$

$$=-\frac{1}{P(\mathbf{x},t)}[\frac{\partial P(\mathbf{x},t)}{\partial t}+\sum _{i=1}^N{\partial }_{x_i}P(\mathbf{x},t){|}_{\mathbf{x}(t)}{\dot{x}}_i]$$

$$=-\frac{1}{P(\mathbf{x},t)}[\frac{\partial P(\mathbf{x},t)}{\partial t}-\sum _{i=1}^N\frac{J_i(\mathbf{x},t)}{D_i(\mathbf{x})}{|}_{\mathbf{x}(t)}{\dot{x}}_i]$$

$$-\beta {\mathbf{F}}^{eff}(\mathbf{x}{)}^{\text{T}}\dot{\mathbf{x}}.$$

$${\dot{S}}_g\left(t\right)=-\frac{{\partial }_{t}P(\mathbf{x},t)}{P\left(x,t\right)}{|}_{\mathbf{x}(t)}-\sum _{i=1}^N\frac{J_i(\mathbf{x},t)}{D_i(\mathbf{x})P\left(x,t\right)}{|}_{\mathbf{x}(t)}{\dot{x}}_i.$$

$$⟨{\dot{S}}_g\left(t\right)⟩\approx \sum _{i=1}^N\int \frac{J_i^2(\mathbf{x},t)}{D_i(\mathbf{x})P(\mathbf{x},t)}d\mathbf{x}\ge 0,$$

$$⟨\dot{S_g}⟩\le ⟨\mathbf{D}\cdot {\left({\mathbf{F}}^{eff}\right)}^2⟩.$$

$$\Theta [\chi ]=\int \mathbf{\Lambda }(\mathbf{x}{)}^{\text{T}}\circ \dot{\mathbf{x}}dt,$$

$$⟨\dot{\Theta }⟩=\int \mathbf{\Lambda }(\mathbf{x}{)}^{\text{T}}\mathbf{J}(\mathbf{x},t)d\mathbf{x}.$$

$$\frac{Var[\Theta ]}{⟨\Theta {⟩}^2}\ge \frac{2}{⟨\Delta S_g⟩},$$

$$q^6(i)=\frac{1}{6}\left|\sum _{j\in N^i}\exp \left(i6{\phi }_{ij}\right)\right|,$$

Full text

I Introduction

Over the past two decades, stochastic thermodynamics has gained extensive attention for describing nonequilibrium thermodynamics of mesoscopic systemsSekimoto (2010); Seifert (2005, 2008); Jarzynski (2011); Seifert (2012). Due to the small size of such systems, fluctuations are significant, so that thermodynamic quantities become stochastic variables. This observation allows ones to generalize laws of thermodynamics at single trajectory levelGomez-Solano et al. (2011); Koski et al. (2015); Martinez et al. (2017), which leads to the study of stochastic energetics and fluctuation theorems (FT). In particular, an important universal inequality between the fluctuation in currents and thermodynamic cost, the thermodynamic uncertainty relation (TUR), has been discoveredBarato and Seifert (2015); Gingrich et al. (2016); Pigolotti et al. (2017); Pietzonka et al. (2016); Polettini et al. (2016); Koyuk et al. (2018); Horowitz and Gingrich (2017); Dechant and Sasa (2018a); Hasegawa (2018); Van Vu and Hasegawa (2019); Potts and Samuelsson (2019); Koyuk and Seifert (2019); Marsland III et al. (2019). Specifically, TURs constrain the Fano factor of an arbitrary observable current by the total entropy production, presenting a trade-off relation between precision and dissipation, and provide an alternative method to obtain a lower bound on the entropy production. Moreover, TURs make an irreplaceable contribution to our understanding of non-equilibrium phenomena (e.g., work extraction under measurement and feedbackVan Vu and Hasegawa (2019); Potts and Samuelsson (2019); Sagawa and Ueda (2012) and biological clocksBarato and Seifert (2016)) which can provide more detailed information about the systems than the second law. TUR was first proposed for biological processes by Barato and SeifertBarato and Seifert (2015) and then extended to many other situations, such as diffusion processPigolotti et al. (2017); Polettini et al. (2016), finite-time generalization Pigolotti et al. (2017); Pietzonka et al. (2016); Dechant and Sasa (2018a), periodically driven systems Koyuk et al. (2018); Koyuk and Seifert (2019), biological oscillators Hasegawa (2018); Marsland III et al. (2019); Nguyen et al. (2018); Cao et al. (2020), time-delayed systems Van Vu and Hasegawa (2018).

Very recently, the frontiers of stochastic thermodynamics have shifted to the active particle systems. Active particles form a class of nonequilibrium systems which have the ability to perform directional motion through self-propulsion by consuming energy from the environmentVicsek and Zafeiris (2012); Marchetti et al. (2013); Bechinger et al. (2016); Fily and Marchetti (2012); Buttinoni et al. (2013); Giomi et al. (2008); Peruani et al. (2006); Redner et al. (2013); Paoluzzi et al. (2016). Study of active particles encompasses a wide variety of biological and soft matter systems, such as schools of fishVicsek and Zafeiris (2012), biological microorganismsCzirók et al. (2001); Sokolov et al. (2009); Frydel (2022); Felderhof (2017); Jin and Marshall (2020) and colloidal particlesIvanov (2018); Shin (2020); Fujitani (2022). Such systems can usually show a range of typical collective behaviors including phase separationStenhammar et al. (2015, 2016); Pu et al. (2017), turbulenceDunkel et al. (2013); Rank and Voigt (2021); Ghaemi (2020), self-assemblyDu et al. (2019). It has been reported that even the purely repulsive active systems can yield a motility induced phase separation (MIPS) where particles spontaneously separate into solid-like and gas phasesFily and Marchetti (2012); Redner et al. (2013), which has attracted much attention. So far, the laws of thermodynamics, the definition of related entropy production or the construction of fluctuation theorem in active particle systems has been discussed in a few studiesFodor et al. (2016); Mandal et al. (2017); Dabelow et al. (2019); Ganguly and Chaudhuri (2013); Speck (2016); Falasco et al. (2016); Chaki and Chakrabarti (2018, 2019); Caprini et al. (2019); Speck (2018); Maggi et al. (2014); Nardini et al. (2017); Shankar and Marchetti (2018); Busiello and Maritan (2019); Szamel (2019); Crosato et al. (2019); Cao et al. (2021). Most studies require not only tracking the particles’ position, but also full information including the self-propulsionGanguly and Chaudhuri (2013); Szamel (2019); Crosato et al. (2019); Cao et al. (2021), which is a great challenge in experimentsHoang et al. (2019). Therefore, how to understand the behaviors of many-body active particle systems from a thermodynamics perspective in most practical scenarios when only partial information is available, such as the establishment of TUR, is a great challenge.

In this article, based on the approximation of mapping the active particles system to an “effective equilibrium” oneFarage et al. (2015); Wittmann et al. (2017); Fox (1986a, b), we have introduced a generalized entropy production (EP) $S_{g}$, which only needs to track the position of particles. We compare the proposed EP with definitions in other studies and demonstrate the hierarchical order between them by analyzing the degree of coarse-graining. Direct simulations help us to identify the generalized EP at each point in the phase diagram. Detailed analysis of the spatial distributions of the proposed quantity allows us to identify the interface and defects of MIPS, which means that it can unambiguously be utilized to measure the dynamic irreversibility on a macroscopic scale. Furthermore, the entropic boundsDechant and Sasa (2018b); Shiraishi et al. (2016) and generalized TURsBarato and Seifert (2015); Gingrich et al. (2016); Pigolotti et al. (2017); Pietzonka et al. (2016); Polettini et al. (2016); Koyuk et al. (2018); Horowitz and Gingrich (2017); Dechant and Sasa (2018a); Hasegawa (2018); Cao et al. (2020); Van Vu and Hasegawa (2019, 2018); Potts and Samuelsson (2019); Koyuk and Seifert (2019); Marsland III et al. (2019); Cao et al. (2021) of active systems based on the demonstrated approximation, providing a convenient tool for entropy production inferenceSeifert (2019).

II Model

We consider a homogeneous system of $N$ active Brownian particles with spatial coordinates $\bm{x}(t)=\{\bm{x}_{i}(t)\}$, self-propelling with constant velocities $v_{0}$ along its direction of orientations $\bm{n}_{i}=(\cos\theta_{i},\sin\theta_{i})$ for $i$-th particle. Assume that the particles move in a viscous medium and hydrodynamic interactions are neglected, the resulting governing equations are:

[TABLE]

[TABLE]

Here, $\bm{F}=-\nabla_{\bm{x}}U$ is the mechanical force generated from the total interactions $U(\bm{x})$ and $\mu$ is the mobility. The stochastic terms $\bm{\xi}_{i}(t)$ and $\zeta_{i}(t)$ are Gaussian white noises with correlations $\langle\xi_{i}(t)\xi_{j}(s)\rangle=2D_{t}\delta_{ij}\delta(t-s)$ and $\langle\zeta_{i}(t)\zeta_{j}(s)\rangle=2D_{r}\delta_{ij}\delta(t-s)$. The translational diffusion coefficient $D_{t}$ satisfies $D_{t}=\mu k_{B}T$ with $k_{B}$ the Boltzmann constant (which is set to be $1$ throughout the paper) and $T$ the ambient temperature. The rotational diffusion coefficient $D_{r}$ relates to persistent time as $\tau_{p}=(2D_{r})^{-1}$. To derive an explicit TUR for ABPs system, we now introduce a coarse-grained active Ornstein-Uhlenbeck process with thermal noise (AOU-T) as a direct mapping model, which reads

[TABLE]

Here, the active term $\bm{\eta}^{A}(t)$ represents the OU active components with zero mean and time correlation

[TABLE]

In the limit $\tau_{p}\rightarrow 0$, the time correlation becomes $\langle\eta_{i}(t)\eta_{j}(s)\rangle=2D_{a}\delta_{ij}\delta(t-s)$ with $D_{a}=v_{0}^{2}\tau_{p}/3$, i.e, the system reduces to an equilibrium one with effective diffusion coefficient $D_{t}+D_{a}$.

Over the past decade, study of ABPs is a very hot topic and has gained extensive research attention. In particular, stochastic thermodynamics of ABPs has become a frontier area very recently. The main motivation of the present work is to address a TUR for many-body ABPs system. Following the scheme proposed by SeifertSeifert (2005), one can define the EP of the system along a given stochastic trajectory $\chi(t)=\{\bm{x}(t)|_{t=0}^{t=t_{f}}\}$ as $S_{sys}(t)=-\ln P(\bm{x},t)$, where $P(\bm{x},t)$ is the configurational probability distribution for the state variable to take the value $\bm{x}$ at time $t$. In order to establish the framework of stochastic thermodynamics of many-body active systems, we need to obtain the Fokker-Planck equation (FPE) which governs the evolution of probability distribution $P(\bm{x},t)$Farage et al. (2015). To proceed, we adopt the Fox methodFox (1986a, b) to get an effective FPE which can best approximate the process of physical interests and make accurate predictionsSharma et al. (2017); Scacchi et al. (2019); Scacchi and Sharma (2018), which reads

[TABLE]

The probability current is given by

[TABLE]

where $\beta=1/T$, and $D_{i}(\bm{x})$ denotes a configuration-dependent diffusion coefficient given by

[TABLE]

with $\tau=\tau_{p}D_{t}/d^{2}$ a dimensionless persistence time and $d$ the typical diameter of a particle. $F_{i}^{eff}(\bm{x})=D_{i}^{-1}(\bm{x})[D_{t}F_{i}(\bm{x})-T\partial_{x_{i}}D_{i}(\bm{x})]$ gives the effective force exerting on the $i$-th particle. For a passive system in the absence of $\bm{\eta}^{A}(t)$, $D_{i}\left(\bm{x}\right)=D_{t}$ and $F_{i}^{eff}\left(\bm{x}\right)=F_{i}\left(\bm{x}\right)$, while in the limit $\tau\rightarrow 0$, $D_{i}\left(\bm{x}\right)=D_{t}+D_{a}$ and $F_{i}^{eff}\left(\bm{x}\right)=D_{t}F_{i}\left(\bm{x}\right)/(D_{t}+D_{a})$. The Fox method is an approximation which is valid in lower powers of the persistence time $\tau_{p}$, as shown in the original paperFox (1986a, b). Nevertheless, it may go beyond this by including contributions to higher orders in $\tau_{p}$Farage et al. (2015); Wittmann et al. (2017). Indeed, detailed studies of different systems demonstrate the validity of Fox approximation over a large range of $\tau_{p}$ valuesFeng and Hou (2017). Thus, one may expect that Fox approximation could be applied in a certain range of $\tau_{p}$, the exact values of which may be system-dependent. In the current system of active particles, there still exists another condition for the Fox approximation to be valid, i.e., $1-\beta\tau\partial_{i}F_{i}\left(\bm{x}\right)>0$, such that $D_{i}\left(\bm{x}\right)$ is positive in the entire area. Thus, the range of accessible $\tau_{p}$ values depends upon the specific form of the bare interaction potential.

III Effective entropy production

Based on the Fox approximation, the AOU-T equation Eq.(3) corresponds to an equivalent Langevin equation. Within this framework, $\bm{F}^{eff}(\bm{x})$ represents the effective interparticle force done on the particle which is related to the heat flux of the system. According to Sekimoto’s suggestionSekimoto (2010), we can define a generalized heat dissipation in the medium along a stochastic path $\chi(t)$ as

[TABLE]

where “$\circ$” stands for the Stratonovich product and the superscript ‘T’ means transposition. Since the effective force $\bm{F}^{eff}(\bm{x})$ is the total force done on the system including the effect of activity, the generalized heat dissipation will recover to the normal heat dissipation $\int_{0}^{t_{f}}\bm{F}(\bm{x})^{\text{T}}\circ\dot{\bm{x}}dt$ in passive systemsSeifert (2005). The difference between these two types of heat dissipation is the activity-induced extra entropy flux.

According to the Eq.(6), the change rate of the system entropy is

[TABLE]

Clearly, the final term in the third equality is related to the generalized heat dissipation in Eq.(8), i.e., $\beta\bm{F}^{eff}(\bm{x})^{\text{T}}\circ\dot{\bm{x}}=\dot{\Sigma}_{m}/T$. Then, Eq.(9) can be rewritten as a balance equation for the trajectory-dependent total EP $\dot{S}_{g}=\dot{\Sigma}_{m}/T+\dot{S}_{sys}$,

[TABLE]

By averaging over the path ensemble, we can obtain the following equation

[TABLE]

where $\int d\bm{x}\partial_{t}P(\bm{x},t)=0$ and $\langle\dot{x}_{i}|\bm{x},t\rangle\approx J_{i}(\bm{x},t)/P(\bm{x},t)$ have been used. All the information about particle activity is contained in $D_{i}\left(\bm{x}\right)$ and $J_{i}\left(\bm{x}\right)$. The second law Eq.(11) ensures that the averaged total EP must increase with time. The equality holds if all the currents $J_{i}\left(x\right)$ vanish, i.e., the system can be exactly mapped to an equivalent equilibrium system.

In general, the generalized EP $S_{g}$ along a stochastic trajectory is an apparent measure of the time-reversal symmetry broken of the system at the scale of observed trajectories. In the current work, the proposed EP can be obtained simply by tracking the trajectory $\chi(t)=\{\bm{x}(t)|_{t=0}^{t=t_{f}}\}$ for particle positions $\bm{x}(t)$. To this end, two steps have been used. Firstly, the orientational degree of freedom $\bm{n}(t)$ has been eliminated and replaced by a colored noise within a mean-field level of description. Secondly, the system with a non-Markovian colored noise is approximated to an “effective equilibrium” one on a coarse-grained time scale via the Fox method. Therefore, the instantaneous entropy production rate (EPR) $\dot{S}_{g}=\bm{F}^{eff}(\bm{x})^{\text{T}}\circ\dot{\bm{x}}/T+\frac{d}{dt}\ln P(\bm{x},t)$ defined in our work can be viewed as a “coarse-grained” measure of dynamic irreversibility of the system (In steady states, $\dot{S}_{g}=\bm{F}^{eff}(\bm{x})^{\text{T}}\circ\dot{\bm{x}}/T$). Based on this effective mapping, a clearcut TUR for the coarse-grained EPR can then be well established. Other frameworks for studying EPR and related properties for the ABPs system have been proposed. Nevertheless, many-body TUR has not been addressed so far. In the following, we elucidate the the precise hierarchy of EPs.

Firstly, in Ref.Szamel (2019), Szamel has proposed an EPR $\dot{S}_{sz}=\dot{\Sigma}_{sz}/T+\frac{d}{dt}\ln P\left(\bm{x},\bm{n};t\right)$ for the active particles with the heat dissipation $\dot{\Sigma}_{sz}=\left(\bm{F}+\mu^{-1}v_{0}\bm{n}\right)^{\text{T}}\circ\dot{\bm{x}}$. $\dot{S}_{sz}$ was constructed at the full dynamics level described by Eqs.(1) and (2), including the information of orientation trajectory with difficulty in tracking. As discussed above, the generalized heat dissipation rate $\dot{S}_{g}$ proposed by us can be regarded as a coarse-grained form of $\dot{S}_{sz}$, since the orientational degree of freedom $\bm{n}(t)$ has been reduced and the memory effects have been coarse-grained. The difference between $\dot{S}_{g}$ and $\dot{S}_{sz}$ is commonly referred to as “hidden EPR”, which can be identified as the loss of information quantifying the correlation between particle trajectory and the active termEsposito (2012); Celani et al. (2012); Kawaguchi and Nakayama (2013); Chun and Noh (2015); Wang et al. (2016); Shankar and Marchetti (2018); Dabelow et al. (2019); Busiello and Maritan (2019); Crosato et al. (2019).

Secondly, based on the AOU-T model Eq.(3), Debalow $et$ $al.$ has proposed an instantaneous EPR $\dot{S}_{da}=\bm{F}_{m}[\chi,t]^{\text{T}}\circ\dot{\bm{x}}/T$ (here $\bm{F}_{m}[\chi,t]$ is defined as the nonlocal “memory forces”) which depends not only on $\bm{x}(t)$, but also on the whole trajectory $\chi(t)=\{\bm{x}(t)|_{t=0}^{t=t_{f}}\}$ of the particles’ positionDabelow et al. (2019). In contrast, according to the definition of generalized EPR in our work, $\dot{S}_{g}=\bm{F}^{eff}(\bm{x})^{\text{T}}\circ\dot{\bm{x}}/T+\frac{d}{dt}\ln P(\bm{x},t)$, one can clearly find that it is only dependent on the current configuration $\bm{x}(t)$ in the steady state, which is easily accessible in experiments. On the other hand, since AOU-T model is a coarse-grained form of Eqs.(1) and (2), we also have another EP hierarchy: $\dot{S}_{sz}>\dot{S}_{da}$.

Thirdly, in the literature, various continuous field theories based on coarse-graining procedures have been proposed to capture the large scale physics of active particles, such as “Active Models” A, B, HCates (2019). In Ref.Nardini et al. (2017), based on Active model B, Nardini $et$ $al.$ proposed an EPR $\dot{S}_{na}$ to quantify the dynamic irreversibility of the many-body active particle systems at a macroscopic scale, even when phase separation happens. Therein, the local steady-state EPR was defined as $\dot{S}_{na}=-\frac{1}{D}\left\langle\mu_{A}\dot{\phi}\right\rangle$, where $\phi(\bm{x},t)$ denotes the fluctuating density field, $\mu_{A}$ is the additional contribution to the chemical potential due to the effect of activity and $D$ is collective diffusivity. Due to its field dependence, $\dot{S}_{na}$ is more “coarse-grained” than our version $\dot{S}_{g}$.

At last, to highlight the effect of activity and effective interactions, it would be instructive to consider a comparative case, where one can treat the active system as another “effective equilibrium” system with a high effective temperature $T_{eff}=\mu^{-1}\left(D_{t}+D_{a}\right)$, corresponding to the case in the limit $\tau_{p}\to 0$. The corresponding total EPR is given by $\dot{S}_{g}^{\left(1\right)}=\bm{F}\left(\bm{x}\right)^{\text{T}}\circ\dot{\bm{x}}/T_{eff}$. Such scheme has been reported to establish the TUR for a single hot Janus swimmer successfullyFalasco et al. (2016). The difference between $\langle\Delta S_{g}\rangle$ and $\langle\Delta S_{g}^{(1)}\rangle$, treated as some kind of the “hidden EP” discussed above, is dominant especially when the gradient of mechanical force $\partial_{x_{i}}F_{i}\left(\bm{x}\right)$ is significant.

Generally, due to the above discussion, the hierarchical structure of the mentioned EPRs can be rationalized as $\dot{S}_{sz}>\dot{S}_{da}>\dot{S}_{g}>\dot{S}_{g}^{(1)}>\dot{S}_{na}$ by clarifying their corresponding degree of coarse-graining. Such a hierarchy has a similar counterpart in Maxwell’s demon systemBarato and Seifert (2014); Horowitz and Sandberg (2014).

In the following, we also present some discussion about the physical quantities by our formulation and the Harada-Sasa relation (HSR), which provides a useful tool to calculate the heat dissipation of the nonequilibrium systemsHarada and Sasa (2005). Specifically, Harada and Sasa proposed an exact equality quantifying heat production in terms of the violation of the fluctuation dissipation relation (FDR), which has been developed for active particle systems. For instance, in Ref. Nardini et al. (2017), Nardini $et$ $al.$ have derived a generalized HSR within a field-theoretical description of active matter. Chaki and Chakrabarti have utilized the HSR to calculate the heat dissipation of a colloidal particle immersed in an active bath Chaki and Chakrabarti (2019). More recently, Jones $et$ $al.$ have analyzed the power dissipation of an active microswimmer ($Chlamydomonas$ $reinhardtii$) based on the HSR jones et al. (2019). Besides, another meaningful application of the HSR is that the seemingly hidden entropy production can be partially probed from the violation spectrum of FDR Wang et al. (2016). First of all, we state that both quantities can provide proper measures of deviation from equilibrium of many-body active particles. The difference is that in the experimental investigation, only the information of the spatial steady-state trajectories is required under our framework, while for HSR, the response spectrum and fluctuation spectrum of the system must be accessible. The amount of information contained in the violation spectrum determines how well the HSR probes the heat dissipation of the system. On the one hand, in small stochastic systems, it is a nontrivial task to directly measure the response functions, whereas details about spatial trajectories are easily observed. In fact, determining the response spectrum requires measuring each frequency separately, which must cover the high frequency region to ensure convergence of the integration in the HSR, substantially increasing the statistical effort Lander et al. (2019). On the other hand, to experimentally test the HSR, one needs to perturb the system. Conversely, our framework is noninvasive in experiments.

IV Entropic bounds and thermodynamic uncertainty relation(TUR)

According to Ref. Dechant and Sasa (2018b); Shiraishi et al. (2016), the entropic bounds on $\dot{S}_{g}$, stronger than the second law, can be obtained by using the Cauchy-Schwarz inequality

[TABLE]

This conclusion worthy of attention implies that the change rate of the generalized EP is bounded by an activity and configuration-dependent term. Actually, activity affects both sides of the equation, mainly through the activity dependent diffusivity $\bm{D}\left(\bm{x}\right)$ and interaction $\bm{F}^{eff}(\bm{x})$.

Now, we turn to an important universal inequality between the fluctuations in current and thermodynamic cost, the thermodynamic uncertainty relation (TUR) Barato and Seifert (2015); Pietzonka et al. (2016); Gingrich et al. (2016); Horowitz and Gingrich (2017); Pigolotti et al. (2017); Polettini et al. (2016); Van Vu and Hasegawa (2018); Potts and Samuelsson (2019); Dechant and Sasa (2018a); Hasegawa (2018); Van Vu and Hasegawa (2019); Koyuk et al. (2018); Koyuk and Seifert (2019); Marsland III et al. (2019); Cao et al. (2020, 2021). To see the TUR of ABPs, we consider a generalized current $\Theta[\chi]$ along a single trajectory $\chi$ defined as Van Vu and Hasegawa (2018); Dechant and Sasa (2018b)

[TABLE]

where $\bm{\Lambda}(\bm{x})=\left(\Lambda_{1}\left(\bm{x}\right),\Lambda_{2}\left(\bm{x}\right),\dots,\Lambda_{N}\left(\bm{x}\right)\right)$ is a projection operator. Using different projection operator, one can get different kinds of current, such as the moving distance of particles or the EP in a time interval. The change rate of $\Theta$ can be written as

[TABLE]

For instance, for the choice $\Lambda_{i}(\bm{x})=\delta_{ik}$ in steady state, the generalized observable current is the drift velocity of the $k-$th active particle and $\Psi_{i}(\bm{x})=\delta_{ik}D_{i}(\bm{x})$ only depends on the current configuration of the system.

Due to the markovity of effective dynamics, one can obtainDechant and Sasa (2018a)

[TABLE]

which serves as a TUR for currents based on information of particle-position trajectories in the steady states. The effect of particle activity is reflected in the activity-dependent total EP $\left\langle\Delta S_{g}\right\rangle$. Mathematically, we have approximated the colored noise by a white one, and obtained the effective Fokker-Planck equation which facilitates following derivation. Only through this mapping, can we then set up a TUR which can be numerically checked for real systems or even by experiments. Several notable points are presented as follows. Firstly, in a steady state, the change of system entropy vanishes and thus $\left\langle\Delta S_{g}\right\rangle=\left\langle\Sigma_{g}\right\rangle/T=T^{-1}\left\langle\int_{0}^{t_{f}}F^{eff}\left(\bm{x}\right)^{\text{T}}\circ\dot{\bm{x}}dt\right\rangle$. Secondly, the TUR $\langle\Delta S_{g}\rangle\geq 2\langle\Theta\rangle^{2}/Var[\Theta]$ gives the lower bound, and the entropic bound $\langle\dot{S_{g}}\rangle\leq\left\langle\bm{D}\cdot\left(\bm{F}^{eff}\right)^{2}\right\rangle$ provides the upper bound of the generalized EP. Further, as the generalized EP will decrease after coarse-grained approximation, Eq.(15) can even be used to infer the exact EP including the self-propulsion’s contribution. Thirdly, in Ref.Van Vu and Hasegawa (2019), Tan Van Vu $et$ $al$. indicated that the current fluctuation is constrained not only by the entropy production but also by the average dynamical activity in athermal AOU model, by mathematically mapping the system into an underdamped one. As with the framework in most studies, the dynamic irreversibility measure they define requires tracking the self-propelled velocities of active particles.

V Numerical Results of ABPs Model

In this part, we have discovered the generalized EP as well as the validity of entropic bound and TUR by direct numerical simulations. We consider a system with $N$ disk-shaped particles in a two-dimensional $xy$ plane. Here, the exclusive-volume pair potentials of ABPs are modeled by the Weeks-Chandler-Anderson (WCA) potential: $U(r)=4\epsilon\left[\left(\frac{d}{r}\right)^{-12}-\left(\frac{d}{r}\right)^{-6}+\frac{1}{4}\right]$ for $r<2^{1/6},$ $\epsilon=k_{B}T$ and $U=0$. Here, $r=\left|\bm{x}_{1}-\bm{x}_{2}\right|$ is the particle separation, and $\epsilon$ is the interaction strength. Proper values of parameters have been chosen to illustrate our main results while $N=5120$ throughout the paper. Here, we focus on the steady-state thermodynamics of the system, where the system entropy change vanishes. We need to emphasize that the thermodynamic quantities are obtained from the trajectories generating from Eqs.(1) and (2).

For a sufficiently large particle density $\phi$, our numerical results show the MIPS: a coexistence between vapor and dense at a critical self-propulsion velocity $v_{0}^{c}$($\phi$), which is phase density dependent. To further demonstrate the phase separation, we introduce the local order parameter with respect to particle $i$Pu et al. (2017); Crosato et al. (2019):

[TABLE]

where $N^{i}$ are the closest six neighboring particles of $i$ and $\varphi_{ij}$ is the angle between the bond vector connecting particle $i$ to $j$.

V.1 TUR and entropic bound

To test the validity of TUR and entropic bound, we explore the ABPs model over $v_{0}$, $\tau_{p}$ and $\phi$. In Fig.1, we begin by choosing the current $\Theta_{1}=S_{g}$ and plot the generalized EP $\langle\Delta S_{g}\rangle$ and $\langle\Delta S_{g}^{(1)}\rangle$ with the TUR bound $B_{1}=B(\Theta_{1})=2\langle\Theta_{1}\rangle^{2}/Var[\Theta_{1}]$ over $\tau_{p}$ by varying $v_{0}$. Indeed, one can see that all the data for $B_{1}$ lie below $\langle\Delta S_{g}\rangle$, demonstrating the validity of our TUR Eq.(15). Nevertheless, if one use $\langle\Delta S_{g}^{(1)}\rangle$ instead of $\langle\Delta S_{g}\rangle$, obvious violation presents while increasing the persistent time $\tau_{p}$. Therefore, the simulation results clearly vindicate that our method to treat the many-body ABPs system correctly establishes the TUR in a large range of persistence time $\tau_{p}$.

In Fig.2(a), we also plot how the generalized EP, $\langle\Delta S_{g}\rangle$ and $\langle\Delta S_{g}^{(1)}\rangle$, and TUR bound behave for different particle density $\phi$. We find that the difference between $\langle\Delta S_{g}\rangle$ and $\langle\Delta S_{g}^{(1)}\rangle$ (normalized by the particles density) increases for larger $\phi$. This means that the generalized EP $S_{g}$ we proposed may partially recover the information loss when simply treating the active particles system as an effective system with high temperature by considering the interparticle correlations at a coarse-grained level via Fox approximation. The difference between $\langle\Delta S_{g}\rangle$ and $\langle\Delta S_{g}^{(1)}\rangle$ becomes significant especially in a high density system.

Furthermore, in Fig.2(b), we numerically calculate the TUR parameter $\eta_{g}=\frac{2\langle\Theta\rangle^{2}}{Var[\Theta]\langle\Delta S_{g}\rangle}$ and $\eta_{g}^{(1)}=\frac{2\langle\Theta\rangle^{2}}{Var[\Theta]\langle\Delta S_{g}^{(1)}\rangle}$ for a large range of $\tau_{p}$ when $v_{0}=50$ and $v_{0}=100$. If $\eta_{g}\leq 1$, our TUR Eq.(15) is established, otherwise, the TUR is invalid. The TUR still holds for a quite large $\tau_{p}=10$, even though the Fox approximation might break down for such a large $\tau_{p}$. Further increasing the value of $\tau_{p}$, it can be observed that our TUR Eq.(15) also fails. In addition, we also plot the persistent time dependence of the generalized EP for larger range of $\tau_{p}$ in Fig.2(c). More interestingly, we find that $\Delta S_{g}$ has a non-monotonic dependence on the persistence time, which is not consistent with the physical expectation that increasing the persistent time displaces the active particle systems progressively away from equilibrium. The non-monotonic persistence time dependence implies that the measure of dynamic irreversibility is not monotonically related to the degree of departure from equilibrium quantified by character of the “effective equilibrium” hypothesis necessarilyFlenner and Szamel (2020). In our opinion, this nontrivial phenomenon is mainly due to the fact that the coarse-grained method may not effectively restore the irreversibility of the system in large persistent-time regime. Actually, the establishment of the breakdown of the time-reversal symmetry for active systems with significant persistent motion is of great challenge. Thus, a deeper investigation of this disconnect is still a open question and deserves further investigation.

At last, we also validate the entropic bound Eq.(12) $\langle\dot{S_{g}}\rangle\leq\left\langle\bm{D}\cdot\left(\bm{F}^{eff}\right)^{2}\right\rangle$. In Fig.2(d), we choose $\tau_{p}=0.0167$ and the entropic bound has been numerically proved by varying the self-propelling velocity $v_{0}$.

V.2 Generalized entropy production and MIPS

Another fundamental question in the context of stochastic thermodynamics is whether entropy production/dynamical irreversibility can act as a tool for typifying phase transitions. Insight into this question has been gained in some recent studies Crochik and Tomé (2005); Tomé and de Oliveira (2012); Shim et al. (2016); Noa et al. (2019); Xiao et al. (2008, 2009); Seara et al. (2021); Cao et al. (2020); Fodor et al. (2020); Tociu et al. (2019). For instance, by analyzing the majority-vote model, Noa $et$ $al.$Noa et al. (2019) have argued that there are specific hallmarks of entropy production for a given transition, whether it is continuous or discontinuous.

We now focus on the relation between the dynamical irreversibility and MIPS in ABPs system. The simulations are performed at a fixed particle density $\phi=0.768$ by varying the velocity $v_{0}$, and the MIPS occurs at a critical value $v_{0}^{c}|_{\phi=0.768}\approx 54.3$. In Fig.3(a), the steady-state generalized EPs, $\langle\Delta S_{g}\rangle$, have been shown. As $v_{0}$ increases, the systems are driven far away from equilibrium with $\langle\Delta S_{g}\rangle$ increasing. Further, we find the $\langle\Delta S_{g}\rangle-v_{0}$ derivative has inflection points and reach the maximum near the phase transition point, providing the evidence that the generalized EP $\Delta S_{g}$ can be used to indicate the large scale MIPS. On the other hand, we also calculate the local entropy production (density), $\langle\Delta S_{g}^{l}(\bm{x})\rangle$, by averaging the particles’ entropy production in a local lattice around given position $\bm{x}$. As shown in Fig.3(b), the local EP shows a strong contribution in the vicinity of the interfaces between phasesNardini et al. (2017). In addition, we find that local entropy production almost vanishes in the high density phase due to the dynamical arrest effectNemoto et al. (2019); Cagnetta et al. (2017). Thus, the coarse-grained entropy production $\Delta S_{g}$ also acts as a reliable measure to determine the boundary of MIPS. Finally, to investigate the connection between EP and defects, typical snapshots for $\phi=0.768$ and $v_{0}=80>v_{0}^{c}$ of the MIPS have been shown in Fig.4. Specifically, the structural information coded in the spatial distribution of the local order parameter for individual particles has been shown in Fig.4(a) with the generalized EP in Fig.4(b) as a contrast. The defects are found to allow for the increase in the generalized EP of active particles. To confirm this conclusion clearly, we calculate the mean EP, $\langle\Delta S_{g}\rangle_{a}$, by averaging $\Delta S_{g}$ for all particles with same $q^{6}$ in high density phase in Fig.4(c). Clearly, the mean EP $\langle\Delta S_{g}\rangle_{a}$ decreases with the local order parameter $q^{6}$ increasing as expected.

VI Conclusions

In conclusion, we focus on the thermodynamic quantities for active systems over a finite time interval in steady states. We establish the stochastic thermodynamics for many-body active particles system based on an approximate FPE obtained via a time-local approximation. By mapping the systems into an equivalent Langevin equation, one can identify a generalized trajectory-dependent EP $\Delta S_{g}$, wherein particle activity comes into play by a configuration-dependent diffusion coefficient and a many-body effective interaction force. The relationship between the generalized EP and the rich collective behaviors of active matter has been illustrated. Precisely, we utilize the generalized entropy production to identify the phase transition point, the interface and the defects in high density phases of MIPS, showing that the generalized EP acts as a tool to quantify the dynamical irreversibility on a macroscopic scale. Furthermore, the TUR and entropic bound for the currents in the steady state can be established successfully for a large range of persistent time. In contrast, we show that simply mapping the system to an equivalent one with an effective temperature does not capture the right bounds, highlighting the suitable coarse-grained approach via Fox approximation. Due to the link between TURs and anomalous diffusionHartich and Godec (2021), our results may help to bound the timescale of anomalous kinetics in active systems. We believe that our work can provide a deeper understanding of the stochastic thermodynamics in many-body active particles system.

Author Contributions

Z.-Y.C. and J.S. contributed equally to this work.

Acknowledgments

This work is supported by MOST(2018YFA0208702), NSFC (32090044, 21973085, 21833007, 21790350).

Author declarations

The authors have no conflicts to disclose.

Data availability

The data that support the findings of this study are available from the corresponding author upon reasonable request.