Effective diffusion of a tracer in active bath: A path-integral approach

Mengkai Feng; Zhonghuai Hou

doi:10.1360/nso/20230069

Open Access

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

National Science Open 3: 20230069, 2024

RESEARCH ARTICLE

Effective diffusion of a tracer in active bath: A path-integral approach

Mengkai Feng^* and Zhonghuai Hou^*

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: This email address is being protected from spambots. You need JavaScript enabled to view it. (Mengkai Feng); This email address is being protected from spambots. You need JavaScript enabled to view it. (Zhonghuai Hou))

Received: 30 October 2023
Revised: 3 January 2024
Accepted: 5 February 2024

Abstract

We investigate the effective diffusion of a tracer immersed in an active particle bath consisting of self-propelled 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 / mean-field 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 self-propelled units that are able to convert stored or surrounding energy into their persistent motion, provide a fresh opportunity for applications of nonequilibrium statistical mechanics [1-4]. 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 [5-12]. 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, 13-16], chemical[17, 18], and physical phenomena[8, 9, 12, 19-32]. Notably, the tracer particle immersed in the active bath exhibits a distinct diffusion profile compared to its equilibrium counterpart[26, 33-39]. 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, 40-46] and transport properties in such media [47-51]. 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 fluctuation-dissipation theorem in thermal equilibrium. Nevertheless, the dynamics of particles in active baths introduce additional complexity[37, 45, 51-57]. 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 fluctuation-response relation for thermal systems driven out of equilibrium [58-63], utilized this method to investigate the fluctuation-dissipation 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 fluctuation-dissipation 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]. Esparza-López 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 run-and-tumble model, found that the short-time mean squared displacement is proportional to the square of the swimming speed while the long-time 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 single-particle probability distribution function, found that the active contribution to the diffusivity scales as U₀ (characteristic swim speed of ABP) for strong swimming and $U_{0}^{2}$ Mathematical equation 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 long-time dynamics of a tracer immersed in a one-dimensional active bath, derived a time-dependent 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 super-diffusion 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 path-integral 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 non-trivial 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 Ornstein-Uhlenbeck (OU) particles (Figure 1), the coordinates of the tracer x and the active bath particles r_i are governed by the overdamped Langevin equations, $\dot{x} = - μ_{t} \nabla_{x} [\sum_{i = 1}^{N} U (| x - r_{i} |) + U_{ext} (x)] + \sqrt{2 μ_{t} T} ξ_{t},$ Mathematical equation (1a) ${\dot{r}}_{i} = - μ_{b} {f_{i} + \nabla_{i} [\sum_{j \neq i} V (| r_{i} - r_{j} |) + U (| r_{i} - x |)]} + \sqrt{2 μ_{b} T} ξ_{i},$ (1b) $τ_{b} {\dot{f}}_{i} = - f_{i} + \sqrt{2 D_{b} / μ_{b}^{2}} η_{i},$ (1c)wherein $μ_{t}$ and $μ_{b}$ 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, $ξ_{t}$ Mathematical equation , $ξ_{i}$ and $η_{i}$ are white noises with zero means and unit variances, f_i is the self-propulsion force acting on bath particle i with time correlation function $〈 f_{i} (t) f_{j} (t^{'}) 〉 = \frac{D_{b}}{μ_{b}^{2} τ_{b}} δ_{i j} e^{- | t - t^{'} | / τ_{b}} I$ Mathematical equation , where I is the unit matrix, $τ_{b}$ is the self-propulsion 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: $U_{k} = \int U (r) e^{i k \cdot r} d r$ Mathematical equation , $V_{k} = \int V (r) e^{i k \cdot r} d r$ .

Figure 1

Schematic diagram of a tracer particle with coordinate x in an active particle bath consisting of self-propulsion particles at r_i, i=1,2,…, N. The tracer-bath interaction is $U (| x - r_{i} |)$ Mathematical equation , and the bath-bath interaction is $V (| r_{i} - r_{j} |)$ . The wave lines label the thermal noises.

Inspired by the idea of Dean’s studies [72, 73], we derive a self-consistent equation for the density profile of bath particles $ρ (r, t) = \sum_{j = 1}^{N} δ (r - r_{j} (t))$ Mathematical equation (See details in Appendix A in Supplementary information), $\begin{array}{l} \frac{\partial}{\partial t} ρ (r, t) = μ_{b} \nabla \cdot {ρ (r, t) [\int ρ (r', t) \nabla V (| r - r' |) d r' \\ + \nabla U (| r - x |)]} + μ_{b} T \nabla^{2} ρ (r, t) + \nabla \cdot {\sqrt{ρ (r, t)} [ξ^{T} (r, t) + ξ^{A} (r, t)]}, \end{array}$ Mathematical equation (2)where the noise terms $ξ^{{T, A}}$ has correlation functions $〈 ξ^{T} (r, t) ξ^{T} (r', t^{'}) 〉 = 2 μ_{b} T δ (r - r') δ (t - t^{'}) I$ , $〈 ξ^{A} (r, t) ξ^{A} (r', t^{'}) 〉 = \frac{D_{b}}{τ_{b}} e^{- | t - t^{'} | / τ_{b}} δ (r - r') I$ . In the Fourier space ( $δ ρ_{k} (t) = \int e^{i r \cdot k} (ρ (r, t) - ρ_{0}) d k$ Mathematical equation ), Eq. (4) has a formal solution, $δ ρ_{k} (t) = \int_{- \infty}^{t} e^{- (t - s) G_{k}} {- μ_{b} k^{2} ρ_{0} U_{k} e^{i k \cdot x_{s}} + \sqrt{ρ_{0}} i k \cdot [{\tilde{ξ}}_{k}^{T} (s) + {\tilde{ξ}}_{k}^{A} (s)]} d s,$ (3)wherein $G_{k} = μ_{b} k^{2} (T + ρ_{0} V_{k})$ Mathematical equation can be considered as a characteristic frequency (a typical illustration of this term is shown in Figure S1A). Noise terms ${\tilde{ξ}}_{k}^{{T, A}} (t)$ are the Fourier transform of $ξ^{{T, A}}$ , respectively, with correlation functions $〈 {\tilde{ξ}}_{k}^{T} (t) {\tilde{ξ}}_{k^{'}}^{T} (t^{'}) 〉 = 2 μ_{b} T {(2)}^{3} δ (k + k') δ (t - t^{'}) I,$ Mathematical equation (4a) $〈 {\tilde{ξ}}_{k}^{A} (t) {\tilde{ξ}}_{k^{'}}^{A} (t^{'}) 〉 = \frac{D_{b}}{τ_{b}} {(2)}^{3} δ (k + k') e^{- | t - t^{'} | / τ_{b}} I,$ (4b)wherein $〈 \dots 〉$ 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 $\nabla_{x} \sum_{i = 1}^{N} U (| r_{i} - x |) = - \frac{1}{{(2 π)}^{3}} \int i k e^{- i k \cdot x} ρ_{k} U_{k} d^{3} k$ Mathematical equation , then has a GLE for x, $\begin{matrix} \dot{x} (t) = - μ_{t} \nabla_{x} U_{ext} (x) + μ_{t} \frac{1}{{(2 π)}^{3}} \int i k e^{- i k \cdot x (t)} δ ρ_{k} U_{k} d^{3} k + \sqrt{2 μ_{t} T} ξ_{t} \\ = - μ_{t} \nabla_{x} U_{ext} (x) - \int_{- \infty}^{t} K (s, t) d s + η_{T} (t) + η_{A} (t) \sqrt{2 μ_{t} T} ξ_{t}, \end{matrix}$ Mathematical equation (5)where the memory kernel $\begin{matrix} K (s, t) = \frac{μ_{t} μ_{b} ρ_{0}}{{(2 π)}^{3}} \int i k^{2} k U_{k}^{2} e^{- (t - s) G_{k}} e^{- i k \cdot (x (t) - x (s))} d^{3} k \\ = \frac{μ_{t} μ_{b} ρ_{0}}{{(2 π)}^{3}} \int Im [e^{i k \cdot (x (t) - x (s))}] k k^{2} U_{k}^{2} e^{- (t - s) G_{k}} d^{3} k, \end{matrix}$ Mathematical equation (6)and the colored noise terms, $η_{T} (t) = - μ_{t} \frac{1}{{(2 π)}^{3}} \int k e^{- i k \cdot x_{t}} \sqrt{ρ_{0}} U_{k} \int_{- \infty}^{t} e^{- (t - s) G_{k}} k \cdot {\tilde{ξ}}_{k}^{T} (s) d s d^{3} k,$ (7) $η_{A} (t) = - μ_{t} \frac{1}{{(2 π)}^{3}} \int k e^{- i k \cdot x_{t}} \sqrt{ρ_{0}} U_{k} \int_{- \infty}^{t} e^{- (t - s) G_{k}} k \cdot {\tilde{ξ}}_{k}^{A} (s) d s d^{3} k,$ Mathematical equation (8)and their correlations are $G_{T} (t, t^{'}) = 〈 η_{T} (t) η_{T} (t^{'}) 〉 = \frac{μ_{t}^{2} ρ_{0} μ_{b} T}{{(2 π)}^{3}} \int e^{- i k \cdot (x_{t} - x_{t^{'}})} k k k^{2} U_{k}^{2} \frac{e^{- | t - t^{'} | G_{k}}}{G_{k}} d^{3} k,$ (9) $G_{A} (t, t^{'}) = 〈 η_{A} (t) η_{A} (t^{'}) 〉 = \frac{μ_{t}^{2} ρ_{0} D_{b}}{{(2 π)}^{3}} \int e^{- i k \cdot (x_{t} - x_{t^{'}})} k k k^{2} U_{k}^{2} \frac{τ_{b} e^{- | t - t^{'} | / τ_{b}} - (e^{- | t - t^{'} | G_{k}} / G_{k})}{[{(τ_{b} G_{k})}^{2} - 1]} d^{3} k .$ Mathematical equation (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 $η_{A, T}$ Mathematical equation , 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 $Z = \int \prod_{t} δ {\dot{x} + μ_{t} \nabla U_{ext} + \int_{- \infty}^{t} K (s, t) d s - η_{A} - η_{T} - \sqrt{2 μ_{t} T} ξ_{t}} P [ξ_{t}] P [η_{T}] P [η_{A}] D x D ξ_{t} D η_{A} D η_{T} .$ Mathematical equation (11)Using the identity of delta function, $δ (x) = \frac{1}{(2 π)} \int e^{i p x} d p$ , we also have $Z = \int e^{i \int p \cdot {\dot{x} + μ_{t} \nabla U_{ext} + \int_{- \infty}^{t} K (s, t) d s - η_{A} - η_{T} - \sqrt{2 μ_{t} T} ξ_{t}} d t} P [ξ_{t}] P [η_{T}] P [η_{A}] D x D p D ξ_{t} D η_{A} D η_{T},$ Mathematical equation (12)where p is an auxiliary real vector field. Next, utilizing $〈 e^{a u} 〉 = e^{\frac{1}{2} a^{2} 〈 u^{2} 〉}$ for a Gaussian random variable u with zero mean, we have the partition function as a function of the action, $Z = \int e^{- S (x, p)} D x D p, S (x, p) = S_{0} (x, p) + S_{int} (x, p),$ Mathematical equation (13)where $S_{0} = - i \int p (t) \cdot [\dot{x} (t) + μ_{t} \nabla_{x} U_{ext}] d t + μ_{t} T \int | p (t {) |}^{2} d t,$ (14)for a free particle and $S_{int} = - i \int p (t) \cdot [\int_{- \infty}^{t} K (s, t) d s] d t + \frac{1}{2} \iint p (t) \cdot [G_{T} (t, s) + G_{A} (t, s)] \cdot p (s) d t d s$ Mathematical equation (15)counts for the tracer-bath interaction. Herein, we have assumed that $η_{A, T}$ 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 inter-particle 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 $x (t), t \in [t_{i}, t_{j}]$ Mathematical equation can be defined as $〈 A 〉 = \frac{\int A e^{- (S_{0} + S_{int})} D x D p}{Z} = \frac{{〈 A e^{- S_{int}} 〉}_{0}}{{〈 e^{- S_{int}} 〉}_{0}},$ (16)where ${〈 \dots 〉}_{0} = Z_{0}^{- 1} \int (\dots) e^{- S_{0}} D x D p$ corresponds to a tracer particle only affected by external potential, not any particle bath. So far, the $e^{- S_{int}}$ Mathematical equation term is still too complicated to handle. A common treatment is the linear truncation when S_int is weak. Herein, we treat the tracer-bath interaction U as a small perturbative quantity by assigning $U (r) = \sqrt{ϵ} u (r)$ , where $ϵ$ 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 $ϵ$ Mathematical equation . Therefore, Eq. (19) can be expand as $〈 A 〉 = {〈 A 〉}_{0} - {〈 A S_{int} 〉}_{0} + {〈 A 〉}_{0} {〈 S_{int} 〉}_{0} + O (ϵ^{2}) .$ (17)For studying the diffusion problem, we only need to consider the free particle situation, U_ext=0. According to the symmetry, we have ${〈 p (t) e^{- i k \cdot [x (t) - x (s)]} 〉}_{0} = - i {〈 p (t) [x (t) - x (s)] 〉}_{0} \cdot k e^{- k^{2} μ_{t} T (t - s)} = 0$ Mathematical equation . Similarly, we also have ${〈 p (t) \cdot G_{A, T} (t, s) \cdot p (t) 〉}_{0} \propto {{〈 | p (t {) |}^{2} 〉}_{0} - {(k \cdot {〈 (x (t) - x (s)) p (t) 〉}_{0})}^{2}} e^{- k^{2} μ_{t} T (t - s)} = 0.$ (18)Therefore, ${〈 S_{int} 〉}_{0} = 0.$ (19)

The next step is to calculate the mean square displacement (MSD), i.e., to calculate $〈 {[x (t_{f}) - x (t_{i})]}^{2} 〉$ Mathematical equation for a long time interval t_f-t_i. Then, the effective diffusion coefficient can be given as $D_{eff} = \lim_{(t_{f} - t_{i}) \to \infty} \frac{1}{2 d (t_{f} - t_{i})} 〈 {[x (t_{f}) - x (t_{i})]}^{2} 〉,$ (20)where d is the dimension of the system. According to Eq. (20), the key step of calculating the MSD is handling ${〈 {[x (t_{f}) - x (t_{i})]}^{2} S_{int} 〉}_{0}$ Mathematical equation , i.e., calculating the following two correlation functions, one is $\begin{matrix} {〈 {[x (t_{f}) - x (t_{i})]}^{2} p (t) e^{- i k \cdot [x (t) - x (s)]} 〉}_{0} \\ = - 2 i k \cdot {〈 [x (t) - x (s)] [x (t_{f}) - x (t_{i})] 〉}_{0} {〈 [x (t_{f}) - x (t_{i})] p (t) 〉}_{0} e^{- k^{2} μ_{t} T (t - s)} \\ = 4 k μ_{t} T L ([t_{i}, t_{f}] \cap [s, t]) χ_{[t_{i}, t_{f})} (t) e^{- k^{2} μ_{t} T (t - s)} \\ = 4 k μ_{t} T [t - \max (t_{i}, s)] χ_{[t_{i}, t_{f})} (t) e^{- k^{2} μ_{t} T (t - s)}, \end{matrix}$ Mathematical equation (21)and the other is $\begin{matrix} {〈 {[x (t_{f}) - x (t_{i})]}^{2} p (t) p (s) e^{- i k \cdot [x (t) - x (s)]} 〉}_{0} \\ = {4 μ_{t} T [t - \max (s, t_{i})] k k - 2 χ_{[t_{i}, t_{f})} (s) 1} e^{- k^{2} μ_{t} T (t - s)} χ_{[t_{i}, t_{f})} (t), \end{matrix}$ (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 $\begin{matrix} {〈 {[x (t_{f}) - x (t_{i})]}^{2} S_{int} 〉}_{0} = 2 μ_{t} T (t_{f} - t_{i}) \frac{μ_{t} μ_{b} ρ_{0}}{{(2)}^{3}} \int \frac{k^{4} U_{k}^{2}}{G_{k} (k^{2} μ_{t} T + G_{k})} d^{3} k \\ + \frac{μ_{t}^{2} ρ_{0} D_{b}}{{(2 π)}^{3}} (t_{f} - t_{i}) \int \frac{2 k^{4} U_{k}^{2}}{{(τ_{b} G_{k})}^{2} - 1} {\frac{τ_{b} μ_{t} T k^{2} - 1}{{(μ_{t} T k^{2} + τ_{b}^{- 1})}^{2}} - \frac{(μ_{t} T k^{2} / G_{k}) - 1}{{(μ_{t} T k^{2} + G_{k})}^{2}}} d^{3} k \\ + o (t_{f} - t_{i}), \end{matrix}$ Mathematical equation (23)where $G_{k} = μ_{b} k^{2} (T + ρ_{0} V_{k})$ reflects the bath properties including background temperature T, number density ρ₀ and interactions between bath particles. At last, taking the long-time 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 $D_{eff} = μ_{t} T {1 - \int_{0}^{\infty} g_{T} (k) d k + \int_{0}^{\infty} g_{A} (k) d k},$ Mathematical equation (24a)to the linear order, wherein $g_{T} (k) = \frac{μ_{t} μ_{b} ρ_{0}}{6^{2}} \frac{k^{6} U_{k}^{2}}{G_{k} (k^{2} μ_{t} T + G_{k})},$ (24b) $g_{A} (k) = \frac{μ_{t} ρ_{0} D_{b} / T}{6^{2}} \frac{k^{6} U_{k}^{2}}{{(τ_{b} G_{k})}^{2} - 1} [\frac{μ_{t} T k^{2} / G_{k} - 1}{{(μ_{t} T k^{2} + G_{k})}^{2}} - \frac{τ_{b} μ_{t} T k^{2} - 1}{{(μ_{t} T k^{2} + τ_{b}^{- 1})}^{2}}] .$ Mathematical equation (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 tracer-bath 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 $μ_{eff} = μ_{t} (1 - \int_{0}^{\infty} g_{T} (k) d k)$ Mathematical equation (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, ρ₀, 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 ρ₀ (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, $V (r) = \frac{κ_{b}}{2} {(r - σ_{b b})}^{2}$ Mathematical equation for $r < σ_{b b}$ , and $U (r) = \frac{κ_{t}}{2} {(r - σ_{t b})}^{2}$ for $r < σ_{t b}$ . We set σ_bb as the unit of length, $κ_{b} σ_{b b}^{2}$ as the unit of energy, and $1 / (μ_{b} κ_{b})$ as the unit of time. The common parameters are set as: $σ_{t b} = 2.0$ Mathematical equation , $κ_{t} = 1.0$ , 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 three-dimensional system with periodic boundary containing (1+4095) particles. The diffusion coefficient is calculated through a long-time simulation (~10⁸ 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 non-monotonic 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 non-monotonic 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 ${(G_{k} τ_{b})}^{2} - 1$ Mathematical equation and $\frac{τ_{b} μ_{t} T k^{2} - 1}{{(μ_{t} T k^{2} + τ_{b}^{- 1})}^{2}}$ 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 self-propulsion 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 non-equilibrium 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 fluctuation-dissipation 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.

References

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All Figures

	Figure 1 Schematic diagram of a tracer particle with coordinate x in an active particle bath consisting of self-propulsion particles at r_i, i=1,2,…, N. The tracer-bath interaction is $U (\| x - r_{i} \|)$ , and the bath-bath interaction is $V (\| r_{i} - r_{j} \|)$ . The wave lines label the thermal noises.
In the text

Figure 2

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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