Tagged active particle: probability distribution in a slowly varying external potential is determined by effective temperature obtained from the Einstein relation
TTagged active particle: probability distribution in a slowly varying external potentialis determined by effective temperature obtained from the Einstein relation
Alireza Shakerpoor, Elijah Flenner and Grzegorz Szamel ∗ Department of Chemistry, Colorado State University, Fort Collins, Colorado 80523, USA (Dated: February 23, 2021)We derive a distribution function for the position of a tagged active particle in a slowly varying inspace external potential, in a system of interacting active particles. The tagged particle distributionhas the form of the Boltzmann distribution but with an effective temperature that replaces thetemperature of the heat bath. We show that the effective temperature that enters the taggedparticle distribution is the same as the effective temperature defined through the Einstein relation, i.e. it is equal to the ratio of the self-diffusion and tagged particle mobility coefficients. This showsthat this effective temperature, which is defined through a fluctuation-dissipation ratio, is relevantbeyond the linear response regime. We verify our theoretical findings through computer simulations.Our theory fails when an additional large length scale appears in our active system. This lengthscale is associated with long-wavelength density fluctuations that emerge upon approaching motility-induced phase separation.
1. INTRODUCTION
Equilibrium statistical mechanics provides us with ex-plicit expressions for many-particle probability distribu-tions for systems that are either isolated or in contactwith one or more reservoirs [1]. Probably the most of-ten invoked distribution is the Boltzmann distribution ∝ exp ( −H /T ) describing an equilibrium system withHamiltonian H at temperature T (here and in the follow-ing we use units such that Boltzmann constant is equal to1, k B = 1). A lot of effort, analytical and/or numerical,is required to obtain from this distribution explicit re-sults for measurable properties of a system of interactingparticles, but at least we are provided with an explicitstarting point for such an effort.In contrast, for out-of-equilibrium stationary states wedo not have such a starting point. If we were to followthe same route as in equilibrium statistical mechanics, wewould need to derive an exact or approximate expressionfor a non-equilibrium steady-state many-particle distri-bution and then use it to calculate measurable propertiesof the non-equilibrium system considered. It is rather un-likely that a general formula for such a distribution exists.Conversely, it is very likely that if it were to be found itwould be more complicated than the many-particle equi-librium distribution.On the other hand, it is not clear that we need the fullmany-particle distribution. Most of the interesting prop-erties of many-particle systems can be expressed in termsof reduced distribution functions, i.e. pair distribution g ( r ) and its generalizations to groups of more than twoparticles. To calculate these properties one can attemptto derive approximate formulas for the reduced distri-bution for specific non-equilibrium steady states. Wenote that in some cases the reduced distributions in non-equilibrium steady states can be measured directly. Forexample, in the iconic scattering experiment of Clark and ∗ Email: [email protected]
Ackerson [2] the static structure factor, i.e. the Fouriertransform of the pair distribution function, of a shearedcolloidal suspension was measured. This experiment in-spired a number of other experimental, computational,and theoretical studies of the pair structure in colloidalsystems under shear.In the present paper we focus on a class of non-equilibrium systems that have attracted a lot of atten-tion in last decade, active matter systems [3–9]. Theconstituents of these systems consume energy and as a re-sult move in a systematic way. Examples include assem-blies of bacteria or of cells, suspensions of Janus colloidalparticles, swarms of insects and flocks of birds. Theseconstituents are often modeled as active or self-propelledparticles, which move in a systematic way on short-timescales and in a diffusive way on long-time scales. Im-portantly, their dynamics breaks detailed balance, andthus their stationary states are profoundly different fromequilibrium states. Needless to say, many-particle proba-bility distributions describing these stationary states arenot known explicitly. Several different approximate ex-pressions for such distributions have been proposed andtested [10–12]. In spite of a considerable body of workit is not yet clear which approximate method is mostpromising.In some limits the problem of finding the many-particlestationary distribution for systems of interacting activeparticles may simplify. For example, in a recent remark-able contribution de Pirey et al. [13] showed that in thelarge dimensional limit, higher-than-two-particle corre-lations are negligible and used this finding to derive anexact expression for the pair distribution function.Here we are interested in a more restricted problem.We consider a system of interacting active particles inthe presence of an external potential that varies slowlyin space and acts on one particle only, the tagged par-ticle. The question we want to answer is, what is thespatial distribution of tagged particle’s position? For anequilibrium system at constant temperature this prob-lem has a simple answer; the tagged particle distributionis the Boltzmann distribution for a single particle in an a r X i v : . [ c ond - m a t . s o f t ] F e b external potential at the temperature of the system. Re-markably, this answer is valid irrespectively of the spatialdependence of the external potential.We show that for a system of interacting active parti-cles in the limit of slowly varying in space external poten-tial the tagged particle distribution also has a form of theBoltzmann distribution. However, in this case the role ofthe temperature is played by a variable that is a ratioof two quantities for which we derive exact albeit formalexpressions. Importantly, we show that these quantitiesare two well-known parameters describing tagged parti-cle dynamics, the self-diffusion coefficient and the taggedparticle mobility. Thus, the role of the temperature inour tagged particle distribution is played by the ratio ofthe self-diffusion and mobility coefficients, which has longbeen recognized as one of the so-called effective temper-atures [14], the Einstein relation temperature.Recall that in equilibrium statistical mechanics thetemperature appears not only in equilibrium probabilitydistributions but also in other relations. In particular,it appears as a proportionality constant in fluctuation-dissipation relations, which connect fluctuations in equi-librium and linear response functions due to weak ex-ternal perturbations [1, 15, 16]. The derivation of theserelations relies upon the equilibrium form of the many-particle distribution and in out-of-equilibrium systemsthese relations are generally not valid. In the ninetiesCugliandolo, Kurchan and Peliti [17] realized that theviolation of fluctuation-dissipation relations can be usedto define temperature-like quantities, which they calledeffective temperatures. These temperatures are definedthrough the fluctuation-dissipation ratios, i.e. the ratiosof the properties characterizing fluctuations and linearresponse/dissipation in non-equilibrium states. Impor-tantly, Cugliandolo, Kurchan and Peliti showed that ina slowly relaxing model system, the effective tempera-ture determines the direction of the heat flow. Followingthis work, a number of different effective temperaturesand their properties have been investigated in globallydriven non-equilibrium stationary states [18, 19] and non-stationary aging systems [20, 21]. Remarkably, in drivenglassy systems it was found that several seemingly differ-ent temperatures have the same value [18], which hintedthat there might be a unique effective temperature, atleast in this case.More recently, the Einstein effective temperature,which is defined as the ratio of the self-diffusion andtagged particle mobility coefficients, has been usedto characterize some properties of active matter sys-tems [22–25]. In particular, some of us argued thatthe difference between the Einstein temperature andthe so-called active temperature, which characterizes thestrength of the self-propulsions, is a good measure of thedeparture of an active system from equilibrium [26].Since effective temperatures are defined through theratio of fluctuations in a steady state to a function de-scribing linear response of this state to a weak externalperturbation, it is not clear whether these temperaturescan also describe any non-linear response of steady states. Two studies showed the usefulness of the Einstein effec-tive temperature for non-linear response. First, Hayashiand Sasa [27] showed that the Einstein temperature de-termines the large scale distribution of a single Brownianparticle moving in a tilted periodic potential. Second,Szamel and Zhang [28] showed that the Einstein temper-ature determines the tagged particle density distributionin a slowly varying in space external potential in a systemof interacting Brownian particles under steady shear. Inboth cases the important assumption was the slow varia-tion in space of the external potential, but there was norestriction on its strength.The present result is similar to that of Ref. [28] inthat we assume that the external potential acting on thetagged particle is slowly varying in space and we showthat the density distribution is determined by the Ein-stein temperature. The important difference with thisearlier work is in that the present system is athermal,locally driven by self-propulsions of individual particles,and isotropic.We verify our theoretical results by performing com-puter simulations of an active system with an externalpotential. We show that the the theory is valid as longas the spatial scale on which the tagged particle densitydistribution varies is the longest relevant length scale inthe problem. When the density correlation length be-comes large, due to the incipient motility-induced phaseseparation, the assumption behind our theory becomesinvalid and numerical results show that the theory fails.The paper is organized as follows. In Sec. 2 we presentour theoretical derivation. In Sec. 3, we describe ourcomputer simulation model and describe our numericalprocedures in Sec. 3.1, and then we present the resultsand discuss the limitations of our theory in Sec. 3.2. Fi-nally, we conclude the paper with an overview of ourresults in Sec. 4.
2. THEORETICAL DERIVATION
To derive the equation describing the tagged activeparticle density distribution in a slowly varying in spaceexternal potential we use a gradient expansion. Specifi-cally, we use a version of the celebrated Chapman-Enskogexpansion that was originally introduced to derive hydro-dynamic equations and the expressions for transport co-efficients from the Boltzmann kinetic equation [29]. Thespecific implementation of the Chapman-Enskog proce-dure that we use is inspired by Titulaer’s [30] deriva-tion of the generalized Smoluchowski equation, which de-scribes diffusive motion of a colloidal particle in an ex-ternal potential, from the Fokker-Planck equation, whichdescribes the motion of the same particle on a shortertime scale, using both particle’s positions and momen-tum. Our present derivation is similar to that used earlier[28] to obtain an equation describing the tagged particledistribution in a sheared colloidal suspension.To make the derivation concrete we need to specifythe active particles model. We consider a system of ac-tive Ornstein-Uhlenbeck particles (AOUPs) [12, 31, 32].These particles move in a viscous medium, without in-ertia, under the combined influence of the inter-particleforces and self-propulsions, with the latter evolving ac-cording to the Ornstein-Uhlenbeck stochastic process.The equations of motions read˙ r i = µ [ F i + f i ] , (1) τ p ˙ f i = f i + η i . (2)In Eq. (1) r i is the position of particle i , µ is the mobilitycoefficient of an isolated particle, which is the inverse ofthe isolated particle’s friction coefficient, µ = ξ − , F i isthe force acting on particle i due to all other particles, F i = (cid:88) j (cid:54) = i F ( r ij ) , (3)where r ij = r i − r j and F ( r ) = − ∂ r V ( r ) with V ( r ) beingthe two-body potential, and f i is the self-propulsion. InEq. (2) τ p is the persistence time of the self-propulsionand η i is the internal Gaussian noise with zero mean andvariance (cid:10) η i ( t ) η j ( t (cid:48) ) (cid:11) noise = 2 ξ k B T a I δ ij δ ( t − t (cid:48) ), where (cid:104) . . . (cid:105) noise denotes averaging over the noise distribution, T a is the “active” temperature, and I is the unit tensor.The active temperature characterizes the strength of theself-propulsion. In addition, it determines the long-timediffusion coefficient of an isolated AOUP, D = T a µ ≡ T a /ξ .We assume that there is a slowly varying in space ex-ternal potential, Φ( r ), acting on particle 1. This particlewill be referred to as the tagged particle. The externalpotential results in an additional term, − ∂ r Φ( r ), in theequation of motion for the tagged particle,˙ r = µ [ F − ∂ r Φ( r ) + f ] , (4) τ p ˙ f = f + η . (5)We assume that the systems described by equationsof motion (1-4) can reach a stationary state. The N -particle stationary state distribution of positions and self-propulsions, P Φs , satisfies the following equation,[Ω s + ∂ r · µ ( ∂ r Φ( r ))] P Φs ( r , f , . . . , r N , f N ) = 0 . (6) Here Ω s is the evolution operator that corresponds to theunperturbed equations of motion,Ω s ( r , f , . . . , r N , f N ) = − µ N (cid:88) i =1 ∂ r i · ( F i + f i )+ N (cid:88) i =1 ∂ f i · (cid:18) τ p f i + T a µ τ p ∂ f i (cid:19) . (7)To make the assumption that the external potentialacting on the tagged particle explicit, we write it asΦ( (cid:15) r ), where (cid:15) is a small parameter. As described before[28], we will use (cid:15) as an expansion parameter and then,at the end of the derivation, we will set it to 1.Our goal is to derive from Eq. (6) a closed equationfor the stationary tagged particle density distribution, n s ( r ), n s ( r ) = (cid:90) d f d r . . . d f N d r N P Φs ( r , f , . . . , r N , f N ) . (8)The tagged particle density is non-uniform due to theexternal potential Φ. Due to the slow variation of theexternal potential, we assume that the tagged particledensity will also be slowly varying. Again, to make thisassumption explicit we write the tagged particle densityas n s ( (cid:15) r ).Due to the inter-particle interactions the N -particledistribution is not a slowly varying function of the taggedparticle position, if the positions of all other particles arekept constant. However, it should be a slowly varyingfunction of r if it is written in terms of the tagged par-ticle position and positions of all other particles relative to the tagged particle position, i.e. in terms of r and r , r etc. To make this assumption explicit we changethe variables and write the stationary state equation interms of R = (cid:15) r , R = r , . . . , R N = r N , (cid:104) Ω (0) s + (cid:15) Ω (1) s + (cid:15) ∂ R · µ ( ∂ R Φ( R )) (cid:105) P Φs ( R , f , . . . , R N , f N ) = 0 , (9)where we separate contributions to the evolution operator of different orders in (cid:15) ,Ω (0) = − µ − N (cid:88) i =2 ∂ R i · N (cid:88) i (cid:54) = j =2 F ( − R j ) + f + N (cid:88) i =2 ∂ R i · N (cid:88) i (cid:54) = j =2 F ( R i ) + f i + N (cid:88) i =1 ∂ f i · (cid:18) τ p f i + T a µ τ p ∂ f i (cid:19) , (10)Ω (1) = − µ ∂ R · (cid:32) N (cid:88) i =2 F ( − R i ) + f (cid:33) − N (cid:88) i =2 ∂ R i · µ ( ∂ R Φ ( R )) . (11)Following Titulaer [30] and Ref. [28], we now look fora special perturbative solution of Eq. (9) P Φs ( R , f , . . . , R N , f N ) = n s ( R ) P (0)s ( f , R , f , . . . , R N , f N )+ (cid:15)P (1)s ( R , f , R , f , . . . , R N , f N )+ (cid:15) P (2)s ( R , f , R , f , . . . , R N , f N ) + . . . . (12)We use the solution postulated in Eq. (12) to deriveperturbatively an equation for the tagged particle densitydistribution, (cid:16) D (0) + (cid:15) D (1) + (cid:15) D (2) + . . . (cid:17) n s ( R ) = 0 . (13)Eq. (13) is obtained by the integration of Eq. (9) over theself-propulsion of the tagged particle and the positions ofall particles other than the tagged particle. For example,the first two terms in Eq. (13) read D (0) n s ( R ) = (cid:90) d f d R d f . . . d R N d f N Ω (0) n s ( R ) P (0)s , (14) D (1) n s ( R ) = (cid:90) d f d R d f . . . d R N d f N Ω (1) n s ( R ) P (0)s + (cid:90) d f d R d f . . . d R N d f N Ω (0) P (1)s . (15)We note that the second term in Eq. (15) vanishes dueto integration by parts.Following the standard Chapman-Enskog procedure[29, 30], the tagged particle density n s ( R ) is not ex-panded in (cid:15) . Moreover, as in the standard Chapman-Enskog procedure [29, 30], there is some freedom inchoosing higher order functions P ( i )s , i ≥
1. This free-dom is eliminated by imposing the usual conditions, (cid:90) d f d R d f . . . d R N d f N P ( i )s = 0 ∀ i ≥ . (16)Conditions (16) imply that the tagged particle densityis completely determined by the zeroth order term inexpansion (9).To find the special solution for the stationary stateprobability distribution we substitute (12) into Eq. (9)and solve order by order. The terms of zeroth order giveΩ (0) P (0)s ( f , R , f , . . . , R N , f N ) = 0 . (17)Thus, P (0)s is the translationally invariant steady statedistribution of the positions and self-propulsions in theabsence of the external potential. The combination ofthe expansion (12) and conditions (16) implies that thisdistribution should be normalized to 1, (cid:90) d f d R d f . . . d R N d f N P (0)s = 1 . (18)The first order terms give an equation for P (1)s in which n s P (0)s plays the role of a source term,Ω (0) P (1)s + Ω (1) n s ( R ) P (0)s = 0 . (19) We can formally solve Eq. (19) for P (1)s , P (1)s = − (cid:104) Ω (0) (cid:105) − Ω (1) n s ( R ) P (0)s . (20)We recall that P (0)s does not depend on R and we get P (1)s = µ (cid:104) Ω (0) (cid:105) − (cid:34) N (cid:88) i =2 F ( − R i ) + f (cid:35) P (0)s · ∂ R n s ( R )+ µ (cid:104) Ω (0) (cid:105) − (cid:32) N (cid:88) i =2 ∂ R i (cid:33) P (0)s · ( ∂ R Φ ( R )) n s ( R ) . (21)We then use these results to derive successive termsin the stationary state equation for the tagged particledistribution, Eq. (13). We note that D (0) , Eq. (14), in-volves Ω (0) P (0)s , and thus it vanishes. Then, we note that D (1) , Eq. (15), consists of two terms and, as we statedearlier, that the second term vanishes due to integrationby parts. In turn, the first term, involving Ω (1) , consistsof two contributions that originate from the two contri-butions to Ω (1) , Eq. (11). The first one is proportionalto the following integral, µ (cid:90) d f d R d f . . . d R N d f N (cid:32) N (cid:88) i =2 F ( − R i ) + f (cid:33) P (0)s , (22)which involves the sum of the total inter-particle forceacting on that tagged particle and of the self-propulsionof the tagged particle. We note that integral (22) is equalto the tagged particle current in the unperturbed station-ary state. We assume that there are no average station-ary currents in the stationary state, and thus integral(22) vanishes. The term contributing to D (1) that orig-inates from the second contribution to Ω (1) , Eq. (11),vanishes due to integration by parts.The lowest order non-vanishing contribution to sta-tionary state equation (13) originates from D (2) , D (2) n s ( R ) = (cid:90) d f d R d f . . . d R N d f N ∂ R · µ ∂ R Φ( R ) n s ( R ) P (0)s + (cid:90) d f d R d f . . . d R N d f N Ω (1) P (1)s + (cid:90) d f d R d f . . . d R N d f N Ω (0) P (2)s . (23)The first term at the right-hand-side gives ∂ R · µ ∂ R Φ( R ) n s ( R ) and the last term vanishes after in-tegration by parts. The second term is a sum of twocontributions that originate from the two contributionsto Ω (1) , Eq. (11). The second one vanishes after integra-tion by parts and the first one can be re-written as ∂ R D · ∂ R n s ( R ) + ∂ R ¯ µ · ( ∂ R Φ ( R )) n s ( R ) (24)where D = − µ d (cid:90) d f d R d f . . . d R N d f N (cid:32) N (cid:88) i =2 F ( − R i ) + f (cid:33) ·× (cid:104) Ω (0) (cid:105) − (cid:34) N (cid:88) i =2 F ( − R i ) + f (cid:35) P (0)s , (25)¯ µ = − µ d (cid:90) d f d R d f . . . d R N d f N (cid:32) N (cid:88) i =2 F ( − R i ) + f (cid:33) ·× (cid:104) Ω (0) (cid:105) − (cid:32) N (cid:88) i =2 ∂ R i (cid:33) P (0)s . (26)We note that while writing Eqs. (25-26) we used the ro-tational invariance of the d -dimensional stationary statewithout the external potential.Combining all non-vanishing contributions to D (2) , set-ting (cid:15) = 1, and re-writing the resulting stationary stateequation in terms of the original coordinate r we get thefollowing equation for the tagged particle distribution, ∂ r · [ D∂ r + µ ( ∂ r Φ ( r ))] n s ( r ) = 0 (27)where, rewritten in terms or the original coordinates r , . . . , r N , D and µ read D = − µ d V (cid:90) d r d f . . . d r N d f N ( F + f ) ·× [Ω s ] − ( F + f ) P (0)s ( r , f , . . . , r N , f N ) (28) µ = µ − µ d V (cid:90) d r d f . . . d r N d f N ( F + f ) ·× [Ω s ] − ∂ r P (0)s ( r , f , . . . , r N , f N ) (29)Eq. (27) implies that the tagged particle distributionhas the Boltzmann form, n s ( r ) ∝ exp (cid:0) − Φ ( r ) /T eff (cid:1) , (30)where the effective temperature is the ratio of D and µ , T eff = D/µ. (31)The final step in the derivation is to assign some phys-ical interpretation to D and µ . This interpretation hasalready been hinted by our choice of the symbols we usedfor these quantities. First, we note that since µ ( F + f )is the tagged particle velocity, D can be formally in-terpreted as the integral of the velocity auto-correlationfunction, D = d − (cid:90) ∞ (cid:104) ˙ r ( t ) · ˙ r (0) (cid:105) , (32)which in turn is the standard expression for the self-diffusion coefficient [1]. Second, we note that if, for asystem initially in a stationary state, a weak spatiallyuniform external force F ext1 is applied to the tagged par-ticle, the tagged particle will start moving. Initially, sincethe distribution of the other particles around the tagged particle is isotropic, its velocity will be equal to µ F ext1 but after some time the the distribution of the other par-ticles will become slightly anisotropic and they will beexerting an additional friction force on the tagged par-ticle. It can be shown that due to the change of theprobability distribution the long-time limit of the taggedparticle velocity will be ( µ + ¯ µ ) F ext1 . Thus, µ = µ + ¯ µ ,Eq. (29), is the tagged particle mobility coefficient.To summarize, we showed in this section that for aslowly varying in space external potential acting on thetagged particle the tagged particle density has the Boltz-mann form with the temperature determined by the ra-tio of the self-diffusion and mobility coefficients, i.e. theEinstein effective temperature.
3. NUMERICAL VERIFICATION3.1. Methods
To test Eq. (30) for the probability distribution andEq. (31) for the effective temperature, we performed aseries of computer simulations of interacting AOUPs inan external potential Φ, evolving according to equationsof motion (1-5), and a parallel series of computer sim-ulations of unperturbed particles, evolving according toequations of motion (1-2). We simulated N = 10 par-ticles at a number density of ρ = 0 . V WCA ( r ) = 4 ε (cid:104)(cid:0) σr (cid:1) − (cid:0) σr (cid:1) (cid:105) + ε ,for r < / σ and zero otherwise. We present the resultsin standard LJ units where ε is the unit of energy, σ isthe unit of length, and σ / ( µ ε ) is the unit of time.We simulated AOUP systems at two different activetemperatures, T a = 0 .
01 and T = 1 . τ p . The values of T a were chosen toroughly represent two different dependencies of the self-diffusion coefficient on the persistence time, which weidentified in an earlier investigation [26, 33]. At the lowertemperature we expected D either to increase with τ p orto have a non-monotonic dependence on τ p . In contrast,at the higher temperature we expected D to decreasewith τ p .For simulations without the external potential we usedtime step dt = 0 .
001 for τ p ≥ .
02 and dt = 0 . τ p = 0 . dt =0 .
001 for T a = 0 .
01 (at which τ p ≥ .
02) and dt = 0 . T a = 1 . To induce a slowly varying, non-uniform density dis-tribution of the tagged particle we used a potential thatis periodic over the simulation box length L , and varyingalong the x , y or z axis,Φ( α ) = Φ sin (2 πα/L ) α = x, y, z. (33)Since for the N = 10 particle system L is much largerthan the particle size, this potential is indeed slowly vary-ing. However, we will see that if the parameter charac-terizing the strength of the potential, Φ , is large enough,the tagged particle density can vary on a smaller lengthscale.Without the external potential, the tagged particle dis-tribution is uniform and equal to 1 /V . We are primar-ily interested in the non-linear response regime, i.e wechose Φ such that the tagged particle distribution isstrongly non-uniform. Specifically, we chose Φ = 0 . T a = 0 .
01 and Φ = 1 . . T a = 1 . .
0% of the particles for T a = 0 .
01 and 0 .
2% ofthe particles for T a = 1 .
0. We note that while selectingthe percentage of particles to which external potential isapplied one has to make sure that these particles are di-lute enough in the whole system to be non-interacting.To check for this we calculated steady state structure fac-tors for the particles on which the external potential acts,in the plane perpendicular to the direction of the exter-nal force and confirmed that with these percentages theparticles were dilute enough. The fact that a smaller per-centage is necessary to fulfill this condition at T a = 1 . T a = 1 . T a = 0 . T fit . The Einstein effective temperature is defined as theratio of the self-diffusion and tagged particle mobilitycoefficients [35, 36], T E = D/µ. (34)To calculate the self-diffusion coefficient we simulateunperturbed systems of AOUPs described before and usethe standard relation, D = (2 dN ) − lim t →∞ t (cid:88) i (cid:68) ( r i ( t ) − r i (0)) (cid:69) (35)where d is the dimensionality of the system.To evaluate the tagged particle mobility coefficient forour out-of-equilibrium systems we use the approach pre-sented in Ref. [23], which involves the application ofMalliavin weights. We define the mobility coefficient in terms of a time-dependent response function χ ( t ) [23], µ = lim t →∞ t χ ( t ) . (36)In turn, the response function is calculated through av-erages involving weighting functions [23], χ ( t ) = ( N d ) − (cid:88) α,i [ (cid:104) α i ( t ) ( q iα ( t ) − q iα (0)) (cid:105) + τ p (cid:104) ˙ α i ( t ) ( q iα ( t ) − q iα (0)) (cid:105) ] , (37)where α = x, y, z and the weighting function q iα ( t ) obeysthe equation of motion,˙ q iα = µ / (2 T a ) − η iα , (38)where η iα is the α component of the Gaussian noise act-ing on particle i . For the external potential Φ varying along α axis, α = x, y, z , the tagged particle distribution varies alongthe same direction and is uniform along the remainingtwo directions. In Fig. 1 we show tagged particle distribu-tions averaged over the three directions of the perturba-tion. More precisely, we show L ¯ n s ( α ) where L = 24 . (cid:82) V d r ¯ n s ( r ) = 1. The dis-tributions shown in Fig. 1 are significantly different fromthe tagged particle density in the absence of the externalpotential, when L V − = 1 /L = 0 . T a = 0 .
01 and Φ = 0 . ∝ exp( − Φ( r ) /T fit ) using T fit as the fit parameter[34]. The resulting values T fit are shown in Fig. 3. Weobserve that the fitted temperatures decrease with in-creasing persistence time, which could have been antici-pated from the persistence time dependence of the taggedparticle densities.To verify our theory presented in Sec. 2 we need tocheck whether temperatures obtained from the fits, T fit , -12 -10 -8 -6 -4 -2 0 α L n s ( α ) τ p = 0.02 τ p = 0.2 τ p = 2 τ p = 20T a = 0.01, Φ o = 0.1(a) -12 -8 -4 0 4 8 12 α L n s ( α ) τ p = 0.002 τ p = 0.02 τ p = 0.2 τ p = 2 τ p = 20T a = 1.0, Φ o = 1.0(b) -10 -9 -8 -7 -6 -5 -4 -3 -2 α L n s ( α ) T a = 1.0, Φ o = 10.0(c) FIG. 1: The tagged particle density distribution along thedirection of the external potential, L ¯ n s , averaged over threedifferent directions of the potential. (a) T a = 0 .
01, Φ =0 . τ p ∈ [0 . , T a = 1 .
0, Φ = 1 . τ p ∈ [0 . , T a = 1 .
0, Φ = 1 . τ p ∈ [0 . , L × V − = L − = 0 . are the same as the Einstein temperatures obtained fromthe ratios of the self-diffusion and tagged particle mo-bility coefficients. Even before calculating the lattertemperatures we can infer from Fig. 3 that the theorydoes not work for the two longest persistence times for T a = 1 .
0. The reason is that the Einstein temperaturedescribes an unperturbed system and thus does not de-pend on Φ whereas for the two longest persistence timesfor T a = 1 . . We will return to this issue at the end ofthis section.In Figs. 4(a-b) we show the MSDs (cid:10) δr ( t ) (cid:11) , where δr ( t ) = ( r ( t ) − r (0)) , for unperturbed systems. The -1 t -6 -5 -4 -3 -2 -1 〈 δ α ( t ) 〉 τ p = 0.02 τ p = 0.2 τ p = 2 τ p = 20 τ p = 20, α = y -1 t -4 -3 -2 -1 〈 δ α ( t ) 〉 T a = 0.01 Φ o = 0.1 T a = 1.0 Φ o = 1.0 FIG. 2: Tagged particle mean squared displacement along thedirection of the external potential. (a) T a = 0 .
01, Φ = 0 . τ p ∈ [0 . , T a = 1 .
0, Φ = 1 . τ p ∈ [0 . , τ p = 20.The motion in the perpendicular direction is unperturbed bythe external potential. -2 -1 τ p T E / T a , T f it / T a T fit T E T a = 0.01 -3 -2 -1 τ p T E / T a , T f it / T a T fit , Φ o = 1.0T fit , Φ o = 10.0T E T a = 1.0 FIG. 3: Comparison of the temperatures obtained from fit-ting Boltzmann distributions to tagged particle density dis-tributions and the Einstein relation effective temperatures.(a) T a = 0 .
01, Φ = 0 . τ p ∈ [0 . , T a = 1 . = 1 .
0, and Φ = 10 .
0, and τ p ∈ [0 . , T a . -1 t -5 -4 -3 -2 -1 〈 δ r ( t ) 〉 τ p = 0.02 τ p = 0.2 τ p = 2 τ p = 20 τ p D (a)T a = 0.01 -1 t -5 -4 -3 -2 -1 〈 δ r ( t ) 〉 τ p = 0.02 τ p = 0.2 τ p = 2 τ p = 20 τ p D (b)T a = 1.0 FIG. 4: Mean squared displacement in unperturbed systems.(a) T a = 0 .
01 and τ p ∈ [0 . , T a = 1 . τ p ∈ [0 . , D is a non-monotonic function of τ p at T a = 0 .
01 and decreases monotonically with increasing τ p at T a = 1 . self-diffusion coefficients are calculated from these MSDsaccording to Eq. (35) and are presented in the insets. Asanticipated and in agreement with earlier investigations[26, 33], we get two different behaviors of the self diffu-sion coefficient at two active temperatures investigated.For the lower active temperature, T a = 0 .
01, we observea non-monotonic dependence of the self-diffusion coeffi-cient on the persistence time and for the higher activetemperature, T a = 1 .
0, we observe that the self-diffusioncoefficient decreases monotonically with increasing per-sistence time.In Figs. 5(a-b) we show the persistence time depen-dence of the time-dependent response function χ ( t ) [23]at the two active temperatures investigated. The insetsshow the mobility coefficients calculated from the longtime limit of χ ( t ) according to Eq. (36). We note thatat the lower active temperature, T a = 0 .
01, the mobilitymonotonically increases with increasing τ p , in contrastto the non-monotonic behavior of the self-diffusion coef-ficient. At the higher active temperature, T a = 1 .
0, themobility monotonically decreases with increasing persis-tence time, and thus exhibits the same τ p dependence asthe self-diffusion coefficient. -1 t -4 -3 -2 -1 T a χ ( t ) τ p = 0.02 τ p = 0.2 τ p = 2 τ p = 20 τ p T a µ (a)T a = 0.01 -1 t -2 -1 T a χ ( t ) τ p = 0.02 τ p = 0.2 τ p = 2 τ p = 20 τ p T a µ (b)T a = 1.0 FIG. 5: Time dependent response function to a weak externalpotential in unperturbed systems calculated using Eq. 37. (a) T a = 0 .
01 and τ p ∈ [0 . , T a = 1 . τ p ∈ [0 . , T a µ decreases monotonically with increasing τ p at both active temperatures. Comparing insets in Figs. 4(a-b) and in Figs. 5(a-b) wecan see that at the smallest persistence times D ≈ T a µ .This behavior is expected since in the limit of the vanish-ing persistence time at constant active temperature, thepresent model active systems become equivalent to Brow-nian systems at temperature equal to the active temper-ature, T = T a . For a Brownian system the fluctuation-dissipation theorem holds and D = T µ .In Fig. 3 we compare the Einstein temperatures T E defined as the ratios D/µ to the temperatures obtainedfrom fits to the Boltzmann distribution, T fit . As men-tioned in the previous paragraph, in the limit of smallpersistence times our active system becomes equivalentto the Brownian system and both T fit and T E becomeequal to the active temperature. With increasing per-sistence time, while keeping the active temperature con-stant, both T fit and T E decrease. We note that the de-crease of the ratio of the Einstein temperature and theactive temperature, T E /T a , was observed before [23, 26].It contrasts with the increase of the ratio of the Einsteineffective temperature to the bath temperature with in-creasing shear rate for colloidal suspensions under steadyshear [28].For the lower active temperature, T a = 0 .
01, we ob-serve a very good agreement between T fit and T E for allpersistence times investigated. In contrast, for the higheractive temperature, T a = 1 .
0, we initially see a verygood agreement between T fit and T E but then, for longerpersistence times we observe that temperatures obtainedfrom the fits deviate from the temperatures from the Ein-stein relation. Notably, it happens first for T fit obtainedfor the more confining potential, Φ = 10 .
0, and then T fit obtained for the less confining potential, Φ = 1 . S ( k ), S ( k ) = 1 + 1 N (cid:42) N (cid:88) i =1 N (cid:88) j (cid:54) = i exp [ − i k · ( r i − r j )] (cid:43) , (39)for unperturbed systems at both active temperatures.We observe that for the lower active temperature, T a =0 .
01, only a modest increase of S ( k ) is observed for smallwavevectors at the longest persistence times. This sug-gests that at this active temperature and in the rangeof the persistence times investigated density correlationsare relatively short-ranged.In contrast, for the higher active temperature, T a =1 .
0, we observe a large small wavevector increase of S ( k )for the two longest persistence times. This suggeststhat at this active temperature at these persistence timesthere are long-ranged density fluctuations. To make thisstatement more quantitative we simulated a larger sys-tem consisting of 8 × particles at τ p = 2 .
0. Thesmall wavevector behavior of the steady state structurefactor for this system is shown in the inset to Fig. 6.To quantify the range of the density correlations we fit-ted the numerical results to the Ornstein-Zernicke form f ( k ) = a/ [1 + ( bk ) ]. We recall that parameter b in theOrnstein-Zernicke fit is a measure of the density correla-tion length. We obtained b = 2 .
19 which is perhaps mod-erate but is larger than the length on which the taggedparticle density varies for Φ = 10 . T a = 1 . τ p = 2 .
4. CONCLUSIONS
We derived an expression for the tagged particle den-sity distribution in a slowly varying in space externalpotential in a system of interacting athermal active par-ticles. The tagged particle distribution has the Boltz-mann functional form, but the role of the temperature isplayed by the ratio of the self-diffusion and tagged parti-cle mobility coefficients. We used computer simulations k S ( k ) τ p = 0.02 τ p = 0.2 τ p = 2 τ p = 20(a) k S ( k ) k S ( k ) (b) FIG. 6: Stationary state structure factors of unperturbed sys-tems. (a) T a = 0 .
01 and τ p ∈ [0 . , T a = 1 . τ p ∈ [0 . , × par-ticle system for T a = 1 . τ p = 2. The solid line shows anOrnstein-Zernicke function, f ( k ) = a/ [1 + ( bk ) ], fitted to thedata. to verify the theoretical result. The theory works wellif the characteristic length of the tagged particle densityvariation is the longest relevant length in the system. Thetheory is inapplicable if the characteristic length of thedensity fluctuations is longer than the the characteristiclength of the tagged particle density variation.The ratio of the self-diffusion and tagged particle mo-bility coefficients has long been known as the Einsteintemperature, one of several effective temperatures ob-tained for non-equilibrium systems from the fluctuation-dissipation ratios. Our result shows that the Einsteintemperature determines the large spatial scale taggedparticle density distribution beyond the linear responseregime. This resembles earlier results that establishedthat the Einstein temperature plays similar role for a sin-gle Brownian particle in a tilted periodic potential and fora tagged particle in a colloidal suspension under steadyshear flow. These three results obtained for very differentsystems suggest that the Einstein temperature may begenerally relevant for the large spatial scale tagged parti-cle density distribution in any stationary non-equilibriumsystem in which the large scale motion is diffusive.0 Acknowledgments
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