Issue 
Natl Sci Open
Volume 3, Number 4, 2024
Special Topic: Active Matter



Article Number  20230069  
Number of page(s)  12  
Section  Physics  
DOI  https://doi.org/10.1360/nso/20230069  
Published online  01 April 2024 
RESEARCH ARTICLE
Effective diffusion of a tracer in active bath: A pathintegral approach
Hefei National Research Center for Physical Sciences at the Microscale & Key Laboratory of Precision and Intelligent Chemistry, University of Science and Technology of China, Hefei 230026, China
^{*} Corresponding authors (emails: fengmk@ustc.edu.cn (Mengkai Feng); hzhlj@ustc.edu.cn (Zhonghuai Hou))
Received:
30
October
2023
Revised:
3
January
2024
Accepted:
5
February
2024
We investigate the effective diffusion of a tracer immersed in an active particle bath consisting of selfpropelled particles. Utilising the Dean’s method developed for the equilibrium bath and extending it to the nonequilibrium situation, we derive a generalized Langevin equation (GLE) for the tracer particle. The complex interactions between the tracer and bath particles are shown as a memory kernel term and two colored noise terms. To obtain the effective diffusivity of the tracer, we use path integral technique to calculate all necessary correlation functions. Calculations show the effective diffusion decreases with the persistent time of active force, and has rich behavior with the number density of bath particles, depending on different activities. All theoretical results regarding the dependence of such diffusivity on bath parameters have been confirmed by direct computer simulation.
Key words: active matter / nonequilibrium statistical mechanics / active bath / meanfield theory / path integral / tracer diffusivity
© The Author(s) 2024. Published by Science Press and EDP Sciences.
This is an Open Access article distributed under the terms of the Creative Commons Attribution License (https://creativecommons.org/licenses/by/4.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
INTRODUCTION
Active matter systems, consisting of selfpropelled units that are able to convert stored or surrounding energy into their persistent motion, provide a fresh opportunity for applications of nonequilibrium statistical mechanics [14]. In particular, active colloidal suspension can serve as an active bath that can significantly influence the motion and dynamics of passive objects submerged within them [512]. Understanding the behavior of tracer particles in the active bath is a fundamental pursuit in statistical physics and plays a crucial role in various biological[10, 1316], chemical[17, 18], and physical phenomena[8, 9, 12, 1932]. Notably, the tracer particle immersed in the active bath exhibits a distinct diffusion profile compared to its equilibrium counterpart[26, 3339]. Furthermore, studying the diffusion behavior of such a tracer is essential for unraveling the intricate dynamics that govern systems away from thermal equilibrium, providing a valuable tool to investigate the collective behavior [8, 4046] and transport properties in such media [4751]. Because of the importance and wide range of applications, understanding the dynamics of the tracer in such an active bath is desirable.
As one already knows, the classical work by Einstein laid the foundation for understanding Brownian motion, providing a framework for diffusive behavior in passive media. Subsequent advancements, such as the Langevin equation, have enriched our understanding of stochastic processes, diffusive phenomena, and the fluctuationdissipation theorem in thermal equilibrium. Nevertheless, the dynamics of particles in active baths introduce additional complexity[37, 45, 5157]. Several studies have investigated this question, often employing analytical and numerical techniques to model and characterize the motion of tracer particles within nonequilibrium media, including the active bath. For instance, Maes et al. established a generalized fluctuationresponse relation for thermal systems driven out of equilibrium [5863], utilized this method to investigate the fluctuationdissipation relation for nonequilibrium bath [35], and further studied the dynamics of a tracer immersed in such bath, gave the friction and noise properties [53], the Langevin description [64], and correlation functions of the tracer variables to study the fluctuation properties[54]. Speck and Seifert et al. formulated a fluctuationdissipation theorem (FDT) within a nonequilibrium steady state of a sheared colloidal suspension system [65, 66], and subsequently investigated the mobility and diffusivity of a tagged particle within this system, determined the velocity autocorrelation functions and response functions with small shear force, found that a phenomenological effective temperature recovers the Einstein relation in nonequilibrium [67]. EsparzaLópez et al. [68] proposed a stochastic fluid dynamic model to describe analytically and computationally the dynamics of microscopic particles driven by the motion of surface attached bacteria, analytically calculated expressions for the effective diffusion coefficient through a runandtumble model, found that the shorttime mean squared displacement is proportional to the square of the swimming speed while the longtime one only depends on the size of the particle. Burkholder and Brady [22] studied the diffusion of a tracer in a dilute dispersion of active Brownian particles (ABPs), by employing the Smoluchowski equation and averaging over bath particles and orientation variables, obtained tracers singleparticle probability distribution function, found that the active contribution to the diffusivity scales as U_{0} (characteristic swim speed of ABP) for strong swimming and for weak swimming. Furthermore, they [69] derived a general relationship between diffusivity and mobility in generic colloidal suspensions, provided a method to quantify deviations from the FDT and express them in terms of an effective SES relation. More recently, Granek et al. [51] studied the longtime dynamics of a tracer immersed in a onedimensional active bath, derived a timedependent friction and noise correlation with power law long tails that depend on the symmetry of tracers, and found that shape asymmetry of the tracer induces ratchet effects and leads to superdiffusion and friction that grows with time.
Numerous theories based on various starting points have demonstrated the importance and attraction of studying tracer behavior in nonequilibrium baths. In this study, we propose an alternative theoretical method based on pathintegral method to investigate the behavior of tracer diffusion in an active particle bath, and subsequent simulation results successfully validate our theory. The starting point of the theory is the generalized Langevin equation (GLE) for the tracer, which utilizes a generalized version of Dean’s equation to describe the active bath. The GLE contains a memory kernel function and complex effective noise terms, reflecting the complex interactions between the tracer and bath particles. We then employ the path integral method [70, 71] to calculate the diffusion coefficient. Numerical calculations show that the effective diffusion has a nontrivial dependence on bath parameters such as the number density and the persistent time of the active bath particle. Finally, we perform extensive computer simulations, which show very good agreement with our theoretical predictions.
This work is organized as follows: In Section “Model and theory”, we introduce the model system and derive the GLE of the tracer. Then, we utilize the path integral method to obtain the effective diffusion of the tracer formally. In Section “Simulation results”, we show the numerical solution of such diffusion and compare it with simulation results. The paper ends with conclusion in Section “Conclusion”.
MODEL AND THEORY
Active bath model
Considering a system consisting of a tracer particle and N active OrnsteinUhlenbeck (OU) particles (Figure 1), the coordinates of the tracer x and the active bath particles r_{i} are governed by the overdamped Langevin equations,(1a)(1b)(1c)wherein and are the mobilities of the tracer and bath particles respectively, U_{ext} is an external potential acting on the tracer particle, U is the interacting potential between tracer and bath particles, and V is potential between bath particles, , and are white noises with zero means and unit variances, f_{i} is the selfpropulsion force acting on bath particle i with time correlation function , where I is the unit matrix, is the selfpropulsion correlation time and D_{b} serves as the amplitude of such force, with the same dimension as diffusivity. We consider the situation that all potentials (U and V) are square integrable, meaning that they have well defined and limited Fourier transforms: , .
Figure 1 Schematic diagram of a tracer particle with coordinate x in an active particle bath consisting of selfpropulsion particles at r_{i}, i=1,2,…, N. The tracerbath interaction is , and the bathbath interaction is . The wave lines label the thermal noises. 
Inspired by the idea of Dean’s studies [72, 73], we derive a selfconsistent equation for the density profile of bath particles (See details in Appendix A in Supplementary information),(2)where the noise terms has correlation functions , . In the Fourier space (), Eq. (4) has a formal solution,(3)wherein can be considered as a characteristic frequency (a typical illustration of this term is shown in Figure S1A). Noise terms are the Fourier transform of , respectively, with correlation functions(4a)(4b)wherein means the average over noises.
Generalized Langevin equation
To achieve an effective equation of motion of the tracer which does not contain any bath particle variables, one can use an identity , then has a GLE for x,(5)where the memory kernel(6)and the colored noise terms,(7)(8)and their correlations are(9)(10)
Path integral and effective diffusion
To calculate the transport coefficients, one needs to calculate several correlation functions first. Considering the coupling between the tracer position and the colored noises , we propose a path integral method to calculate them.
We consider a path of the tracer in the time interval [t_{i}, t_{f}]. The partition function of such trajectory can be written as(11)Using the identity of delta function, , we also have(12)where p is an auxiliary real vector field. Next, utilizing for a Gaussian random variable u with zero mean, we have the partition function as a function of the action,(13)where(14)for a free particle and(15)counts for the tracerbath interaction. Herein, we have assumed that are both Gaussian, and their deviations from the Gaussian distribution are only weakly present in regions far from the mean. Since this approximation primarily reflects the properties of the second moments, it is reasonable in this context. In addition, since we have used the perturbative expansion, the interaction between tracer and bath particles should be weak. Therefore, we choose soft harmonic potentials as the interparticle potentials. The perturbation also demands that the activity is not very large, which constitutes a condition for the application of our theoretical framework.
After introducing the partition function over the trajectory, the average over any operator A as function of can be defined as(16)where corresponds to a tracer particle only affected by external potential, not any particle bath. So far, the term is still too complicated to handle. A common treatment is the linear truncation when S_{int} is weak. Herein, we treat the tracerbath interaction U as a small perturbative quantity by assigning , where is a dimensionless factor that scales the interacting strength, and can be used to the following perturbative expansion. The memory kernel k and correlators G_{A,T}(t) are order 1 of . Therefore, Eq. (19) can be expand as(17)For studying the diffusion problem, we only need to consider the free particle situation, U_{ext}=0. According to the symmetry, we have . Similarly, we also have(18)Therefore,(19)
The next step is to calculate the mean square displacement (MSD), i.e., to calculate for a long time interval t_{f}t_{i}. Then, the effective diffusion coefficient can be given as(20)where d is the dimension of the system. According to Eq. (20), the key step of calculating the MSD is handling , i.e., calculating the following two correlation functions, one is(21)and the other is(22)for t>s (the omitted mathematical details can be found in Appendix B in Supplementary information). After the calculation in Eq. (S9) (Supplementary information), we get(23)where reflects the bath properties including background temperature T, number density ρ_{0} and interactions between bath particles. At last, taking the longtime limit of t_{f}t_{i}, the o(t_{f}t_{i}) term can be neglected, the effective diffusion coefficient of a tracer in an active bath is obtained(24a)to the linear order, wherein(24b)(24c)This is the main result of the present work. In this equation, “1” in the brace of Eq. (24a) denotes the bare diffusion of a free tracer particle. The second term in the brace denotes the “passive part” of the tracerbath interaction which is always a negative contribution to effective diffusion and recovers the results in Ref.[70]. In the absence of activity, the FDT holds since the effective mobility of the tracer satisfies μ_{eff}=D_{eff}/T [70], wherein they have obtained an effective mobility coefficient (expressed with notations of the present work for convenience). The third term is a pure “active” contribution on the diffusion, which is a positive contribution and explicitly gives the nontrivial dependence of D_{eff} on the bath parameters, ρ_{0}, D_{b}, τ_{b} and interactions U(r) and V(r). For this linear truncation, D_{eff} is a linear function of D_{b}. However, the dependence of τ_{b} and ρ_{0} (which is also contained in G_{k}) is illegible, which require numerical calculations to determine (see Section “Simulation results”). Further mathematical analysis of this expression is shown in Appendix C in Supplementary information.
SIMULATION RESULTS
In this section, we show numerical calculations of Eq. (24a) with the persistent time of active force τ_{b} and the number density of bath particles ρ, at small activity D_{b} region. Then, by comparing these results with computer simulations, the validity and applicability of the theory can be verified.
In the present work, we choose V(r) and U(r) as both harmonic potentials, for , and for . We set σ_{bb} as the unit of length, as the unit of energy, and as the unit of time. The common parameters are set as: , , μ_{t}=0.333, T=1.0. The other parameters are set as variables which are explicitly given in the following figures. In computer simulations, we construct a threedimensional system with periodic boundary containing (1+4095) particles. The diffusion coefficient is calculated through a longtime simulation (~10^{8} steps with 10^{3} as the time step) and averaged over 20 samples with random initial configurations.
Firstly, we focus on the contribution of bath number density ρ on D_{eff}, shown in Figure 2. In Figures 2 and 图3, dots and corresponding error bars are the direct simulations, and lines are numerical calculation of Eq. (24a). In general, D_{eff} shows a diverse dependence on ρ. For small active force amplitude D_{b}, D_{eff} decreases with ρ as the tracer’s behavior in a passive particle bath. For large activity situation, D_{eff} shows the opposite behavior. Consequently, one may expect that there is a moderate activity region that D_{eff} has a nonmonotonic dependence on ρ.
Figure 2 The dependence of the effective diffusion on the number density of active bath particle ρ. Dots and corresponding error bars are simulation results of D_{eff}. Solid lines are the numerical calculations of D_{eff}. Both results show that, with the increase of activity D_{b}, effective diffusion D_{eff} gradually changes from monotonic decrease to monotonic increase with the number density of bath particle ρ, including a nonmonotonic interval around D_{b}=2.0. Herein, τ_{b} is set as 0.1. 
Figure 3 The dependence of the effective diffusion on the active force persistent time τ_{b}. Dots and corresponding error bars are simulation results of D_{eff}. Solid lines are the numerical calculations of D_{eff}. Both show that D_{eff} monotonically decreases with the persistent time τ_{b}. The horizontal dash line denotes the bare diffusion coefficient of the tracer particle. 
We then investigate the dependence of D_{eff} on persistent time τ_{b}. According to Eq. (27), D_{eff} decreases with τ_{b} since both and increase with τ_{b} monotonically. This prediction has been confirmed in simulations, as shown in Figure 3. Physically, when the persistent time of the active force tends to zero, the active OU particle can be reduced to an ordinary Brownian particle under a higher temperature. If the activity amplitude D_{b} is constant, the longer τ_{b} means a larger deviation of equilibrium. Based on the results here, we might conclude that an active OU particle bath that stays closer to equilibrium, is more conducive to the tracer diffusion, when the activity amplitude is given.
CONCLUSION
This study aims to shed light on the intricacies of tracer diffusion in an active particle bath. By employing the generalized Dean’s equation, incorporating the path integral method, and utilizing computer simulations, we characterize the impact of selfpropulsion on the diffusion behavior of a passive tracer particle. In summary, the effective diffusion decreases with persistent time τ_{b}, and exhibits a variety of dependencies on bath density, depending on D_{b}. The obtained insights expand our understanding of collective dynamics and transport phenomena in nonequilibrium systems, with potential applications in diverse scientific disciplines. For further studies, an extension of the active bath situation is straightforward since it has been confirmed to calculate the mobility of a tracer in particle bath [70, 71]. Additionally, after the effective mobility is achieved, the fluctuationdissipation theorem can be further investigated to determine its validity or deviations with respect to the activity parameters.
Funding
This work was supported by the Strategic Priority Research Program of the Chinese Academy of Sciences (XDB0450402), and the National Natural Science Foundation of China (32090040 and 22373090).
Author contributions
H.Z. directed the project. F.M. derived the theory, wrote codes, and wrote the manuscript. All authors commented on the manuscript.
Conflict of interest
The authors declare no conflict of interest.
Supplementary information
The supporting information is available online at https://doi.org/10.1360/nso/20220069. The supporting materials are published as submitted, without typesetting or editing. The responsibility for scientific accuracy and content remains entirely with the authors.
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All Figures
Figure 1 Schematic diagram of a tracer particle with coordinate x in an active particle bath consisting of selfpropulsion particles at r_{i}, i=1,2,…, N. The tracerbath interaction is , and the bathbath interaction is . The wave lines label the thermal noises. 

In the text 
Figure 2 The dependence of the effective diffusion on the number density of active bath particle ρ. Dots and corresponding error bars are simulation results of D_{eff}. Solid lines are the numerical calculations of D_{eff}. Both results show that, with the increase of activity D_{b}, effective diffusion D_{eff} gradually changes from monotonic decrease to monotonic increase with the number density of bath particle ρ, including a nonmonotonic interval around D_{b}=2.0. Herein, τ_{b} is set as 0.1. 

In the text 
Figure 3 The dependence of the effective diffusion on the active force persistent time τ_{b}. Dots and corresponding error bars are simulation results of D_{eff}. Solid lines are the numerical calculations of D_{eff}. Both show that D_{eff} monotonically decreases with the persistent time τ_{b}. The horizontal dash line denotes the bare diffusion coefficient of the tracer particle. 

In the text 
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