Dynamical fluctuations of a tracer coupled to active and passive particles Ion Santra

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Dynamical ﬂuctuations of a tracer coupled to active

and passive particles

Ion Santra

Raman Research Institute, Bengaluru 560080, India

Abstract. We study the induced dynamics of an inertial tracer particle elastically

coupled to passive or active Brownian particles. We integrate out the environment

degrees of freedom to obtain the exact eﬀective equation of the tracer—a generalized

Langevin equation in both cases. In particular, we ﬁnd the exact form of the

dissipation kernel and eﬀective noise experienced by the tracer and compare it with

the phenomenological modeling of active baths used in previous studies. We show that

the second ﬂuctuation-dissipation relation (FDR) does not hold at early times for both

cases. However, at ﬁnite times, the tracer dynamics violate (obeys) the FDR for the

active (passive) environment. We calculate the linear response formulas in this regime

for both cases and show that the passive medium satisﬁes an equilibrium ﬂuctuation

response relation (FRR), while the active medium does not—we quantify the extent

of this violation explicitly. We show that though the active medium generally renders

a nonequilibrium description of the tracer, an eﬀective equilibrium picture emerges

asymptotically in the small activity limit of the medium. We also calculate the mean

squared velocity and mean squared displacement of the tracer and report how they

vary with time.

1. Introduction

The eﬀective behavior of a system depends crucially on the dynamical nature of its

environment. It is well established that if a system is in an equilibrium environment,

the linear response of an observable in the presence of small external perturbation is

governed by the equilibrium correlations of the same observable. This is popularly

called the ﬂuctuation-response relation (also sometimes called the ﬂuctuation-dissipation

theorem of the ﬁrst kind). As a corollary, it is also known that the random force driving

the velocity ﬂuctuations of the system is related to the friction (dissipation) experienced

by the system. This is known as the ﬂuctuation-dissipation relation (also known in

the literature as the ﬂuctuation-dissipation relation of the second kind) [1, 2, 3, 4, 5].

Descriptions of such eﬀective motion of a system in an equilibrium environment can be

derived starting from the microscopic descriptions of the environment and the system

environment coupling and thereafter integrating out the fast environmental degrees of

freedom [6, 7]. Popular and famous ways of doing this include the Feynman-Vernon

model [8] or the Caldeira-Leggett model [9], which describe the dynamics of a system

(particle) elastically coupled to an environment consisting of a set of non-interacting

arXiv:2210.05139v2 [cond-mat.stat-mech] 15 Aug 2023

Dynamical ﬂuctuations of a tracer coupled to active and passive particles 2

harmonic oscillators. Integrating out the oscillator degrees of freedom, one can arrive

at a generalized Langevin equation for the time evolution of the position of the system.

Further, by using the equilibrium correlations of the oscillators, the validity of the

ﬂuctuation relations can be explicitly shown.

An interesting question is what happens if the environment is out of equilibrium.

The examples of such nonequilibrium environments are widespread in the world of

complex systems—a collection of active particles [10, 11], glassy medium [12], sheared

ﬂuid [13], inter-cellular medium [14] etc. It is known that the familiar forms of the

equilibrium relations connecting the ﬂuctuation-dissipation and ﬂuctuation-response of

the system do not hold anymore [15, 16, 17, 18, 19]. In fact, there are no general

forms for the ﬂuctuation relations and they depend on the exact dynamical nature

of the environment and how the system is coupled to it [20]. Thus, providing an

eﬀective description of a system in a nonequilibrium environment is a problem highly

sought-after. Moreover, transport properties of extended systems connected to active

environments have also gained interest recently [21], and an accurate description of such

environments has thus become very important. This is evident from the huge body of

works involving experiments in microrheology and biophysics, whose main objective is to

understand how the nonequilibrium features of the active environment are reﬂected on

the dynamics of a tracer particle [10, 16, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32].

These experiments have, in turn, triggered a number of numerical and theoretical

works [33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 52, 53,

54, 55, 56, 57]. Importantly, in most of the studies involving ﬂuctuating environments,

the eﬀect of the environment is modeled phenomenologically, motivated by experiments,

simulations [36, 41, 40, 34, 55] and the derivations of the eﬀective motion of a tracer in

a nonequilibrium environment starting from the microscopic dynamics and interaction

has been limited [37, 49, 54, 57]. However, such studies provide great insight into what

roles the parameters of the ﬂuctuating environment have on the dynamics of the tracer.

In this paper, we investigate the dynamics of a tracer in a ﬂuctuating environment

by explicitly integrating the degrees of freedom of the environment. We consider a simple

microscopic model where the system, a tracer particle, is connected to a collection of

non-self-interacting particles (environment), following their own stochastic dynamics,

by harmonic springs. In particular, we consider two cases where these stochastic

dynamics are Markovian and non-Markovian—modeled by Brownian (passive) and

active particles, respectively; each active particle is modeled by a Langevin equation

with colored noise, having an exponentially decaying autocorrelation. The harmonic

springs render the equations of motion of the tracer and environmental particles to be

linear, which helps in integrating out the environmental degrees of freedom exactly to

get an eﬀective equation of motion—a generalized Langevin equation for the tracer.

We ﬁnd the exact forms of the eﬀective noise and dissipation experienced by the tracer

and show that, though the eﬀective noise correlations contain terms that are not time

translation invariant, they decay very fast, and the random force (noise) on the tracer

reaches a stationary state. We show that the familiar phenomenological models used

Dynamical ﬂuctuations of a tracer coupled to active and passive particles 3

to describe active environments in earlier studies are recovered in the strong coupling

limit (coupling time-scale much smaller than the active time-scale). Using the exact

forms of the dissipation kernel and the stationary noise correlation of the tracer, we ﬁnd

that the traditional forms of the equilibrium FDT are obtained when the environment

is made of passive particles. On the other hand, the traditional forms of the FDR

are signiﬁcantly modiﬁed for the active particle environment. We compute the linear

response of the tracer in the presence of external perturbations explicitly to ﬁnd that the

traditional equilibrium description ﬁts when the environment is passive, while there are

stark modiﬁcations for the active environment. We calculate all these relations explicitly.

Finally, we measure the two most common and easily measurable observables, namely,

the mean squared velocity (msv) and the mean squared displacement (msd) of the tracer,

and compare the diﬀerences in the active and passive environments. In particular, we

ﬁnd that at times shorter than that set by the coupling time-scale, the tracer obeys

the usual underdamped behavior. However, beyond the coupling time-scale, the tracer

follows the motion of the environment—they show a diﬀusive motion similar to that

of the individual environmental particles with the same diﬀusion constant, albeit with

damped oscillations. These oscillations are a result of the visco-elastic nature of the

set-up (viscous eﬀects produced by the environmental particles and the elastic eﬀects

caused by the coupling springs). Interestingly, these oscillations vanish for highly active

environments (active time-scales much larger than the coupling time-scales). Compared

to the previous analytical studies, which use a weak coupling limit [49, 57], adiabatic

perturbation theory [54] the calculations presented in this paper are exact. Moreover,

our derivation of the eﬀective equation of the tracer applies to any choice of the active

motion of the environmental particles, unlike the previous works, which take a particular

choice of dynamics for the environmental particles. However, it must be mentioned that

though the procedure is exact and simpler compared to the trajectory-based response

formalism and adiabatic perturbation theory used in previous works, its applicability

beyond harmonic interactions is very diﬃcult.

The paper is organized as follows. We introduce the model and derive the eﬀective

generalized Langevin equation for the tracer in Sec. 2 . In Sec. 3, we compute the

autocorrelations of the eﬀective noise appearing in the generalized Langevin equation of

the tracer for both active and passive environments. We investigate the linear response

formula for the tracer in the presence of small external perturbation and derive the

modiﬁed response formulas in Sec. 4. In Sec. 5 we discuss the temporal dependence

of mean squared velocity and displacement of the tracer. Finally, we summarize and

conclude in Sec. 6.

2. Model

Let us consider a particle of mass mcoupled elastically to Noverdamped particles,

which can be either active or passive, via isotropic harmonic springs of spring constant

ki. The model is inspired by the typical microrheology experiments which study the

Dynamical ﬂuctuations of a tracer coupled to active and passive particles 4

motion of a tracer bead in myosin-activated actin networks [16, 25, 58, 59, 60]. The

harmonic interaction can also be viewed as an approximation for some other complicated

potential U[x(t)−yi(t)], with a short range, about its minimum. This can be seen by

expanding U(x) in a Taylor series about its minimum and keeping terms up to second

order; U′′(x0) [x0denotes the minimum of U(x)] serves as an eﬀective spring constant

for the medium [61]. In this paper, we consider the dynamics along one direction

only; however, generalization to higher dimensions is straightforward. The equations of

motion of the combined system are given by,

m¨x(t) = −X

kix(t)−yi(t)(1)

˙yi(t) = −ki

γyi(t)−x(t)+ζi(t)i= 1,2...,N (2)

where xdenotes the tracer displacement while yidenotes the displacement of the ith

environment particle; γdenotes the damping coeﬃcient of the bath particles; and

ζi(t) denotes the random noise of the environemental particles. Note that there is

no interaction among the degrees of freedom of the environmental particles.

We consider two cases where the environmental particles are (i) Brownian particles

and (ii) active particles. When the environmental particles are Brownian, ζ(t) is modeled

by a delta-correlated noise with zero mean,

⟨ζb

i(t)ζb

j(t′)⟩= 2Dδ(t−t′)δij (3)

where the diﬀusion coeﬃcient D=KBT/γ. Active particles, on the other hand,

are modeled by stochastic noises which have zero mean and exponentially decaying

autocorrelation,

⟨ζa

i(t)ζa

j(t′)⟩=v2

0e−α|t−t′|δij.(4)

The exponentially decaying autocorrelation is a hallmark of active particle motion [62,

63]. Popular models like run-and-tumble particles [64, 65, 66] and active Ornstein-

Uhlenbeck particles [67] exhibit exponentially decaying correlation of the propulsion

velocity at all times. Moreover, the active Brownian particles [68, 69] and direction

reversing active Brownian particles [66, 70] also follow such similar exponentially

decaying noise autocorrelation in the stationary state, see also Appendix A. The

characteristic time scale α−1of this decay is a measure of the activity of the particle—

smaller αimplies larger correlation time, which, in turn, implies higher activity.

In the following, we ﬁrst derive the equations of motion of the tracer by explicitly

integrating out the environmental degrees of freedom yi(t). To this end, we ﬁrst note

that the equation of motion of the bath particles (2) are ordinary ﬁrst-order linear

diﬀerential equations with inhomogeneous terms. It has a general solution,

yi(t) = yi(0)e−λit+e−λitZt

ds eλisλix(s) + ζi(s),(5)

Dynamical ﬂuctuations of a tracer coupled to active and passive particles 5

ith environmental

particle

Figure 1. A schematic representation of the model: the red shaded circle denotes

the tracer of mass mmodeled (1) and the solid green circles denote the environmental

particles modeled by (2).

where λi=ki/γ. Putting (5) in (1), we get,

m¨x(t) = X

kiyi(0)e−λt −X

ihkix(t)−kiλie−λitZt

ds eλisx(s)−kie−λitZt

ds eλisζi(s)i

(6)

Thus, the eﬀective dynamics of the tracer depend on the initial condition of the bath

particles in the form of a time-dependent velocity that decays exponentially and hence

does not aﬀect the long time dynamics of the tracer and we have,

m¨x(t) = −X

kix(t) + X

i k2

γe−λitZt

ds eλisx(s) + kie−λitZt

ds eλisζi(s)!.(7)

Performing an integration by parts on the second integral on the rhs of the above

equation, we arrive at a generalized Langevin equation for the tracer dynamics,

m¨x(t) = −Zt

ds Γ(t−s) ˙x(s) + Ω(t),(8)

where

Γ(τ) = X

kie−λiτ(9)

denotes the memory kernel and

Ω(τ) = Zt

ds X

kie−λi(t−s)ζi(s) (10)

denotes the eﬀective noise experienced by the tracer. It is important to note that the

calculations (5) to (8) holds for both passive and active environment. Moreover, (8),(9)

and (10) are exact and can be applied to any choice of ζi(t) corresponding to diﬀerent

active dynamics [see Appendix A]. However, the autocorrelation of the eﬀective noise

Ω(t) is diﬀerent for both cases, which we investigate in the following section.

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DynamicalfluctuationsofatracercoupledtoactiveandpassiveparticlesIonSantraRamanResearchInstitute,Bengaluru560080,IndiaAbstract.WestudytheinduceddynamicsofaninertialtracerparticleelasticallycoupledtopassiveoractiveBrownianparticles.Weintegrateouttheenvironmentdegreesoffreedomtoobtaintheexacteffectivee...

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Dynamical fluctuations of a tracer coupled to active and passive particles Ion Santra

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