Effective equilibrium states in the colored-noise model for active matter I. Pairwise forces in the Fox and Unified Colored Noise Approximations Ren´e Wittmann,1,a) C. Maggi,2 A. Sharma,1 A. Scacchi,1 J. M. Brader,1 and U. Marini Bettolo Marconi3 1)Department of Physics, University of Fribourg, CH-1700 Fribourg, Switzerland 2)NANOTEC-CNR, Institute of Nanotechnology, Soft and Living Matter Laboratory, Piazzale A. Moro 2, I-00185, Roma, Italy 3)Scuola di Scienze e Tecnologie, Universita` di Camerino, Via Madonna delle Carceri, 62032, Camerino, INFN Perugia, Italy (Dated: 1 February 2017) 7 The equationsof motionof active systems canbe modeled in terms of Ornstein-Uhlenbeckprocesses(OUPs) 1 with appropriatecorrelators. Forfurther theoreticalstudies, these shouldbe approximatedto yieldaMarko- 0 vian picture for the dynamics and a simplified steady-state condition. We perform a comparative study of 2 the Unified Colored Noise Approximation(UCNA) and the approximationscheme by Fox recently employed n within this context. We review and discuss in detail the approximations associated with effective interaction a potentials that aredefined in the low-density limit. We study the conditions for whichthe behaviorobserved J in two-body simulations for the OUPs model and Active Brownian particles can be represented. Finally, we 1 discuss different ways to define a force-balance condition in the limit of small activity. 3 ] t I. INTRODUCTION adiabatic elimination on the level of the Langevin equa- f o tions, (ii) the Fox approximation15,16, for which an ap- s Active Brownian particles (ABPs) provide a simple, proximate Fokker-Planck equation is developed. . t minimal model system with which to study the collec- When applied in the context of active matter, both a m tivebehaviorofactivematter. Many-bodyBrowniandy- the UCNA and Fox approximations are referred to as namics simulations of these systems have provided con- ‘effective equilibrium’approaches,owingto their Marko- - d siderable insight into a range of interesting nonequilib- vian character. Indeed, the possibility of mimicking the n rium phenomena, such as the accumulation of particles behavior of nonequilibrium ABPs using an equilibrium o atboundaries1–5andmotility-inducedphaseseparation6. systemofpassiveparticles,interactingviaeffectiveinter- c MuchofthephenomenologyofABPscanbecapturedus- actions, was suggested by severalresearchers (see e.g.17) [ ing coarse-grained, hydrodynamic theories6–9, however, whoobservedthatthephaseseparationinducedbyactiv- 1 theseapproachesarebasedongeneralphysicalarguments ity in systems of repulsive ABPs closely resembled that v whichdonotenableadirectconnectiontobemadetothe familiar from passive systems with an attractive inter- 2 microscopic interparticle interactions. Although some action. Despite its appeal, several years were required 3 progress has recently been made in this direction10, this before this observation could be turned into something 0 remains work in progress. moreconcrete. Bystartingfromthe simplerOUPmodel 9 0 Duetotheinherentdifficultyofdealingsimultaneously itbecamepossible,viaapplicationoftheUCNAandFox . with both the translational and orientational degrees of approximations,to putthe notionofaneffectiveequilib- 1 freedom in active systems, attempts to develop a first- rium description on a firmer footing11,18–24. 0 7 principles theory have largely focused on a simpler, re- Inthispaper,wewillcompareandcontrastthetwodif- 1 lated model, in which the particle dynamics are repre- ferent approaches to effective equilibrium. We will high- : sented by a set of coupled Ornstein-Uhlenbeck processes v light the main approximations involved and assess the (OUPs). Within this model an exponentially correlated i validity of the effective-potential approximation (EPA), X noiseterm,withagivencorrelationtime,servesasproxy whichhasbeen employedinpreviousworkto investigate r for the persistent trajectories of ABPs (connections be- activity induced modifications of the microstructureand a tween the two models were explored in Ref. 11). While motility-induced phase separation11. This analysis both the removal of orientational degrees of freedom does in- clarifies the nature of the approximations involved and deed simplify the problem, it comes at the cost that one suggests ways in which the description can be improved. has to deal with the non-Markovian dynamics of the Thepaperislaidoutasfollows: InSec.IIwefirstspec- translational coordinates. Fortunately, there exist ap- proximation methods12–16 which enable the OUP model ify the model under consideration. The UCNA and Fox approachestoobtaininganeffectiveequilibriumarethen to be represented using an effective Markovian, and described in Sec. III, highlighting similarities and differ- therefore tractable, dynamics. There are two different ences betweenthem. InSec. IV we describe indetail the approaches to doing this, (i) the Unified Colored Noise Approximation (UCNA) of Ha¨nggi et al.13,14, based on EPA,where the emphasis is placedonthe UCNA due to its simpler structure. The resulting approximate effec- tive potentials are compared to computer simulations in Sec. V using a standard soft-repulsive and a non-convex a)Electronicmail: [email protected] (Gaussian core) potential. In Sec. VI we consider an al- 1 ternativeapproachtoobtainpairwiseforces,i.e.,thelow- of the self propulsion. We now aim to clarify some nota- activity limit, and make contact to the EPA in Sec. VII. tional differences in the literature. The persistence time Finally, we conclude in Sec. VIII. τ can be explicitly related to the equations of motion a of run and tumble particles25 or active Brownian parti- cles11,26. The above definitions correspond to the latter II. COLORED-NOISE MODEL case with τa=Dr−1/(d−1), where Dr is the rotational Browniandiffusioncoefficient. Anothercommonchoice21 amounts to consider τ =D−1 and D =v2τ /(d−1)/d. In this section, we introduce the common starting R r a 0 R In the following, we use τ˜ := τ /γ ≡ βτd2, where d is point of both the UCNA and Fox approach. Since parti- a the typicaldimensionofaparticleandthe dimensionless cles drivenby Gaussiancolorednoise originallywere not persistence time τ has been introduced in Refs.11,23. intended as a model for an active system, the choice of Since the dimensionless variable D :=D /D implic- parameters in the literature may depend on the dimen- a a t sionality and on whether contact to ABPs is made11 or itly depends on the persistence time, it constitutes the not18. We will also clarify some notational issues. most general measure for the activity. At vanishing propulsionvelocityv =0,wehaveD =0andEq.(1)re- We consider the coupled stochastic (Langevin) differ- 0 a duces to the equation of motion of a passive (Brownian) ential equations particle with finite reorientation time τ . In the limit a r˙i(t)=γ−1Fi(r1,...,rN)+ξi(t)+vi(t) (1) τa→0 we also recover a passive (non-Brownian) system, as the velocity correlation (4) reduces to a white noise. ofN particles. Themotionofeachparticlei∈{1,...,N} Inorderto connectto anactiveBrowniansystem11,23, it at position ri(t) is determined by conservative Fi and is convenient to consider instead of Da a dimensionless stochastic forces γξi and γvi. The friction coefficient velocity Pe = v0d/Dt, i.e., the Pecl´et number. In the γ = (βD )−1 is related to the translational Brownian literature, some other definitions of a Pecl´et number are t diffusivity D and β = (k T)−1 is the inverse tem- used, which we will not consider here. t B perature. We assume that the total interaction force F (rN)=−∇ U(rN) can be written as the gradient of i i a pairwise additive many-body potential III. EFFECTIVE EQUILIBRIUM APPROACH N 1 U(rN)= V (r )+ u(r ,r ) , (2)  ext i 2 i k  The most important step towards a theoretical study Xk6=i of the OUPs model is to derive from the non-Markovian   stochastic process (1) an equation of motion for the N- consistingoftheone-bodyexternalfieldsV (r )andthe ext i particle probability distribution f (rN,t). In this sec- interparticle potentials u(r ,r )=u(|r −r |). N i k i k tion, we will discuss the differences between the multi- The vector ξ (t) represents the translational Brown- i dimensional generalizations of the UCNA13,14 and the ian diffusion by a Gaussian (white) noise of zero mean Fox15,16 approachestoeffectiveequilibriumandexpound and hξ (t)ξ (t′)i = 2D 1δ δ(t − t′) with the unit ma- i j t ij the surprising similarities between these two approxima- trix 1. Here and in the following the dyadic product of tions in the (current-free) steady state. two vectors with d components results in a d×d matrix. Asacentralquantityemerginginbothcases,wedefine Anycontractionasinascalarproductoramatrix-vector the dN ×dN friction tensor Γ with the components product will be explicitly indicated by a “·”. The OUPs [N] v (t) defined by i Γ (rN)=1δ −τ˜∇ F ij ij i j v˙ (t)=−vi(t) + ηi(t) (3) =δijΓii(rN)+(1−δij)τ˜∇i∇ju(ri,rj) (5) i τ τ a a resultingintheHessianofU andthediagonald×dblock withhη (t)η (t′)i=2D 1δ δ(t−t′)describeafluctuating i j a ij propulsion velocity as a non-Gaussian (colored) noise of N zero mean and Γ (rN):=1+τ˜∇ ∇ V (r )+ u(r ,r ) (6) ii i i ext i i k (cid:18) k6=i (cid:19) hvi(t)vj(t′)i= vd021δije−|t−τat′| = Dτaa1δije−|t−τat′| . (4) not to be confused with Γ[1](r1) forXN =1 particle. In the following, we briefly denote by Γ−1 the ijth block Here we introduced the active time scale τ atwhich the ij a component of the inverse tensor Γ−1. orientationrandomizesandtheactivediffusioncoefficient [N] D =v2τ /d, where v2=hv2(t)i is the average squared The UCNA18,19 amounts to explicitly inserting the a 0 a 0 i self-propulsion velocity and d the spatial dimension. OUPs (3) into the overdamped limit of the time deriva- The colored-noise model for active particles contains tive of (1), resulting in the modified Langevin equation twoparametersdescribingthemagnitudeandpersistence r˙ (t)=Γ−1(rN) γ−1F (rN)+ξ (t)+η (t) . It is now i ij j j j (cid:0) (cid:1) 2 straight-forwardtoobtainforthis(approximate)Marko- introducing the effective force Feff(rN). It has been ar- k vian system driven by white-noise the Smoluchowski gued19 thatthe contributionofthe off-diagonalelements equation∂fN(rN,t)/∂t=− Ni=1∇i·Ji(rN,t) with the to D[N] becomes increasingly irrelevant for high particle probability current (the superscript (u) denotes that the numbers N, which means that we may set Dij→δijDij. UCNA has been used) P Assumingthisdiagonalform,thedeterminantinEq.(12) needs to be replaced according to detD →detD as [N] kk J(iu) = DtΓ−ik1· βFkfN−(1+Da) ∇j· Γ−jk1fN . wbeefofirnedrewirjiDtinj−kg1·t∇hej·eDxipjr≡esDsik−ok1n·∇inkt·hDekkla=st∇stkelpn.(detDkk) Xk (cid:18) Xj (cid:16) (cid:17)(cid:19) PuttiPng aside the dynamical behavior, the only differ- (7) ence between the UCNA or Fox approximation is man- NotethattheUCNAremainsvalidaslongasthefriction ifest in the definitions, (10) and (11), of D . Using tensor (5) is positive definite. [N] UCNA the active diffusivity D only appears as part of The Fox approximation scheme applied to (1), on the a a prefactor in (10), so that the friction matrix Γ , rep- other hand, only makes use of the correlator (4) of the [N] resenting a correction due to activity, contributes to the OUPs, which, in turn, may also be interpreted as the steady-state result even in the case D =0, that is when correlator of v (t)≃v p (t) corresponding to a coarse- a i 0 i v =0andτ 6=0. Hence,thelogicalparametersuggested grained equation of motion representing active Brown- 0 a by the UCNA to tune the activity is τ , with the passive ian particles with a constant velocity v in the direction a 0 system (D = 1δ ) restored only in the limit τ → 0. of their instantaneous orientation p that is subject to ij ij a i Brownian rotational diffusion11. This method directly Notethatinthiswaytheconnectiontotheexperimental situation with Brownian particles is lost, where it seems yields the approximate Smoluchowski equation (super- script (f))24,27 more natural to tune v0 at constant τa. The derivation of the Fox result (11), on the other hand, aims to ap- proximately represent active Brownian particles11. This J(f) =D βF f −∇ f −D ∇ ·(Γ−1f ) , (8) i t i N i N a j ji N reflects that we recover the (same) passive system for (cid:18) Xj (cid:19) both v0=0 and τa=0. IgnoringthecontributionoftheBrowniandiffusionfor where the regime of validity is the same as for UCNA. D ≫D ,the UCNAandFoxapproximationspractically The major difference between Eq. (7) and (8) only af- a t describe the equivalent effective steady states. The ma- fects the effective descriptionofthe dynamics as a result of the additional factor Γ−1 arising on the level of the joradvantageofthisapproximation,ortheUCNAresult ik in general, is that the inverse D−1 ∝ Γ is pairwise Langevin equation within the UCNA. Note that in the [N] [N] originalgeneralizationoftheFoxresult11 thetensorfrom additive, even if Dt6=0. Then the effective many-body Eq.(5) was obtained as Γij≈δij(1−τ˜∇i·Fi), which we potential HN defined as Fekff(rN)=−∇kHN(rN) can be written in a closed form18,19, admitting the explicit so- willlateridentifyasthe(diagonal) Laplacian approxima- lution P (rN) ∝ exp(−βH (rN)) of Eq. (9). Due to tion. It will turn out that this (or another) approxima- N N the more nested form of Eq. (11) the Fox approximation tion is necessary to obtain physical expressions for the does in general not admit an analytic result. As H effective interaction potentials. N is not pairwise additive in either approach, some further In contrast to the dynamical problem, the (current- approximationswillbecomenecessarytoconstructapre- free) steady-state conditions dictivetheory,whichwediscussinthefollowingsections. βF P − ∇ ·(D P )=0 (9) i N j ji N j X IV. EFFECTIVE-POTENTIAL for the stationary distribution P (rN) can be cast in APPROXIMATION (EPA) N a coherent form, defining the effective diffusion tensor D (rN)=D D (rN), suchthat onlythe components Regarding the possible applications using standard [N] t [N] methods of equilibrium liquid-state theory a desirable D(u)(rN):=(1+D )Γ−1(rN), (10) strategy is to approximate Feff in Eq. (12) in terms of ij a ij k pairpotentials. Thisapproachhas beenusedto describe D(f)(rN):=1δ +D Γ−1(rN). (11) ij ij a ij the phase behavior of active Brownian particles (ABPs) approximated as particles propelled by a set of coupled differ between the UCNA (u) and Fox (f) results. OUPs have been discussed in detail11,23 for passive soft- Multiplying Eq. (9) with D−1 and summing over re- ik repulsiveandLennard-Jonesinteractionsinthreedimen- peatedindices,thesteady-stateconditiontakesthemore sions. However, it can be criticized that (IV.i) a sys- instructive form19 tem which obeys detailed balance is used to represent 0= D−1·βF P −∇ P −P ∇ ln(detD ) the interactions in an active system, (IV.ii) the validity ik i N k N N k [N] criteria of underlying theory might be violated so that i X further approximations are required and (IV.iii) higher- =:βFeffP −∇ P (12) k N k N order particle interactions are neglected, which are be- 3 lievedto be important for the phase separationin anac- β/(1+D )to absorbthe factor(1+D )19,21. We refrain a a tive system. In the following,we define the effective pair to do so as this interpretation would not be consistent interactionandmotivatedifferentapproximations,which with the the way D enters within the Fox approach. a wecomparetocomputersimulationsoftwoactiveBrow- Studying Eq. (13) more carefully, we notice that the nianparticlesandtwoparticlespropelledby OUPs. Itis effective potential does not always behave in a physi- our objective to comment on the aforementioned points cal way. This is because, in violation of the validity andillustratethequalitativedifferencesbetweentheFox condition of both the Fox and the UCNA, the diffusion andUCNA.Forthesakeofsimplicity,wewillrestrictthe tensor D is not positive definite for a large number [2] presentation of technical aspects to the UCNA results. of relevant potentials. Inspection of (16) shows that To identify an effective pair potential ueff(r), we con- we may expect a divergence of the term ln(detD ) in [2] sider N = 2 interacting particles, i.e., the low-density the effective potential whenever one of the Eigenvalues limit ofEq.(12). Ignoringthe externalforcesfornow by E [u(r)]:=1+2τ˜rn−2∂nu(r) for n∈{1,2} vanishes. We n r setting V (r)≡0, it is easy to verify that further identify ∇ u(r) in the first term of (13) as the ext 1 Eigenvector of D−1 −D−1 corresponding to the Eigen- ∇1βueff(r)= D1−11−D2−11 ·∇1βu(r)+∇1ln(detD[2]) value E2. 11 21 (13) Given a positive and convex bare potential u(r)>0, (cid:0) (cid:1) the Eigenvalue E is strictly positive, which means that 2 and analog equation for ∇2ueff(r), where we used the effective attraction solely arises from the last term ∇2u(r)=−∇1u(r) and r=|r1 −r2|. Keeping in mind in (13). However, as we have ∂ru(r) < 0 in this case, thatweseektoemploythiseffectivepotentialtoapprox- the Eigenvalue E will vanish at a certain value of r and 1 imately represent the interaction of many particles, it we require a further approximation to remedy the un- appears undesirable that an equal statistical weight is physical behavior of ueff(r) in d> 1 dimensions. At a put to both the diagonal D−1 and the off-diagonal com- highly non-convex or negative region of the bare poten- 11 ponents D−1 of the diffusion tensor. As an alternative tial, the same problem occurs for E . Interestingly, if 21 2 we propose the effective potential we only require knowledge of an effective potential on a finite interval where the eigenvalues are positive, its ∇kβuedffiag(r)=Dk−k1·∇kβu(r)+∇kln(detDkk), (14) overall unphysical behavior is irrelevant24. Also, note that there is a broader range of admissible bare poten- withk ∈{1,2},obtainedforadiagonalformofD with [2] tials when the diagonal form of the effective potential V (r)≡0. Forcompleteness wefindin the one-particle ext is used. In the following, we propose different ways to limit a quite similar formula rid the effective potential (13) of possible artifacts due to a vanishing determinant. Correcting the other term ∇βVeff(r)=D−1·∇βV (r)+∇ln(detD ) (15) ext [1] ext [1] is neither necessary nor has any noticeable effect. The presentedmethods canbe similarlyused to approximate fortheeffectiveexternalfieldVeff(r),asforN=1wehave ext the Fox results, in a way that the equivalent behavior in D =D . A quite different expression for an effective [1] 11 the high-D limit is retained. a external potential has been derived for active Brownian Letusfirstassumethatu(r)>0isconvex. Thenasuf- particles23,28. ficient criterion for the matrix D to have strictly posi- Integration of the above equalities yields the desired tive Eigenvalues would require th[2e]operator ∇ ∇ to be i j formulas for the effective potentials depending only on elliptic. Duetothepresenceoftheterm1δ ,someother ij the bare potential u(r) (or V (r)) and the activity pa- ext potentialsareallowedthatareonlyslightlynegativeand rameters D and τ 11. Alternatively, we could have a a slightlynon-convex. Averygeneralapproximationwould directly defined19,21 Veff(r) := H (r) and ueff(r) := ext 1 thus be to redefine Eq. (5) by an elliptical operator, the H2(r1,r2) from the many-body potential HN(r) identi- simplest example of which is the Laplacian ∆=∇·∇. fiedinthesolutionof (9),whichis,however,inconvenient Uponrepresenting∇ ∇ →1∇ ·∇ the effectivepoten- i j i j when the Fox approachis used. Assuming a bare poten- tial becomes tial u(r) obeying lim u(r)=0, the integrated form r→∞ of (13) reads u(r)+τ˜(∂ u(r))2−2(d−1) ∞dsτ˜(∂su(s))2 βueff(r)=β r r s ∆ βueff(r)=βu(r)+τ˜(∂ru(r))2 1+Da R ∂ u(r) 1+Da −ln 1+2τ˜∂2u(r)+2(d−1)τ˜ r (17) ∂ u(r) (d−1) (cid:18) r r (cid:19) −ln 1+2τ˜ r 1+2τ˜∂2u(r) ,(16) (cid:12) r (cid:12) r ! wheretheadditionaltermcomparedto(16)cannotbein- (cid:12) (cid:12) (cid:12) (cid:12) tegratedingeneral. ThisLaplacian approximation based (cid:12) (cid:12) (cid:12) (cid:12) where ∂r=∂/∂(cid:12)r. Assuming (cid:12)a diagonal diffusion tensor, on the Fox approach has already been employed in the we obtain ueff (r) from Eq. (16) by rescaling all terms diagonal form in explicit calculations11,23. In d=1 di- diag proportionaltoτ˜withafactor1/2. Notethatin(16)we mensionsbothdifferentialoperatorsreducetothesecond could equally introduce an effective energy scale β = derivative and Eq. (17) is equal to Eq. (16), which pro- eff 4 effu(r) (a) UCNA, Eq. (16), Da=4.8 effu(r) (b) UCNA, τ=0.1, Da=4.8 effu(r) (a) 1d, τ=0.025, Da=4.8 effu(r) (b) 3d, τ=0.05, Da=2.4 β β β β 2 321ddd 2 33dd, diagonal 2 UFUoCCxNNAA/Fox, no noise 0,5 OOAUUBPPPsss, no noise OUPs ABPs, no noise Eq. (16) 1 OUPs, no noise 0 0 0 0 -2 τ=0.025 τ=0.1 -2 inv. τ Eq. (17) -1 -0.5 -0,5 0,9 1 1,1 1,2 1,3 r/d 0,9 1 1,1 1,2 1,3 r/d 0,9 1 1,1 r/d 1 1,2 1,4 1,6 r/d effu(r) (c) UCNA, τ=0.025, Pe=24 effu(r) (d) UCNA, τ=0.025, Da=4.8 FIG.2. Effectivepotentialswithandwithoutthermalnoise. β inv. τ Da=4.8 (3d) β (a) As Fig. 1d for d = 1 including the Fox approximation Da=7.2 (2d) Da=14.4 (1d) 0 (dashed lines). Ignoring thermal noise, UCNA and Fox are 0 equivalent (dot-dashed line) and the modified simulation re- simulations inv. τ simulations sult of active OUPs is labeled with dots. (b) Simulations of --21 Eq. (17) -1 Eq. (17) PPPeee=≈≈112934.. 69(3 ((d21)dd)) iAnBdP=s3(edmimpteynssiyonmsb.ols) and OUPs for τ=0.05 and Da=2.4 1 1,2 1,4 r/d 1 1,2 1,4 r/d FIG. 1. Effective potentials from the UCNA in d=1 (dotted V. COMPUTER SIMULATIONS AND lines), d=2 (dashed lines) and d=3 (solid lines) dimensions. THEORY OF TWO ACTIVE PARTICLES (a)Fullresult,Eq.(16),fortheactivediffusivityD =4.8and a persistence times τ =0.025 (thick, brighter lines) or τ =0.1 In Sec. IV we introduced different strategies to define (thin, darker lines). (b) Comparison to the diagonal form a suitable effective interaction potential in the effective- (dot-dashedlines) ind=3forD =4.8andτ=0.1. Thethick a equilibrium approximation for the colored-noise model. linescorrespondtotheinverse-τ approximation(seetext)and Nowweillustrateunderwhichconditionsanapproximate the thin lines to Eq. (17) in the Laplacian approximation. treatment becomes necessary and compare the theoreti- (c) Approximate results compared to simulations of active OUPS(lineswithtriangles)atτ=0.025andaconstantPecl´et cal results to computer simulations. The easiest way to number Pe=pdDa/τ=24, as Da increases with decreasing determineaneffectivepotentialnumericallyistosetupa d. (d)Approximateresultscomparedtosimulationsofactive two-particle simulation, measure the radial distribution OUPS at D =4.8 and τ=0.025. function g(r) and calculate βueff (r)=−lng(r). By do- a sim ing so, we make the same approximation (IV.iii) as in theorytoignorethemany-particlecharacteroftheinter- action. However, the simulations for ABPs and OUPs, detailed in appendix A, take into account the orienta- vides a good account of active particles interacting with tiondependenceandthe non-Markoviancharacterofthe a soft-repulsive potential21. dynamics, respectively. An alternative approach is to empirically rectify the We first discuss some general observations in the explicit formula for ueff(r) in Eq. (16). Most intuitively, UCNAforasoft-repulsivesystemwiththebarepotential one may expand the determinant in (16) up to the first βu(r)=(r/d)−12. ThebehavioroftheFoxresultsisqual- orderin τ. In fact, this small-τ approximation turns out itatively similar. As expected, the full expressionfor the to be quite similar to the Laplacian approximation (17) effectivepotentialinEq.(16)isimpracticalasitdiverges (andcompletelyequivalentinonedimension), butwe do at a certain distance rdiv, determined by the condition not recover the additional term involving the integral. rdiv = (24τ)−1/12d, which is when the first Eigenvalue Performing the small-τ approximation of the full effec- E1 within the logarithm vanishes, whereas E2 is always tive potential appears too crude, as an expansion of the positive. As suggested by Fig. 1a, this behavior is most Tlohgearriethsumltidnogeesffnecottivceonpvoetregnetiafolrwτ˜illthl>u1s∆beuc(orm,rel)to>tall2y. pbreorbalethmeartnicegaattliavreg,earsviatluisesthoef τc,aswehienredu=eff1(rddimiv)enshsioounlsd. uncontrolledforshortseparationsofPhighly-repulsivepar- We further see in Fig. 1a that this effect becomes more ticles. severewithincreasingdimension. Boththeinverse-τ and Laplacian approximations successfully cure this unphys- An approximation that may be applied also to highly ical divergence, which we see in Fig. 1b. As employing non-convexpotentialsisthe inverse-τ approximation, an the diagonal form ueff (r) of the effective potential sim- diag empirical strategy maintaining the leading order in τ ply amounts to a rescaling of τ, we observe in Fig. 1b while not disregarding higher-order terms. This can be that it results in a smaller effective diameter of the re- achievedby replacing an Eigenvalue E with 1/(2−E ) pulsivepartbutaflatterpotentialwell. Accordingly,r n n div wheneveritbecomessmallerthanone. Themajoradvan- becomes smaller. tage of this approximation is that it yields quite similar Since the definition of the active diffusivity D de- a results to the full potential whenever the validity condi- pends on the dimension d, not all parameters τ, D and a tion is only slightly violated. Pe can be kept constant upon varying the dimension- 5 effu(r) simulations, τ=0.05 effu(r) Eq. (17), τ=0.05 effu(r) inverse τ, τ=0.05 effu(r) Eq. (16), τ=0.05 β0 β0 Fox β0 Fox β0 Fox OUPs UCNA UCNA UCNA -0,2 -2 -1 -1 increasing Da and Pe increasing Da and Pe increasing Da and Pe ABPs increasing Da and Pe -0.2 -0.2 -0.2 1 1.2 1.4 1.6 r/d 1 1.2 1.4 1.6 r/d 1 1.2 1.4 1.6 r/d 1 1.2 1.4 1.6 r/d effu(r) simulations, Pe=12 effu(r) Eq. (17), Pe=12 effu(r) inverse τ, Pe=12 effu(r) Eq. (16), Pe=12 β β Fox β Fox β Fox OUPs -0,1 -1 -0,5 -0,5 ABPs increasing τ and Da increasing τ and Da UCNA increasing τ and Da -0,2 increasing τ and Da -2 -1 UCNA -1 UCNA 1 1.2 1.4 1.6 r/d 1 1.2 1.4 1.6 r/d 1 1.2 1.4 1.6 r/d 1 1.2 1.4 1.6 r/d effu(r) simulations, Da=2.4 effu(r) Eq. (17), Da=2.4 effu(r) inverse τ, Da=2.4 effu(r) Eq. (16), Da=2.4 β β β Fox β Fox 0 0 0 0 Fox OUPs -0,1 increasing τ -1 idneccrreeaassiinngg τP aend -0,5 increasing τ and -0,5 and decresing Pe decreasing Pe UCNA -0,2 ABPs -2 UCNA -1 UCNA -1 increasing τ and decreasing Pe -0.2 1 1.2 1.4 1.6 r/d 1 1.2 1.4 1.6 r/d 1 1.2 1.4 1.6 r/d 1 1.2 1.4 1.6 r/d FIG. 3. Effective potentials in three dimensions for soft-repulsive spheres, βu(r)=(r/d)−12, obtained within the Fox (thick, brighterlines)andUCNA(thin,darkerlines)andbynumericalsimulationsofABPs(emptytriangles) andactiveOUPs(filled triangles). Columnsfrom left toright: simulations; Laplacian approximation,Eq.(17);inverse-τ approximation; Eq.(16)only forr>r (seeFig.1aforthefullresult). Rowsfromtoptobottom: increasingtheactivediffusivityD atconstantpersistence div a time τ =0.05; increasing Da and τ at constant Pecl´et number Pe=p3Da/τ =12; increasing τ at constant Da=2.4. The sequence from the solid to the dotted lines corresponds to increasing the respective parameter(s) D from D =1.2 or τ from a a τ=0.025 bya factor of two in each step (dashed lines are always for thesame set of parameters). ality. For a constant reorientation time τ and propul- higher spatial dimensions, the quantitative agreement sion velocity Pe the effective attraction in Fig. 1c is with the simulation results in Figs. 1c and d becomes strongerinlowerspatialdimensionsforbothapproxima- worse. For d=1, a remarkable agreement between the tions considered, which coincide with the full expression UCNA and simulation results for the radial distribution in d=1. This behavior appears sensible, as two particles of two particles has been reported in Ref. 18, where the have less possibilities to avoid each other upon collision, Brownian noise ξ (t) in Eq. (1) was set to zero. Doing i and is in qualitative agreement with computer simula- so alsoin our simulations,we observein Fig.2a that the tions of active OUPs. Moreover, we understand that effective potential deepens and its repulsive barrier be- motility-induced phase separation is harder to observe comes steeper. This curve is in excellentagreementwith in higher dimensions29. Keeping D constant instead of thetheoreticalresultforzeronoise,obtainedbyboththe a Pe(whichthendecreaseswithdecreasingdimension)the UCNAandFoxapproachupondroppingthefirsttermin sametrendisobservedinFig.1dfortheinverse-τ approx- Eq.(10)andEq.(11),respectively. ThefullUCNAresult imationandcomputersimulations,whereastheresultfor is only slightly different in the repulsive regime. Intrigu- theLaplacianapproximationbarelychangeswithdimen- ingly, we also recognize in Fig. 2a that the simulation sionality. Comparingthe qualitativebehaviorinFigs.1c data including the noisetermare excellentlyrepresented and d, we recognize in all spatial dimensions that the by the Fox approach. To further study the influence of numerical effective potential is of longer range than the thetranslationalnoise,wealsoperformedcomputersim- theoreticalpredictionsinanyapproximation. Thisobser- ulations of ABPs (described in appendix A) for d=3. vationconfirmsthecriteriondiscussedinRef.14thatthe Figure 2b reveals that the numerical effective potentials UCNAisexpectedtobecomelessaccurateforlargersep- for the two considered models with and without trans- arationswhereatypicallengthscaleoftheactivemotion, lational noise are nearly identical over the full range of closely related to the effective diffusion tensor, Eq. (10), separations. As for d=1, the effective potential of active exceedsthe spatialscaleoverwhichtheforcefieldvaries. OUPs (and ABPs) in the absence of thermal noise has a deeper well and a larger repulsive diameter. Quantita- As the involved approximations become cruder in 6 tdiivmeleyn,stiohniss.difference is much more pronounced in three βeffu(r) (a) UCNA, Da=4.8 βeffu(r) inv. τ (b) UCNA, τ=0.05 In the following, we restrict ourselves to a three- Eq. (16) τ=0.05 τ=0.1 dimensional system with translational Brownian noise. τ=0.4 0,5 increasing Da and Pe 2 AsinFig.2b,oursimulationsofABPsandOUPs,shown βu=exp(-(r/d)2) in the first column of Fig. 3, are in nice agreement for 0 Eq. (17) OUPs all sets of parameters. This is quite surprising since on 0 the many-particle level ABPs and OUPs have different 0 0,5 1 1,5 r/d 0 0,5 1 1,5 r/d pstleisatdicytswtaot-ebso3d0.y sOynstetmhewbeascisouolfdoruarthdeartacofnocrluthdee sthimat- effu(r) (c) UCNA, Pe=12 effu(r) (d) UCNA, Da=2.4 OUPs including translational Brownian noise should be β inv. τ β inv. τ increasing τ and Da an excellent model for ABPs at moderate activity11,23. 1 1 increasing τ and decreasing Pe We see in Fig. 3 that the depth of the attractive well of all theoreticalversions of the effective potential is signif- 0,5 0,5 icantly overestimatedwhen comparedto the simulations OUPs βu=exp(-(r/d)2) OUPs βu=exp(-(r/d)2) 0 0 ofbothABPsandOUPs. Roughly,thisdeviationisbya 0 0,5 1 1,5 r/d 0 0,5 1 1,5 r/d factorof10fortheLaplacianapproximation(secondcol- umn)and(only)5fortheinverse-τ approximation(third FIG.4. EffectivepotentialsfromtheUCNAinthreedimen- column). To facilitate a qualitative comparisonwe chose sions for Gaussian-core particles, βu(r)=exp(−(r/d)2). (a) the y axes accordingly. Moreover,the inverse-τ approxi- ComparisonofthedivergencesinthefullresultfromEq.(16) mation appears to provide the best guess of the point at and theLaplacian approximation, Eq.(17),for different per- which the effective potential changes its sign. The cho- sistence times τ. (b)-(d) Inverse-τ approximation and com- sen approximationsexhibit a similar behavior as the full puter simulations for active OUPs for the same parameters as in rows 1-3 of Fig. 3, respectively. The bare potential is theoretical results of Eq. (16) in the physical region for shown as thethick dot-dot-dashed line. r>r , shown in the last column. div For the considered soft-repulsive bare potential, we observe in Fig. 3 some notable quantitative differences we stress that it is not clear in how far the effective pair between the UCNA and Fox results, even at relatively potentials can accurately describe the many-body situa- high Da. The effective diameter of the repulsive part is tion, as there are no higher-order interactions present in generally smaller than in the UCNA, whereas the over- a two-body simulation. all attraction is weaker in the Fox approach. This be- The most compelling argument in support of the comes most apparent in the Laplacian approximation. inverse-τ approximation arises from considering non- The first row of Fig. 3 contains the effective potentials convexbarepotentials,acaseinwhichtheLaplacianap- evolving for a constant persistence time τ when the ac- proximation becomes useless above a certain value of τ. tivediffusivityDa(orthePecl´etnumberPe)isincreased. To illustrate this behavior, we discuss the more general All approachesaccordinglypredict anincreasedeffective potential βu(r) = exp(−(r/d)2) of an active Gaussian- attraction and the minimum of the potential is shifted core fluid. Although this model has not received much to smaller separations11,23. The Fox results exhibit a attention,itisquiteappealingfromatheoreticalperspec- strongervariationwithDa,whichismoreconsistentwith tive. Most prominently, this bare potential is a known the numerical data. Similarly, the effective potentials exceptional case in which a simple mean-field theory is in the second row deepen with increasing τ at constant particularly accurate31,32, which might, in a way, also Pe, where the location of the minimum is nearly un- hold for the effective potential of the active system. affected. The most interesting behavior is observed in TheeffectivepotentialofanactiveGaussian-corefluid the third column at constant Da. Again, all approaches is discussed in Fig. 4 within the UCNA. As the absolute agree that the minimum is shifted to larger separations value of both the curvature and slope of this model po- with increasing τ, but the effective attraction predicted tential is bounded, the Fox results (not shown) are quite by the simulations is nearly constant, as simultaneously similar, even for the moderate values of D considered a the magnitude of the self-propulsion is decreased. This here. When the persistence time τ is sufficiently small, observation is not consistent with the UCNA results. theeffectivepotential,Eq.(16),doesnotdivergeandthe Based on the presented simple comparison, our con- different approximations behave in a quite similar way. clusionisthatthebestchoiceforthetheoreticaleffective Interestingly,weobserveinFig.4athatthedivergenceof potential is the Fox approach in the inverse-τ approxi- the Laplacian approximation sets in at an even smaller mation. Uponfurtherincreasingtheactivity(notshown) value of τ≥1/12 than for the full potential. The latter thequantitativediscrepancyoftheLaplacianapproxima- diverges at two points, each related to one of the two tionbecomesevenmorepronounced. RecallfromFig.1b Eigenvalues, if τ≥1/4. that the additional assumption of the diagonal form of In Fig. 4b, c and d we discuss the only suitable form the effective potential also results in a slightly better of the effective potential, i.e., in the inverse-τ approxi- (quantitative)agreementwiththesimulations. However, mation. Intriguingly, the predicted behavior depends on 7 in which way, i.e., by means of which parameter, the ac- sityoperator. Approximatingtheinversemobilitymatrix tivity is modified. Increasing the active diffusivity (or inEq.(10)asΓ−1(rN)≈1δ −τ˜∇ ∇ U andintegrating ij ij i j the Pecl´et number) at a constant value of τ results in Eq.(9) overN−1 coordinates,we find the firstmember a less repulsive core. Upon increasing τ, however, the −ρ(r)h∇βUi=(1+D )(∇·(1ρ(r)−τ˜ρ(r)h∇∇Ui)) , height of the maximum increases and an attractive well a (18) develops at larger separations. Quite counterintuitively, we observe that in this case the effective interaction be- ofaYBG-likehierarchy19,20 fortheactivesystem,where comes more repulsive than in the passive case, also for the inverse-τ approximation. Our computer simulations ρ(2)(r,r′) hDUi=DV (r)+ dr′ Du(r,r′) (19) (also carried out for values of τ much larger than shown ext ρ(r) inFig.4)confirmthatthisisanartifactofthetheory,re- Z for any differential operator D. In the derivation of lated to the negative curvature of the bare potential. At (18) it turns outthat the off-diagonalcomponents ofthe constant τ the evolution of the theoretical results agrees mobility tensor do not contribute at first order in τ20. qualitatively with the simulations. The simulation data Hence,we mightaswellhaveassumedthe diagonalform are, however, not very sensitive to changes in the per- Γ ≈δ Γ at linear order in τ beforehand. sistence time. At constant D the theory predicts the ij ij ii a In order to connect to the EPA, we derive a YBG-like correct trend upon increasing τ, whereas this is not the hierarchy from Eq. (12). Assuming the diagonal form case at constant Pecl´et number. Γ ≈ δ Γ , the integration over N −1 coordinates of For a discussion of other non-convex bare potentials ij ij ii the first equality is carried out i n appendix B. Making withattractivepartssee,e.g.,Refs.24and27. However, use of the equilibrium versionof the YBG hierarchyand the understanding of the behavior of particles interact- expanding the ln(detΓ (rN)) up to first order in τ the ing with a bare potential which has a negativecurvature ii result is remains one of the most urgent open problems in our theoretical framework. Further numerical and theoreti- 0=−D−1(r)ρ(r)h∇βUi−∇ρ(r)+τ˜ρ(r)h∇·∇∇Ui cal analysis will be needed to fully clarify this issue. I ρ(r) ρ(2)(r,r′) + τ˜ dr′(∇∇u(r,r′))·∇ (20) 1+D ρ(r) a Z VI. LOW-ACTIVITY APPROXIMATION introducing the averagedinverse diffusion tensor Γ 1+τ˜h∇∇Ui A second strategy to simplify the steady-state condi- D−1(r):= D−1 = [1,·] = . (21) tionistoperformanexpansionintheactivityparameter I [1,·] (cid:10)1+D(cid:11)a 1+Da τ. Atlinearorder,theeffectivediffusiontensorD[N](rN) Multiplying EDq. (20)Ewith D ≈(1+D )(1−τ˜h∇∇Ui) I a becomes pairwise additive. In this low-activity approx- it is easy to verify in appendix B that at first order in τ imation, a YBG-like hierarchy can be obtained by suc- it becomes equivalent to (18) up to a term proportional cessively integrating Eq. (9) over N −n coordinates19, to the expression in the second line, which we consider which allows defining a mechanical pressure and interfa- as a higher-order contribution. cial tension20. Moreover, for the active system evolving InordertoderiveEq.(20)intheFoxapproach,anad- according to Eqs. (1) and (4) it has been demonstrated (f) ditional approximationis required, as the inverse of D thatthereexistsaregimeforsmallvaluesofτ,wherethe ij principleofdetailedbalanceisrespected33. Thissuggests from Eq. (11) is not proportional to Γij. We would thus need to redefine D in Eq. (21) according to Eq. (11) that, at leading order in this parameter, the approxima- I tions resulting in Eq. (9) are perfectly justified. where Γ −1 takesthe roleofΓ−1. Regardingin gen- [1,·] ij Knowing, however, that Eq. (9) contains the same in- eral the presented alternative derivation of Eq. (18), its (cid:10) (cid:11) formation as Eq. (12), which depends logarithmically on validity appears to be in question. This is because to the parameter τ, we should clarify whether (VI.i) the derive the intermediate result in Eq. (20) it is necessary low-activityexpansionconverges,(VI.ii)itissufficientto to expand a logarithmic term, the Taylor series of which only consider the leading order and (VI.iii) one can ob- only has a finite radius of convergence. We further ex- tain similar results when employing the EPA. To do so, plicitlyassumedthediagonalformofthemobilitytensor wedemonstratehowthefirstmember(n=1)oftheYBG to avoid further terms that are not present in the origi- hierarchycanbe rederivedfromEq.(12) anddiscuss the nalresult. EmployinginthenextsteptheEPAwillshed consequences of approximating Feff in terms of pair in- more light on these issues. k teractions. Again, we only discuss the UCNA, where, without any further approximation,the inverse diffusion tensor is found to be pairwise additive. VII. LOW-ACTIVITY APPROXIMATION By saying we integrate over N − 1 coordinates we OF EFFECTIVE POTENTIALS understand calculating the average hXi := hhρˆXii/ρ(r), where hhXii denotes the full canonical ensemble average Having established a connection between (9) and ofanobservableX(rN)andρˆ= N δ(r−r )istheden- (12) also at linear order in τ, we now turn to the i=1 i P 8 case in which the second equality in (12) does not small τ˜ according to hold. This is when we assume Feff ≈ −∇ Ueff = k k −∇k Veexfft(rk)+ l6=kueff(rk,rl) alongthe linesof (2) ∇lndet 1+τ˜∇∇ u(r,rl) but w(cid:16)ithin the EPPA using the resu(cid:17)lts derived in Sec. IV. Xl>1 ! As detailed in appendix C, the obvious result is that all →∇· lndet(1+τ˜∇∇u(r,r )) correlation functions between more than two particles l vanish in the approximate integrated version Xl>1 →τ˜ ∇∆u(r,r )+O(τ2) (24) l 0=−∇ρ(r)−ρ(r) ∇βUeff (22) l>1 X (cid:10) (cid:11) eventually results in full consistency with the respective of (12). Ignoring the interparticle interactions the ap- proximation involving only Veff(r) becomes exact. This term in Eq. (18), stated as point (VII.iii). This suggests ext that the expansion to first order in τ ”implies” making situation is the same as discussed in Ref. 19. theEPAwhenEq.(12)isourstartingpoint. Despitethe Considering the interacting system, we multiply Eq. (22) with D as done previously for (20). As de- aforementioned crudity of this expansion, we argue that I Eq.(18)isvalid,asitscleanderivationfromEq.(9)does tailed in appendix C, we can only approximately repro- not require dealing with a logarithmic term. The last duce Eq. (18) by doing so. This reflects both the limita- step in Eq. (24) is required to recover Eq. (18) without tions of the EPA and an inconsistency between Eq. (9) inducing undesired higher-order terms in τ, as, similar and Eq. (12) when being subject to the same type of to point (VII.ii), the integrated version is incompatible approximation, as we discuss in the following. We ob- withthechosenD . However,westressthatthisapprox- serve that (VII.i) the coupling between external and in- I imationshouldcertainlybeavoidedwhencalculatingthe ternal interactions is ignored by Eq. (22) (VII.ii) spuri- fluid structure. ous three-body correlations appear on the left-hand-side Finally, we demonstrate in appendix C that the off- of Eq. (18) (VII.iii) the second term on the right-hand diagonal elements of the diffusion tensor entering in side of Eq. (18) is recovered but involves a seemingly Eq. (12) contribute to Eq. (22). Hence, the present ap- unjustified expansion and (VII.iv) if we do not explic- proach would be even more inconsistent with Eq. (18) if itly assume a diagonal diffusion tensor, the last term in we did not assume the diagonal form, as noted in point Eq. (18) changes by a factor two. As we are mainly in- (VII.iv). We also note that the same problem occurs for terested in bulk systems, the first point is only briefly the according generalization of Eq. (20). In principle we commented on in appendix C. could define in this case an additional averageddiffusion The term including the bare interaction force in tensorD , correspondingto the off-diagonalelements, Eq.(12) depends onthe positionofthree bodies. Hence, I,od which could counteract this inconsistency. Such a calcu- the pairwise approximation,which amounts to setting lation would, however, not be useful when an effective pair potential is employed. (∇ ∇ u(r ,r ))(∇ u(r ,r )) k k k l k k j l,j6=k X → (∇ ∇ u(r ,r ))(∇ u(r ,r )) , (23) VIII. CONCLUSIONS k k k j k k j j6=k X In this paper we studied different ways to define an should not be too crude. Moreover, we have discussed effective pair interaction potential between active par- in Sec. IV that this contribution to the effective force is ticles. Our numerical investigation reveals that a two- usuallypurelyrepulsiveandthus playsonlyaminorrole particlesystemofABPsandactiveOUPsexhibitsaquite in characterizing a possible phase transition. However, similar behavior. These results serve as a benchmark we show in appendix C that definition (21) of the aver- to test the approximations involved in recent effective ageddiffusion tensor DI is notfully compatible with the equilibrium approaches, which have been reviewed and EPA,resultinginpoint(VII.ii). Thisisincontrasttothe compared in detail. For spatial dimensions higher than clean derivation of Eq. (18), where Eq. (9) is recovered one we introduced an empirical way to rid the theoreti- from Eq. (12) by multiplication with the many-body ef- cal result of possible divergences, which also appears to fectivediffusiontensorD beforeintegratingoverN−1 yield the best agreement with the simulation data, al- [N] positions. thoughtheeffectiveattractionisstillsignificantlyoveres- ThelasttermP ∇ ln(detD−1(rN))in(12),although timated. Regarding the quite accurate one-dimensional N k kk considered here for a diagonal diffusion tensor, consti- results and the qualitative features of the effective po- tutesafullN-bodyquantity. Recallingtheconclusionsof tentials in three dimensions, the Fox approximation is Sec. IV, the approximationas a pairwise quantity might superiorto the UCNA whenthe thermalBrowniannoise be quite poor and an expansion of the logarithmic term cannotbeneglected. Intheabsenceofthermalnoiseboth is not justified. However,we demonstrate in appendix C approximation schemes admit the same steady-state so- that successively employing the EPA and expanding for lution. 9 Further analysis is needed to better understand the ties11,21,23,whereasthe low-activityexpansionofEq.(9) role ofthe neglected many-body interactionsin both the provides a direct way to define mechanical properties20. two-body simulations and the theory, which are thought Moreover,ouranalysissuggeststhatthe thermodynamic to be imperative for a quantitative description of active results obtained from Eq. (12) can be rescaled in or- systems27. The presented theoretical approach follows der to obtain a workable definition of mechanical ac- two major approximate steps to define the effective pair tive pressure and surface tension. Arguably, the most potential. First,wemapthe equationofmotion(1)onto simplistic scaling factor would be the diffusivity 1+D a a system respecting detailed balance (effective equilib- of an ideal gas, which can be absorbed into an effective rium picture) and then we define pair forces from the temperature19,20,23. Amoregeneralapproachwillbede- two-particle limit. It could well be that the mapping in tailed in the second paper of this series. the first step breaks down parts of the many-body na- ture of the interactions in the active system, such that theeffectiveattractioninthemany-bodysystembecomes Appendix A: Simulation details accessible already on the level of pair interactions. As a logical next step, it seems worthwhile to study the effec- We performed Brownian Dynamics simulations of a tive potential extracted from computer simulations of a system composed of two particles of unit diameter d=1 many-particle system, in order to clarify in how far the interacting through soft-repulsive potential or Gaussian strong attraction of the effective potential needs to be soft-core potential. The potential is truncated at a dis- seen as the result of a fortuitous cancellation of errors. tance of r = 2d. In the simulations of active OUPs, The low-activity limit in the effective equilibrium pic- evolving according to Eq. (1), each particle is subjected ture also results in pairwise forces. Under this assump- to Gaussian thermal noise and a non-Gaussian(colored) tion, we revealed some minor inconsistencies between noise24. Theintegrationtimestepisfixedtodt=10−4τ B the two equivalent steady-state conditions in Eqs. (9) where τ =d2/D is the time scale of translational diffu- B t and(12),althoughthelattercontainsalogarithmicterm. sion. Thetotalruntimeofsimulationis104τ . Forevery B Relatedly, it was recognized in Ref. 20 that different dt, we calculate the distance between the two particles. routestodefinetheactivepressureonlycoincideatlowest The pair-correlation function is obtained in a standard order in the activity parameter τ. We suspect that fur- way from the distance distribution. therdifferenceswilloccurathigherordersinτ andwhen We also performed Brownian Dynamics simulation employing further approximations,suchas the EPA. We of ABPs, for which the colored-noise variable v (t) in i concludethatitshouldbecarefullychosenforeachprob- Eq. (1) is replaced with the vector v p (t) describing 0 i lem which equality to consider under which approxima- a constant velocity v of the self-propulsion in the di- 0 tions. rection of the instantaneous orientation. The equation The obvious purpose of both the low-activity ap- p˙ (t)=η (t)×p (t) for the time evolution for the orien- i i i proximation and the EPA is to allow for an analyti- tation vector p (t) of each particle i is evaluated as an i cally tractable theory. It appears that the condition Ito integral, where η (t) is a white noise describing ro- i given by Eq. (12) supported by effective pair poten- tational diffusion. The integration time step is fixed to tials is most convenient for accessing structural proper- dt=10−4 and the total run time is 104τ . B Appendix B: Integration of the first equality in Eq. (12) ThederivationofEq.(18)byintegratingEq.(9)overN−1coordinatesisquitesimilartothatoftheYBGhierarchy in a passive system. The first member 0=∇ρ(r)+ρ(r)∇βV (r)+ dr′ρ(2)(r,r′)∇βu(r,r′)=ρ(r)∇µ (B1) ext Z is recovered from (18) when setting τ = D = 0. The second equality reflects the interpretation of the term on the a left-hand side as the gradient of a chemical potential µ, which is constant in equilibrium. The second member reads 0=∇ρ(2)(r,r′′)+ρ(2)(r,r′′)∇(βV (r)+βu(r,r′′))+ dr′ρ(3)(r,r′,r′′)∇βu(r,r′)=ρ(2)(r,r′′)∇µ. (B2) ext Z andis relatedvia the secondequalityto the firstmember. With the helpof these exactequilibriumsumrules we will now derive Eq. (20) by integrating Eq. (12) over N −1 coordinates. Our presentation follows closely the derivation ofa dynamicaldensity functionaltheoryincluding a tensorialdiffusivity34, whereaswe only considerthe steady-state condition. 10