Dynamical density functional theory for "dry" and "wet" active matter
DDynamical density functional theory for ”dry” and ”wet” active matter
Hartmut L¨owen Institut f¨ur Theoretische Physik II: Weiche Materie,Heinrich-Heine-Universit¨at D¨usseldorf, D-40225 D¨usseldorf, Germany (Dated: February 26, 2021)In the last 50 years, equilibrium density functional theory (DFT) has been proven to be a powerful,versatile and predictive approach for the statics and structure of classical particles. This theory canbe extended to the nonequilibrium dynamics of completely overdamped Brownian colloidal particlestowards so-called dynamical density functional theory (DDFT). The success of DDFT makes it apromising candidate for a first-principle description of active matter. In this lecture, we shall firstrecapitulate classical DDFT for passive colloidal particles typically described by Smoluchowski equa-tion. After a basic derivation of DDFT from the Smoluchowski equation, we discuss orientationaldegrees of freedom and the effect of hydrodynamic interactions for passive particles. This bringsus into an ideal position to generalize DDFT towards active matter. In particular we distinguishbetween ”dry active matter” which is composed of self-propelled particles that contain no hydrody-namic flow effects of a surrounding solvent and ”wet active matter” where the hydrodynamic flowfields generated by the microswimmers are taken into account. For the latter, DDFT is a tool whichunifies thermal fluctuations, direct particle interactions, external driving fields and hydrodynamiceffects arising from internal self-propulsion discriminating between ”pushers” and ”pullers”. A num-ber of recent applications is discussed including transient clustering of self-propelled rods and thespontaneous formation of a hydrodynamic pump in confined microswimmers.
I. INTRODUCTION
Density functional theory (DFT) relies on the fact thatthere is a functional of the one-particle density whichgives access to the equilibrium thermodynamics when itis minimized with respect to this density. This impor-tant theory can be both applied to quantum-mechanicalelectrons and to classical systems.In this lecture we shall consider nonequilibrium sit-uations for completely overdamped Brownian dynamicsof colloids. A dynamical version of DFT, the so-calleddynamical density functional theory (DDFT), is avail-able and makes dynamical predictions which are in goodagreement with computer simulations. Here we shall de-rive DDFT for Brownian dynamics in a tutorial way fromthe microscopic Smoluchowski equation. The theory willthen be generalized towards hydrodynamic interactionsbetween the particles and to orientational degrees of free-dom describing e.g. rod-like colloids. This poises us intoan ideal position to generalize the DDFT towards ac-tive matter systems. For many interacting active Brow-nian particles without any hydrodynamic interactions(”dry active matter”), we derive the DDFT approachand discuss confinement-induced clustering as one exam-ple. Finally we develop a generic model for microswim-mers (”wet active matter”) which includes the hydrody-namic flow field and discriminates between ”pushers” and”pullers”. In this context, DDFT is a tool which unifiesthermal fluctuations, direct particle interactions, exter-nal driving fields with the hydrodynamic effects arisingfrom internal self-propulsion. A number of recent exam-ples relevant for microswimmers has been explored withinDDFT ranging from the formation of a hydrodynamicpump in confined system to collections of circle swim-mers and binary mixtures of pushers and pullers. For parts of this tutorial we follow the ideas outlined in Ref.[1]. For more technical aspects, we refer to the recentreview [2]. In contrast to Ref. [2] these lecture notes are not a balanced review, they are rather a biased tutorial,strongly biased with respect to recent works publishedby the author.
II. DENSITY FUNCTIONAL THEORY (DFT) INEQUILIBRIUMA. Basics
We shall consider density functional theory (DFT) herefor classical systems at finite temperature which are inter-acting via a radially-symmetric pair-wise potential v ( r ).The basic variational principle of density functional the-ory establishes the existence of a unique grand canoni-cal free energy-density-functional Ω( T, µ, [ ρ ]), which getsminimal for the equilibrium density ρ ( r ) and then coin-cides with the real grand canonical free energy, i.e. δ Ω( T, µ, [ ρ ]) δρ ( r ) (cid:12)(cid:12)(cid:12)(cid:12) ρ ( r )= ρ ( r ) = 0 . (1)Here, T is the imposed temperature and µ the prescribedchemical potential of the system. However, the functionalΩ( T, µ, [ ρ ]) is not known explicitly, in general. One cansplit the functional Ω( T, µ, [ ρ ]) asΩ( T, µ, [ ρ ]) = F ( T, [ ρ ]) + (cid:90) V d r ρ ( r ) ( V ext ( r ) − µ ) (2)where F ( T, [ ρ ]) is a Helmholtz free energy functionaland V denotes the system volume. a r X i v : . [ c ond - m a t . s o f t ] F e b The knowledge of the functional F ( T, [ ρ ]) for a givenpair potential v ( r ) provides a lot of information (muchmore than just a bulk equation of state, for instance)since it can be applied to any inhomogeneous system inan external potential V ext ( r ). For example, the secondfunctional derivative taken in the homogeneous bulklimit is proportional to the direct fluid pair correlationfunction. B. Approximations for the density functional
Let us first recall the exact functional for the idealgas where the pair interaction v ( r ) between the particlesvanishes, v ( r ) = 0. In three spatial dimensions, it readsas F ( T, [ ρ ]) = F id ( T, [ ρ ]) = k B T (cid:90) V d r ρ ( r ) (cid:2) ln( ρ ( r )Λ ) − (cid:3) (3)where Λ is the irrelevant thermal wave length and k B the Boltzmann constant. In this case, the minimizationcondition0 = δ Ω δρ ( r ) (cid:12)(cid:12)(cid:12)(cid:12) = k B T ln( ρ ( r )Λ ) + V ext ( r ) − µ (4)leads to the generalized barometric law ρ ( r ) = 1Λ exp (cid:18) − V ext ( r ) − µk B T (cid:19) (5)for the inhomogeneous density. More training in the cal-culation of functional derivatives will be shifted to theexercises. For non-vanishing pair interactions v ( r ), onecan split F ( T, [ ρ ]) =: F id ( T, [ ρ ]) + F exc ( T, [ ρ ]) (6)which defines a so-called excess free energy density func-tional F exc ( T, [ ρ ]) which typically needs to be approxi-mated. One important approximation is the mean-fieldapproximation where F exc ( T, [ ρ ]) ≈ (cid:90) d r (cid:90) d r (cid:48) v ( | (cid:126)r − (cid:126)r (cid:48) | ) ρ ( (cid:126)r ) ρ ( (cid:126)r (cid:48) ) . (7)Other approximations are the perturbativeRamakrishnan-Yussouff (RY) approach or non-perturbative fundamental measure theory for stericinteractions [3, 4]. Similar approximations can beformulated in two spatial dimensions, e.g. for hard disks[5]. III. CLASSICAL DYNAMICAL DENSITYFUNCTIONAL THEORY (DDFT) FOR PASSIVEBROWNIAN PARTICLESA. Brownian dynamics and Smoluchowski equation
DFT can be made time-dependent for passive over-damped Brownian particles [2] leading to dynamicaldensity functional theory (DDFT). where the time-dependent density field is the central quantity. It willfollow a deterministic diffusion-like equation.
1. Noninteracting Brownian particles
For noninteracting particles with an inhomogeneoustime-dependent particle density ρ ( r , t ), Fick’s law for thecurrent density j ( r , t ) states j ( r , t ) = − D ∇ ρ ( r , t ) (8)where D is a phenomenological diffusion coefficient.The continuity equation of particle number conserva-tion ∂ρ ( r , t ) ∂t + ∇ · j ( r , t ) = 0 (9)then leads to the well-known diffusion equation for ρ ( r , t ): ∂ρ ( r , t ) ∂t = D ∆ ρ ( r , t ) (10)In the presence of an external potential V ext ( r , t ), theforce F = −∇ V ext ( r , t ) acts on the particles and willinduce a drift velocity v D giving rise to the additionalcurrent density j D = ρ v D with the drift velocity v D = F ξ = − ξ ∇ V ext ( r , t ). Here, ξ denotes the friction co-efficient (for a sphere of radius R in a viscous solventof viscosity η Stokes law tells us that ξ = 6 πηR ).With the Stokes-Einstein relation D = k B Tξ we get j = − ξ ( k B T ∇ ρ + ρ ∇ V ext ) and the continuity equationyields ∂ρ ( r , t ) ∂t = 1 ξ ( k B T ∆ ρ ( r , t )+ ∇· ( ρ ( r , t ) ∇ V ext ( r , t ))) (11)which is the Smoluchowski equation for non-interactingparticles. Note that the external force can even be time-dependent.
2. Interacting Brownian particles
Now we consider N interacting particles at positions (cid:126)r i ( i = 1 , ..., N ). The total potential energy is U tot ( r N , t ) = N (cid:88) i =1 V ext ( r i , t ) + N (cid:88) i,j =1 i 1. Phenomenological derivation of DDFT The general Fick’s law assumes that the particle cur-rent density is proportional to the gradient of the chemi-cal potential [6] and proportional to the time-dependentdensity ρ ( r , t ): j = ξρ ( r , t ) ∇ µ (14)In equilibrium, when the chemical potential is constant,there is no such current. We now take a functional deriva-tive with respect to the density in Eq. (2) and obtain inthe absence of an external potential δ F ( T, [ ρ ]) δρ ( r ) (cid:12)(cid:12)(cid:12)(cid:12) ρ ( r )= ρ ( r ) = µ (15)When combining this with the continuity equation of par-ticle number conservation we get the important DDFTequation ξ ∂ρ ( r , t ) ∂t = ∇ ρ ( r , t ) ∇ δ F [ ρ ] δρ ( r , t ) (16)which is obviously generalized to the presence of an exter-nal potential V ext ( r , t ) by replacing F [ ρ ] with Ω[ ρ ]. Thisis a deterministic time evolution equation for ρ ( r , t ). Foran ideal gas, it reduces to the exact Smoluchowski equa-tion which is standard diffusion equation, see exercise.For an interacting system, the DDFT equation is ap-proximative. 2. Derivation of DDFT from the Smoluchowski equation The DDFT equation can be derived from the Smolu-chowski equation [7] but one essential additional approx-imation, the so-called adiabatic approximation, needs tobe performed here as well. In more detail, one integratesout degrees of freedom from the Smoluchowski equationto obtain the following exact equation ξ ∂∂t ρ ( r , t ) = k B T ∆ ρ ( r , t ) + ∇ ( ρ ( r , t ) ∇ V ext ( r , t )+ ∇ (cid:90) d r ρ (2) ( r , r , t ) ∇ v ( | r − r | )(17) In equilibrium, necessarily ∂ρ ( r ,t ) ∂t = 0 which implies0 = ∇ ( k B T ∇ ρ ( r ) + ρ ( r ) ∇ V ext ( r ) (18)+ (cid:90) d r (cid:48) ρ (2) ( r , r (cid:48) ) ∇ v ( | r − r (cid:48) | ) (cid:19) (19)menaing that the divergence of a current density mustvanish. The current density itself is imposed to vanishfor r → ∞ in equilibrium and thus the curent density isidentical to zero everywhere. Therefore0 = k B T ∇ ρ ( r )+ ρ ( r ) ∇ V ext ( r )+ (cid:90) d r (cid:48) ρ (2) ( r , r (cid:48) ) ∇ v ( | r − r (cid:48) | )(20)which is also known as Yvon-Born-Green-relation(YBG). Here, ρ (2) ( r , r (cid:48) ) is the two-body joint probabilitydensity in nonequilibrium. We now take a gradient ofthe density functional derivative of Eq. (2) and combineit with YBG-relation. Then we obtain (cid:90) d r (cid:48) ρ (2) ( r , r (cid:48) ) ∇ V ( | r − r (cid:48) | ) = ρ ( r ) ∇ δ F exc [ ρ ] δρ ( r ) (21)We postulate that this argument holds also in nonequi-librium. In doing so, non-equilibrium correlations are ap-proximated by equilibrium ones at the same ρ ( r , t ) (iden-tified via a suitable time-independent V ext ( r ) in equilib-rium). Equivalently, one can say that it is postulatedthat pair correlations decay much faster to their equi-librium one than the one-body density [8]. This resultsafter all in the DDFT equation: ξ ∂ρ ( r , t ) ∂t = ∇ ρ ( r , t ) ∇ δ Ω[ ρ ] δρ ( r , t ) (22)For further alternate derivation of the DDFT equation,see [8–10]. IV. POLAR PARTICLESA. DFT of polar particles We now consider polar particles which possess anadditional rotational degree of freedom in the two-dimensional plane which can be described by a simpleangle φ or a unit vectorˆ n = (cos φ, sin φ ) (23)relative to a prescribed axis. Having applications toswimmers on a substrate in mind, we consider motionin two-dimensions only. Equilibrium DFT can readilybe extended to polar particles. A configuration of N particles is now fully specified by the set of positions ofthe center of masses and the corresponding orientations { r i , ˆ n i , i = 1 , . . . , N } . Pairwise interactions are describedby a pair-potential v ( r i − r j , ˆ n i , ˆ n j ) that depends on thedifference vector r i − r j between the centers of the par-ticle i and another particle j plus their two orientationsˆ n i and ˆ n j . In the general context of active matter, ifthis function only depends on r i − r j , the interactionsare called non-aligning . An example are spherical self-propelled Janus particles which do not change their ori-entation when bouncing into each other. If it is energet-ically favorable to have parallel orientations, the interac-tions are called aligning . In the rare case that neighbour-ing particle tend to stay anti-parallel these interactionsare called anti-aligning . Clearly the external potential V ext ( r , ˆ n , t ) can also depend on the particle orientation.As in the case of spherical particles, DFT establishesthe existence of a functional of the one-particle density ρ ( r , ˆ n ) which gets minimal in equilibrium δ Ω( T, µ, [ ρ ]) δρ ( r , ˆ n ) (cid:12)(cid:12)(cid:12)(cid:12) ρ = ρ ( r , ˆ n ) = 0 (24)Again, the functional can be decomposed as followsΩ( T, µ, [ ρ ]) = k B T (cid:90) d r (cid:90) π d φ ρ ( r , ˆ n )[ln(Λ ρ ( r , ˆ n )) − (cid:90) d r (cid:90) π d φ ( V ext ( r , ˆ n ) − µ ) ρ ( r , ˆ n ) + F exc ( T, [ ρ ]) (25)The first term on the right hand side of equation (25)is the functional F id [ ρ (1) ] for ideal rotators. The excesspart F exc ( T, [ ρ (1) ]) is in general unknown and requiresapproximative treatments. Again nonperturbative fun-damental measure theory for hard cylinders is available[11]. B. DDFT for polar particles The Smoluchowski equation for the joint probabilitydensity p ( r , · · · , r N ; ˆ n , · · · , ˆ n N , t ) = p ( r N , ˆ n N , t ) is ∂p∂t = ˆ O S p (26)with the Smoluchowski operatorˆ O S = N (cid:88) i =1 (cid:20) ∇ r i · ¯¯ D (ˆ n i ) · (cid:18) ∇ r i + 1 k B T ∇ r i U ( r N , ˆ n N , t ) (cid:19) + D r ˆ R i · (cid:18) ˆ R i + 1 k B T ˆ R i U ( r N , ˆ n N , t ) (cid:19)(cid:21) (27)where U ( r N , ˆ n N , t ) is the total potential energy. Herethe rotation operator ˆ R i is defined as ˆ R i = ∂/∂φ andthe anisotropic translational diffusion tensor is given by¯¯ D (ˆ n i ) = D (cid:113) ˆ n i ⊗ ˆ n i + D ⊥ ( − ˆ n i ⊗ ˆ n i ) (28)The two diffusion constants D (cid:113) and D ⊥ , parallel andperpendicular to the orientations reflect the fact that thetranslational diffusion is anisotropic. The quantity D r is called rotational diffusion constant and sets the Browniandynamics of the orientations.Integrating the Smoluchowski equation yields the fol-lowing DDFT equation for the time-dependent ρ ( r , φ, t )[12] ∂ρ ( r , φ, t ) ∂t = ∇ r · ¯¯ D ( φ ) · (cid:20) ρ ( r , φ, t ) ∇ r δ Ω[ ρ ( r , φ, t )] δρ ( r , φ, t ) (cid:21) + D r ∂∂φ (cid:20) ρ ( r , φ, t ) ∂∂φ δ Ω[ ρ ( r , φ, t )] δρ ( r , φ, t ) (cid:21) (29) V. DYNAMICAL DENSITY FUNCTIONALTHEORY FOR ACTIVE BROWNIANPARTICLES (”DRY” ACTIVE MATTER) A simple classification of active matter can be doneinto ”dry active matter” where solvent flow does notplay any role and ”wet active matter” where hydrody-namic effects are important. In this chapter we shallstudy the simpler case of dry active matter first andtreat wet active matter in the next chapter. Ignoringhydrodynamic interactions, these swimmers can simplybe modeled by polar particles which are driven by a con-stant internal effective force along their orientations [13];this force corresponds to an effective drift velocity andmimics the actual propulsion mechanism. On top of theintrinsic propulsion, the particles feel Brownian noise ofthe solvent. The corresponding motion is intrinsicallya nonequilibrium one and even the dynamics of a sin-gle Brownian swimmer was solved only in this century[14–16].For dry active matter, the many-body Smoluchowskiequation now reads ∂p∂t = ˆ O a p (30)with the ”active” Smoluchowski operatorˆ O a = ˆ O S + N (cid:88) i =1 (cid:20) ∇ r i · ¯¯ D (ˆ n i ) · (cid:18) k B T v ˆ n i (cid:19)(cid:21) (31)The active part involves a particle current along the par-ticle orientation with a strength proportional to v whichis the self-propulsion velocity a single particle assumes.This Smoluchowski equation is stochastically equivalentto the Langevin equations of active Brownian motion[17]. For ideal particles ( U ( r N , ˆ n N , t ) = 0), the activeSmoluchowski equation has been the starting point tocalculate the intermediate scattering function of an ac-tive Brownian particle [18].DDFT for dry active matter can be derived using thesame adiabatic approximation (21) as in the passive case.The resulting equation of motion for the one-particle den-sity ρ ( r , φ, t ) then reads [19]: k B T ∂ρ ( r , φ, t ) ∂t = ∇ r · ¯¯ D ( φ ) · (cid:20) k B T v ˆ n ρ ( r , φ, t )+ ρ ( r , φ, t ) ∇ r δ Ω[ ρ ( r , φ, t )] δρ ( r , φ, t ) (cid:21) + D r ∂∂φ (cid:20) ρ ( r , φ, t ) ∂∂φ δ Ω[ ρ ( r , φ, t )] δρ ( r , φ, t ) (cid:21) (32)For a non-interacting system, it is important to note herethat this equation is exact under self-propulsion and anyexternal forces, see exercises. It is therefore an idealstarting point to study a single active Brownian particleunder gravity [20–23]. As a result, polar order was dis-covered for an ideal gas of sedimentating particles in thesteady state even if the particle are not bottom-heavy.As an application, for active particles in a channelwith aligning interactions, the time-dependent densityprofiles were found to be in agreement with Browniandynamics computer simulations [19]. In Ref. [19], a crudemean-field Onsager-like density functional approxima-tion [24] was used. Qualitatively, a transient formation ofhedgehog-like clusters near the channel boundaries wasfound in simulations and reproduced by the DDFT.Finally we remark that DDFT was generalized towardsthree spatial dimensions for swimmers of arbitrary shapewith complicated friction tensors [25]. Moreover, supera-diabatic DDFT which goes beyond the adiabatic approxi-mation has been applied to active Brownian systems withnon-aligning interactions [26, 27]. A special applicationwas performed for motility-induced phase separation ofactive particles [26, 28]. VI. DYNAMICAL DENSITY FUNCTIONALTHEORY FOR MICROSWIMMERS (”WET”ACTIVE MATTER) The most general DDFT framework for microswim-mers can be found in Ref. [29] which we closely followhere. This approach includes simultaneously thermalfluctuations, external forces, interparticle interactions bybody forces and hydrodynamic interactions as well asself-propulsion effects. In principle it includes all pre-vious cases in special limit of vanishing self-propulsion(”limit of ”passive particles”) and dry active matter(limit of neglected hydrodynamic interactions). A. The swimmer model To derive the DDFT, we consider a dilute suspen-sion of N identical self-propelled microswimmers at lowReynolds number [30] in two dimensions in an unboundedthree dimensional fluid. Following Ref. [29], the self-propulsion of a microswimmer is concatenated to self-induced fluid flows in the surrounding medium. This FIG. 1. Individual model microswimmer. The sphericalswimmer body of hydrodynamic radius a is subjected to hy-drodynamic drag. Two active point-like force centers exertactive forces + f and − f onto the surrounding fluid. Thisresults in a self-induced fluid flow indicated by small light ar-rows. L is the distance between the two force centers. Thewhole set-up is axially symmetric with respect to the axis ˆn . If the swimmer body is shifted along ˆn out of the geo-metric center, leading to distances αL and (1 − α ) L to thetwo force centers, it feels a net self-induced hydrodynamicdrag. The microswimmer then self-propels. In the depictedstate (pusher), fluid is pushed outward. Upon inversion ofthe two forces, fluid is pulled inward (puller). We considersoft isotropic steric interactions between the swimmer bod-ies of typical interaction range σ , implying an effective stericswimmer radius of σ/ 2. From Ref. [29]. then represents a major source of hydrodynamic inter-action between different swimmers. To proceed we con-sider a minimal model for an individual microswimmeras depicted in Fig. 1. Each microswimmer consists of aspherical body of hydrodynamic radius a . The swimmerbody is subjected to hydrodynamic drag with respectto surrounding fluid flows including self-convection. Thelatter is generated by two active force centers which arelocated at a distance L from each other, see Fig. 1, andexert two antiparallel forces + f and − f , respectively, ontothe surrounding fluid and set it into motion. Summingup the two forces, we find that the microswimmer ex-erts a vanishing net force onto the fluid. Moreover, since f (cid:107) ˆn , there is no net active torque [31]. Self-propulsionis now achieved by shifting the swimmer body along ˆn out of the geometric center. We introduce a parameter α to quantify this shift, see Fig. 1. The distances be-tween the body center and the force centers are now αL and (1 − α ) L , respectively. We confine α to the interval]0 , . α = 0 . 5, no net self-induced motion occursby symmetry. For α (cid:54) = 0 . 5, the swimmer body feels anet self-induced fluid flow due to the proximity to one ofthe two force centers. Due to the resulting self-inducedhydrodynamic drag on the swimmer body, the swimmerself-propels. It is important to note that the two forcecenters are completely fixed or attached to the particlecenter. So they propel the particle and are moving withthe particle at the same time. In the depicted state ofoutward oriented forces, the swimmer pushes the fluidoutward and is called a pusher [32]. Inverting the forces,the swimmer pulls fluid inward and is termed a puller [32].We now consider an assembly of N interacting identicalself-propelled model microswimmers, suspended in a vis-cous, incompressible fluid at low Reynolds number [30].The flow profile within the system then follows Stokes’equation [33], η ∇ v ( r , t ) + ∇ p ( r , t ) = N (cid:88) i =1 f i ( r i , ˆn i , t ) . (33)Here, t denotes time and r any spatial position in the sus-pension, while v ( r , t ) gives the corresponding fluid flowvelocity field. η is the viscosity of the fluid and p ( r , t ) isthe pressure field. On the right-hand side, f i denotes thetotal force density field exerted by the i th microswim-mer onto the fluid. r i and ˆn i mark the current positionand orientation of the i th swimmer at time t , respec-tively. This each microswimmer contributes to the over-all fluid flow in the system by the force density it exertson the fluid. In this way, each swimmer can transportitself via active self-propulsion since the point force cen-ters are firmly attached to the swimmer body. Moreover,all swimmers hydrodynamically interact with each othervia their induced flow fields.The linearity of Eq. (33) and the incompressibility ofthe fluid, i.e. ∇ · v ( r , t ) = 0, implies a linear relation be-tween velocities (angular velocities) and forces (torques).We denote by F j and T j the forces and torques, re-spectively, acting directly on the swimmer bodies ( j =1 , ..., N ), except for frictional forces and torques resultingfrom the surrounding fluid. The non-hydrodynamic bodyforces and torques may for example result from externalpotentials or steric interactions and will be specified be-low. From them, in the passive case, i.e. for f = , theinstantly resulting velocity v i and angular velocity ω i ofthe i th swimmer body follows as (cid:20) v i ω i (cid:21) = N (cid:88) j =1 M ij · (cid:20) F j T j (cid:21) = N (cid:88) j =1 (cid:20) µ ttij µ trij µ rtij µ rrij (cid:21) · (cid:20) F j T j (cid:21) . (34)Here M ij are the mobility matrices, the componentsof which ( µ ttij , µ trij , µ rtij , µ rrij ) likewise form matrices.They describe hydrodynamic translation–translation,translation–rotation, rotation–translation, and rotation–rotation coupling, respectively.The mobility matrices can approximately be calculatedas µ ttii = µ t , µ rrii = µ r , µ trii = µ rtii = (35) for entries i = j (no summation over i in these expres-sions) and µ ttij = µ t (cid:18) a r ij (cid:16) + ˆr ij ˆr ij (cid:17) + 12 (cid:16) ar ij (cid:17) (cid:16) − ˆr ij ˆr ij (cid:17)(cid:19) , (36) µ rrij = − µ r (cid:18) ar ij (cid:19) ( − ˆr ij ˆr ij ) , (37) µ trij = µ rtij = µ r (cid:18) ar ij (cid:19) r ij × , (38)for entries i (cid:54) = j . Here, we have introduced the abbrevi-ations µ t = 16 πηa , µ r = 18 πηa . (39)Because of the linearity of Eq. (33), the effect of the ac-tive forces can be added to the swimmer velocities andangular velocities on the right-hand side of Eq. (34). (cid:20) v i ω i (cid:21) = N (cid:88) j =1 (cid:18)(cid:20) µ tt ij µ tr ij µ rt ij µ rr ij (cid:21) · (cid:20) F j T j (cid:21) + (cid:20) Λ tt ij rt ij (cid:21) · (cid:20) f ˆ n j (cid:21)(cid:19) (40)Note that there are no active torques here, i.e. we areconsidering a linear swimmer here. For circle swimmers,a constant torque must be included to describe the cir-cling. Moreover, Λ tt ij , Λ rt ij , summarize effect of both + f ˆ n j and − f ˆ n j such that the total swimmer is force-free. Indetail, Λ tt ij = µ tt+ ij − µ tt − ij , Λ rt ij = µ rt+ ij − µ rt − ij For i = j , the term Λ tt ii contains the self-propulsion ofthe particles. B. Derivation of the DDFT for microswimmers We now aim at a statistical description for full jointprobability density P = P ( r N , ˆ n N , t ) and start from thedynamical Smoluchowski equation ∂P∂t = − N (cid:88) i =1 {∇ r i · ( v i P ) + (ˆ n i × ∇ ˆ n i ) · ( ω i P ) } (41)Integrating out all degrees of freedom except for oneswimmer, we get the following exact relation for the dy-namics of the swimmer one-body density ∂ρ (1) ( X , t ) ∂t = −∇ r · (cid:0) J tt + J tr + J ta (cid:1) − (ˆ n × ∇ ˆ n ) · (cid:0) J rt + J rr + J ra (cid:1) (42)where X = ( r , ˆn ) is a compact notation for both translational and orientational degrees of freedom. The six currentdensities are given by J tt = − µ t (cid:18) k B T ∇ r ρ (1) ( X , t ) + ρ (1) ( X , t ) ∇ r V ext ( r ) + (cid:90) d X (cid:48) ρ (2) ( X , X (cid:48) , t ) ∇ r v ( | r − r (cid:48) | ) (cid:19) − (cid:90) d X (cid:48) µ tt r , r (cid:48) · (cid:16) k B T ∇ r (cid:48) ρ (2) ( X , X (cid:48) , t ) + ρ (2) ( X , X (cid:48) , t ) ∇ r (cid:48) V ext ( r (cid:48) )+ ρ (2) ( X , X (cid:48) , t ) ∇ r (cid:48) v ( | r − r (cid:48) | ) + (cid:90) d X (cid:48)(cid:48) ρ (3) ( X , X (cid:48) , X (cid:48)(cid:48) , t ) ∇ r (cid:48) v ( | r (cid:48) − r (cid:48)(cid:48) | ) (cid:19) J tr = − (cid:90) d X (cid:48) k B T µ tr r , r (cid:48) (ˆ n (cid:48) × ∇ ˆ n (cid:48) ) ρ (2) ( X , X (cid:48) , t ) J ta = f (cid:18) Λ tt r , r · ˆ n ρ (1) ( X , t ) + (cid:90) d X (cid:48) Λ tt r , X (cid:48) · ˆ n (cid:48) ρ (2) ( X , X (cid:48) , t ) (cid:19) J rt = − (cid:90) d X (cid:48) µ rt r , r (cid:48) (cid:16) k B T ∇ r (cid:48) ρ (2) ( X , X (cid:48) , t ) + ρ (2) ( X , X (cid:48) , t ) ∇ r (cid:48) V ext ( r (cid:48) )+ ρ (2) ( X , X (cid:48) , t ) ∇ r (cid:48) v ( | r − r (cid:48) | ) + (cid:90) d X (cid:48)(cid:48) ρ (3) ( X , X (cid:48) , X (cid:48)(cid:48) , t ) ∇ r (cid:48) v ( | r (cid:48) − r (cid:48)(cid:48) | ) (cid:19) J rr = − k B T µ r ˆ n × ∇ n ρ (1) ( X , t ) − (cid:90) d X (cid:48) k B T µ rr r , r (cid:48) · (ˆ n (cid:48) × ∇ n (cid:48) ) ρ (2) ( X , X (cid:48) , t ) , J ra = f (cid:90) d X (cid:48) Λ rt r , X (cid:48) ˆ n (cid:48) ρ (2) ( X , X (cid:48) , t ) (43)Here ρ (3) ( X , X (cid:48) , X (cid:48)(cid:48) , t ) is the nonequilibrium triplet density.We close these exact equation approximatively by using the DDFT relations on the pair and triplet level (cid:90) d r (cid:48) dˆ n (cid:48) ρ (2) ( r , r (cid:48) , ˆ n , ˆ n (cid:48) , t ) ∇ r (cid:48) v ( | r − r (cid:48) | ) = ρ (1) ( r , ˆ n , t ) ∇ r δ F e xc δρ (1) ( r , ˆ n , t ) (44) ∇ r (cid:48) ρ (2) ( r , r (cid:48) ˆ n , ˆ n (cid:48) t ) + ρ (2) ( r , r (cid:48) ˆ n , ˆ n (cid:48) , t ) ∇ r (cid:48) v ( | r − r (cid:48) | )+ (cid:90) d r (cid:48) dˆ n (cid:48) ρ (3) ( r , r (cid:48) , r (cid:48)(cid:48) , ˆ n , ˆ n (cid:48) , ˆ n (cid:48)(cid:48) , t ) ∇ r (cid:48) u ( r (cid:48) r (cid:48)(cid:48) )= ρ (2) ( r , r (cid:48) ˆ n , ˆ n (cid:48) , t ) (cid:18) ∇ r (cid:48) ln (cid:16) λ ρ (1) ( r (cid:48) , ˆ n (cid:48) , t ) (cid:17) + ∇ r (cid:48) δ F exc δρ (1) ( r (cid:48) , ˆ n (cid:48) , t ) (cid:19) The remaining input is a concrete approximation for the equilibrium density functional where we adopt the mean-field approximation F exc = 12 (cid:90) d r d r (cid:48) dˆ n dˆ n (cid:48) ρ (1) ( r , ˆ n , t ) ρ (1) ( r (cid:48) , ˆ n (cid:48) , t ) v ( | r − r (cid:48) | ) (45)and a high-temperature factorization approximation for the remaining nonequilibrium pair correlation ρ (2) ( r , r (cid:48) , ˆ n , ˆ n (cid:48) , t ) = ρ (1) ( r , ˆ n , t ) ρ (1) ( r (cid:48) , ˆ n (cid:48) , t ) exp ( − βv ( | r − r (cid:48) | )) (46)Then the full set of equations is closed. They only involve the dynamical one-body density field and can be solvednumerically. C. Applications of DDFT to microswimmers As a first application, we consider the motion of mi-croswimmers which are moving in two spatial dimensionssurrounded by a three-dimensional bulk fluid [29]. They are confined to an quartic external potential V ext ( r ) = k | r | . (47)with k defining a confinement strength and exhibit non-aligning interactions embodied in the steric pair potential v ( r ) = (cid:15) exp (cid:18) − r σ (cid:19) . (48)here, (cid:15) sets the strength of this potential and σ an ef-fective interaction range.The calculation protocol is to turn the activity f offfirst and equilibrate the particles in the quartic poten-tial. The parameters are chosen in such a way that theequilibrium system is in the fluid phase but exhibits den-sity peaks due to the steric potential. Then the self-propulsion f is switched on and the density evolutionis followed by solving the DDFT equations numerically.For small self-propulsion strengths | f | , a stationary high-density ring is formed both for pushers ( f > 0) andpullers ( f < 0) which is extended relative to the typ-ical extension of the equilibrated system. If the self-propulsion strength is getting larger a tangential insta-bility occurs and the system forms spontaneously a hy-drodynamic pump as predicted earlier by simulations[34, 35]. For even higher | f | the pump gets unstableresulting in a continuous dynamic ”swashing” of the den-sity cloud. The behaviour is similar for pushers andpullers but details are different, see Figure 2.The swimmer model can be generalized to forcecenters which are not collinear with the swimmer. Thisresults in circe-swimming which was further analyzedwithin DDFT in [36]. Moreover the DDFT approachwas applied to global polar ordering in pure pusher orpuller suspensions [37]. As a result, at sufficient highconcentrations polar ordering was found for pullers butnot for pushers. Finally the versatility of the DDFT isdocumented by its generalization to binary mixtures andto dynamics in a sheared fluid which were consideredand elaborated in Ref. [38].Let us finish with a remark: If one has dry active mat-ter in mind from the very beginning, it is better to startwith the approach described in Section V. The limit ofvanishing hydrodynamic interactions is not a simple oneif one does uses the swimmer model of this section, sincehydrodynamics and self-propulsion are intrinsically cou-pled here. VII. CONCLUSIONS In conclusion, dynamical density functional theorywhich has been known as a successful theory for interact-ing Brownian colloidal particles can be applied to activematter as well. In particular, both ”dry active matter”and ”wet active matter” (microswimmers) can be treatedon different levels of complexity.Future research will be directed along the followingtopics:i) Density functional theory provides an ideal frame-work to tackle aligning interactions . This strength needs to be exploited further to establish a first-principle ap-proach to the Vicsek-model of swarming [39] and to theimpact of alignment effects on motility-induced phaseseparation [40–42].ii) So far we discussed swimmers in a viscous Newto-nian fluid, but in many situations there is a viscoelas-tic solvent . Then memory effects of the solvent playa role which modify and affect the swimming process.One basic example for a viscoelastic medium is a poly-mer solution. For colloids swimming in a polymer solu-tion, a strongly enhanced rotational diffusion was foundin experiment [43] and simulation [44]. It is challengingto treat these memory effects of the viscoelastic solventwithin dynamical density functional theory.iii) Density functional theory is ideal for the calcula-tion of interfaces and wetting effects [45]. So it should beapplied to study interfaces between coexisting states foractive particles. This can be both fluid-fluid and fluid-solid interfaces. For an effective equilibrium model fordry active particles, this was done by Wittmann andBrader [46] but an extension towards the full DDFT isstill needed.iv) As density functional theory describes freezing andcrystallization in equilibrium, dynamical density func-tional theory should be applied to freezing of active par-ticles [47]. A simplified approach based on the phase-field-crystal model has been proposed [48] but this needsextension towards a theory which includes microscopiccorrelations.v) Particles with time-dependent pair interactions suchas breathing particles whose interaction diameter changesperiodically as a function of time. There should beno principle obstacle to formulate a DDFT for thesenonequilibrium systems which play an important role formodelling dense biological tissue [49].vi) Bacteria subjected to simultaneous growth and di-vision establish a complex dynamical phenomenon whenstrongly interacting [50]. A DDFT approach seems to beparticularly promising to described nematic order on theparticle-scale for growing bacterial colonies [51]. VIII. EXERCISES Exercise 1 : Calculate the first two functional deriva-tives δ F [ ρ ] δρ ( r ) and δ F [ ρ ] δρ ( r ) δρ ( r (cid:48) ) for1. F [ ρ ] = (cid:82) d r (cid:82) d r w ( | r − r | ) ρ ( r ) ρ ( r ),2. F [ ρ ] = (cid:82) d r Ψ( ρ ( r ))in three spatial dimensions.Here, w ( r ) and Ψ( ρ ) are prescribed given functions. Exercise 2 : Show that for the ideal gas in an ex-ternal potential the dynamical density functional theoryreduces to the exact Smoluchowski equation. Exercise 3 : Show that the DDFT for dry active mat-ter is equivalent to the underlying Smoluchowski equa-tion is the particles are non-interacting ( v ( r ) = 0) but FIG. 2. Time evolution of the density profiles (color maps) and orientation profiles (white arrows) (a) for pushers and (b) forpullers. The snapshots were obtained at times t = 0 . , . , . , . 8. From Ref. [29] exposed to a general external potential V ext ( r , ˆ n , t ). Bythe way it has been erroneously claimed in the literaturethat DDFT is approximative in this case [20]. 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