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D 1 Collective behavior of active Brownianparticles: From microscopic clustering tomacroscopic phase separation

T. Speck

Institute of Physics – KOMET

Johannes Gutenberg-Universitat Mainz

Contents1 Introduction 2

2 Statistical physics of simple fluids 3

3 Active Brownian particles 43.1 Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43.2 Clustering . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53.3 Mean-field dynamical equation . . . . . . . . . . . . . . . . . . . . . . . . . . 53.4 Effective hydrodynamic equations . . . . . . . . . . . . . . . . . . . . . . . . 7

4 Large-scale behavior and phase separation 74.1 Linear stability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 74.2 Weakly non-linear analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . 84.3 Effective free energy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9

5 Numerical results 11

6 Conclusions 11

Lecture Notes of the Summer School “Microswimmers – From Single Particle Motion to Collective Behaviour”,organised by the DFG Priority Programme SPP 1726 (Forschungszentrum Julich, 2015). All rights reserved.

D1.2 T. Speck

1 Introduction

Active matter has become a paradigm for a wide range of phenomena at the interface of chem-istry, physics, and biology. For reviews on various aspects, see Refs. 1–3. Active matter de-scribes the emerging collective behavior of autonomous components that constantly convert(free) energy into directed motion. The systems that fall under this description are as diverse asflocks of birds [4], dense suspensions of living bacteria [5], granular shaken rods [6], and col-loidal particles propelled through phoretic forces [7]. The collective behavior can be dynamicalsuch as swarming and flocking, or structural such as the formation of bands and the separationinto dense and dilute regions, or both.From a modelling perspective, the components typically become particles with an orientation,along which the particles are propelled at (constant) speed. The details of the model then de-fine the forces (or, more generally, interaction “rules”) between particles. For example, in thefamous Vicsek model [8] there is an alignment of orientations along the average orientation ofneighboring particles within a fixed distance but no direct interactions (i.e., no volume exclu-sion). Analytical large-scale treatments roughly allow for the distinction between “dry” and“wet” active matter [3]. The difference is that for wet active matter the dynamical coupling be-tween the particles and a medium (typically the solvent in which bacteria and colloidal particlesmove) is explicitly taken into account.In these notes we will be interested in active matter in its simplest form: N disks (or spheresin three dimensions) with an orientation along which they are propelled at constant speed v0.This model is referred to as active Brownian particles (ABPs) or spheres. It is motivated byexperimental observations in suspensions of colloidal microswimmers, i.e., colloidal particlesthat are propelled in a solvent due to phoretic forces [9–13]. In a nutshell, a local concentrationgradient of a molecular small solute (e.g. lutidine for local demixing [10] or hydrogen peroxidesplit at a catalytic surface [14]) is maintained by different surface properties of the colloidalparticles (in the simplest case Janus particles with two distinct hemispheres). Such a gradientimplies a current of the molecular solute and thus a flow of the solvent, which then propels thecolloidal big solute (typically towards the dense region of the molecular solute), see Fig. 1(a).The precise mechanisms are currently under investigation but not important in the following.For sufficiently high density and speeds, one observes a large-scale instability (both in simula-tions [15–18] and experiments [13]) strongly resembling gas-liquid phase-separation in passivesuspensions with attractive interactions. For purely repulsive particles, or even hard spheres, theobserved phase separation is thus a genuine non-equilibrium phenomenon. In the following, wewill outline how the large-scale behavior of active Brownian particles can be treated analyti-cally starting from the microscopic dynamics. Our results are consistent with the scenario of a“motility induced phase separation” [19, 20].The spirit of the approach presented here has a long tradition in statistical mechanics. Alreadymuch of the physics of passive colloidal suspensions can be modelled as hard spheres evolvingthrough overdamped Brownian dynamics mimicking the interactions with the solvent. Such amodel neglects many things (such as hydrodynamic interactions due to the solvent) but includesthe salient ingredients: (i) packing due to volume exclusion and (ii) the exchange of energy witha heat bath at fixed temperature T . The only ingredient that we will add to this scenario is (iii)the directed motion. As so often, the success of these simplifications depends on the questionsthat we ask: for example it will fail if lubrication forces between colloidal particles becomerelevant. Luckily, ABPs can explain already some of the features seen in experiments withcolloidal particles that are propelled by self-phoresis [11–13], see Fig. 1(b). Moreover, they

Collective behavior of active Brownian particles D1.3

allow detailed physical insights into the mechanisms leading to emergent collective behaviorand the possible foundations of a statistical physics of active matter [21, 22].

2 Statistical physics of simple fluidsBefore beginning we will recall some concepts and results for simple monatomic fluids to theextent needed in the following.Phase diagram. The thermodynamics of hard spheres modeling volume exclusion is deter-mined by entropy alone. At high density, hard spheres freeze because the entropy of the crystalhas become higher (every particle has a bit of space to wiggle) than that of the disordered liquid.The phase diagram is shown in Fig. 1(c) (in two dimensions there is also an intermediate hex-atic phase [23]). The combination of volume exclusion and attractive interactions introduces acritical point and a critical temperature Tc below which the fluid can separate into a dilute gasand a denser liquid phase, see Fig. 1(d).Ginzburg-Landau theory. Given a dimensionless order parameter φ(r) that distinguishes thetwo phases, the arguably most simple phenomenological ansatz for the free energy in the vicin-ity of a critical point is

F [φ] = kBT

∫dr

[1

2|R∇φ|2 + f(φ)

](1)

F+S

T

density ρ

(c)

Tc

G L

F

SG+L

L+S

G+S

(b)

density ρ

F ST(a)

(d)

Fig. 1: (a) Schematic of self-diffusiophoresis. The colloidal Janus particle maintains a concen-tration gradient of a molecular solute (small disks) that leads to a hydrodynamic flow propellingthe particle so that the combined system remains force-free. (b) Experimental snapshot of adense suspension of colloidal Janus particles that are phoretically propelled in a binary solvent(from Ref. 13). Demixing of the solvent occurs after illumination of the suspension, by whichthe hemispheres sputtered with carbon are heated above the lower critical temperature of thebinary mixture (water and lutidine). Although particles are (nearly) hard spheres, they startto separate into a dilute phase of freely moving particles and dense clusters. These clustersare fully reversible, i.e., after turning off the illumination the particles becomes passive againand the clusters resolve. (c) Phase diagram for passive monodisperse hard spheres with a fluid(F) and a solid (S) phase, and coexistence region (F+S). (d) Attractive interactions introducea critical point at critical temperature Tc below which the fluid can separate into gas (G) andliquid (L) phases.

D1.4 T. Speck

with bulk free energy density f(φ) = r2φ2 + u

4φ4 to lowest order. While the coefficient u > 0

is taken to be a positive constant, r ∝ (T − Tc) changes sign at the critical temperature and be-comes negative for T < Tc. The function f(φ) then takes on a double well shape [cf. Fig. 4(b)]with two minima at φ = ±φb with φb =

√−r/u corresponding to a spontaneous symmetry

breaking. The squared-gradient term |∇φ|2 in Eq. (1) is necessary for inhomogeneous systemsin which both phases coexist separated by interfaces. This term effectively penalizes sharpinterfaces, where the length R can be interpreted as an interaction range.Inhomogeneous system. Minimizing the full free energy functional Eq. (1) leads to

δF

δφ=kBT

V

[−R2∇2φ+ rφ+ uφ3

]= 0. (2)

We now consider an inhomogeneous system, φ = φ(x), with boundary conditions φ(x)→ ±φb

for x → ±∞, respectively. Formally, this problem corresponds to a particle with mass R2

moving at zero energy E = 12R2[φ′(x)]2 − f(φ) = 0 from one maximum at −φb to the other at

+φb in infinite time. The solution is known as instanton leading to the profile

φ(x) = φb tanh

(x− x0

2ξb

), (3)

where ξb = R/√−2r ∝ (T − Tc)

−1/2 is the correlation length of order parameter fluctuationsin the bulk homogeneous phases and 2ξb is the “intrinsic” mean-field width of the interface.

3 Active Brownian particles

3.1 ModelThe model consists of N particles moving in two dimensions. Every particle has an orientationdescribed by the normalized vector ek ≡ (cosϕk, sinϕk)

T with the angle ϕk enclosed by theorientation and the x-axis. For particles with diameter a a convenient quantity is the packingfraction η ≡ πa2ρ/4, where ρ is the number density. Particles are propelled along their orien-tation with constant speed v0. The volume exclusion between particles is modelled through arepulsive pair potential u(r) giving rise to the potential energy U({rk}) =

∑k<l u(|rk − rl|).

Since the particles move in a solvent, i.e. a viscous medium with which they exchange kineticenergy, the equations of motion are

rk = −µ0∇kU + v0ek + ξk (4)

with bare mobility µ0 neglecting hydrodynamic interactions (the influence of hydrodynamicinteractions has been studied numerically [24,25], typically by prescribing a boundary conditionon the surface of the active particles). The noise modelling the medium has zero mean 〈ξk〉 = 0and correlations 〈ξk,α(t)ξl,β(t′)〉 = 2D0δklδαβδ(t− t′). These correlations obey the equilibriumfluctuation-dissipation theorem D0 = µ0kBT and we assume that the self-propulsion does notmodify them.The orientational dynamics is modelled as a free random walk on the unit circle, ϕk = ξk,whereby the correlations 〈ξk(t)ξl(t)〉 = 2Drδ(t − t′) are fixed by the rotational diffusion coef-ficient Dr. The salient property of active particles is their directed motion, which can be char-acterized by the “persistence length” `p = v0τr. Here, τr is the correlation time of orientations,which for ABPs is simply given by τr = 1/Dr.


y

x

(a) (b)

-2

-1

0

1

2

-2 -1 0 1 2 0 1 2 3 4 5 6 7 8

ρζ/

v *

v0/v*

0

1

2

3

4

0 1 2 3 4

rθ

eδF

(b) (c)(a)

Fig. 2: Clustering and force imbalance. (a) Experimental observation of a small cluster ofmutually blocked particles due to the persistence of motion (from Ref. 13). (b) Force imbalancefor interacting active particles. Shown is a tagged active particle and one of the surround-ing particles at distance r and enclosing the angle θ with the tagged particle’s orientation.(c) Anisotropic pair distribution function g(r, θ) for the tagged particle (white circle, the arrowindicates its orientation). There is a larger probability to find another particle in front of thetagged particle compared to finding it behind. (Shown is simulation data for hard disks at areafraction η = 0.5 and speed v0 = 20. From Ref. 26.)

3.2 ClusteringA striking observation in dilute active suspensions is the formation of small clusters even forpurely repulsive interactions, see Fig. 2(a) for an experimental snapshot. This can be under-stood qualitatively rather easily: when two particles collide they block each other due to thepersistence of motion. To resolve this two-particle cluster, one orientation has to point away,which will take on the order∼ τr. The mean time between collisions is controlled by the densityand speed v0. Hence, depending on speed/density another particle might collide before the twoparticles have resolved, forming a three particle cluster. This leads to situations as depicted inFig. 2(a), where transient clusters of a few particles form even in the homogeneous suspension.If the mean collision time falls below a certain value a dynamical instability sets in, which leadsto a growth of clusters.

3.3 Mean-field dynamical equationEven if the microscopic mechanism is known, the collective large-scale and phase behavior isstill highly non-trivial. To make progress in this direction we turn to a description equivalent tothe coupled stochastic differential equations (4), which is the Smoluchowski equation governingthe temporal evolution of the joint probability ψN({rk, ϕk}; t) of all particle positions and theirorientations. Obviously, this is a lot of information. Of more interest are the reduced one-particle density

ψ1(r1, ϕ1; t) ≡∫

dr2 · · · drN∫

dϕ2 · · · dϕN NψN (5)

and two-particle density

ψ2(r1, ϕ1, r2, ϕ2; t) ≡∫

dr3 · · · drN∫

dϕ3 · · · dϕN N(N − 1)ψN , (6)

where we have accounted for the particles being identical and thus indistinguishable.

D1.6 T. Speck

From now we drop the index for the particle at r. In Refs. 26,27 the dynamical equation for theone-particle density has been derived. It involves an average force F(r, ϕ) due to interactionsof a tagged particle fixed at r with its neighbors distributed according to the two-point densityψ2. In contrast to passive suspensions, even in a homogenous suspension of active particlesthis force is not zero since, on average, there will be more particles in front than behind, seeFig. 2(c). For a closure we need to approximate this force. To this end we split

F = (e · F)e + δF ≈ Fee + F‖∇ψ1, (7)

where Fe ≡ e · F is the projection along the particle orientation and δF is the force componentperpendicular, see Fig. 2(b). The component Fe then describes a force that acts against the self-propulsion while δF leads to an “evasive” motion. Typically Fe � |δF| holds, and we onlyretain the part of δF that is parallel to the density gradient∇ψ1. The physical picture is that theevasive motion due to the blocking by the surrounding particles leads to an effective diffusionDe on larger scales. Employing this closure, the dynamical equation for the one-particle densitybecomes

∂tψ1 = −∇ · [v(ρ)e−De∇]ψ1 +Dr∂2ϕψ1. (8)

The projected force Fe = −ρζψ1 along the orientation leads to the effective speed

v(ρ) ≡ v0 − ρζ, (9)

which depends on the local density

ρ(r, t) ≡∫ 2π

0

dϕ ψ1(r, ϕ, t). (10)

The force imbalance [26] is quantified by the coefficient

ζ ≡ µ0

∫ ∞0

dr r[−u′(r)]∫ 2π

0

dθ cos θg(r, θ), (11)

which is a microscopic expression involving the pair distribution g(r, θ), cf. Fig. 2(c). In writingdown this expression we have assumed a homogeneous suspension such that the pair distributionfunction does not depend on the position of the tagged particle and only on the displacementvector. In the following this will suffice since we are mainly interested in the onset of theinstability. For inhomogeneous (phase-separated) systems one has a spatially dependent ζ(r),which becomes constant in the two bulk phases.We briefly discuss an alternative derivation [28] of the density-dependent speed, which is closerto the picture of blocked particles and based on time scales. It starts from the distance ` travelledduring τr, which for free particles equals the persistence length `p. For interacting particles, weassume that during τr particles on average are blocked nc times for a time τc, whence ` =v0(τr − ncτc). Since τr is constant for ABPs, with τMF � τc the effective speed becomes

v(ρ) =`(ρ)

τr≈ v0

(1− τc

τMF

)= v0(1− v0σsτcρ), (12)

where τMF = (v0ρσs)−1 is the mean free time between collisions with effective cross section

σs. This string of arguments leads to qualitatively the same form as Eq. (9) with a speed thatdecreases linearly with the local density.


3.4 Effective hydrodynamic equationsTo proceed further we introduce the conditional moments of the orientations (with the localdensity Eq. (15) being the zeroth moment)

p(r, t) ≡ 〈e〉 =

∫ 2π

0

dϕ eψ1(r, ϕ, t), (13)

Q(r, t) ≡ 〈[eeT − 121]〉 =

1

2

⟨(cos 2ϕ sin 2ϕ

sin 2ϕ − cos 2ϕ

)⟩. (14)

Note that the tensor Q is symmetric and traceless. The hierarchical dynamical equations forthese moments are obtained easily from Eq. (8): The continuity equation for the density ρ reads

∂tρ = −∇ · [v(ρ)p−De∇ρ] (15)

while the polarization p(r, t) is governed by

∂tp = −∇P (ρ)−∇ · [v(ρ)Q] +De∇2p−Drp. (16)

It should now become clear why these equations are called effective hydrodynamic equationswhen interpreting p as a velocity. The first three terms in Eq. (16) can be interpreted to stemfrom a stress with pressure P (ρ) = 1

2v(ρ)ρ, where the second term is a deviatoric stress and

the third term is formally simular to a viscosity term. The last term describes the relaxationof the coarse-grained polarization due to the decorrelation of particle orientations. The finalapproximation is to neglect Q, which closes the dynamical equations. As before, this closureassumes a weak perturbation of the homogeneous suspension.Although colloidal self-propelled particles of course move in a solvent, the decision to neglecthydrodynamic interactions places the resulting effective hydrodynamic theory into the fieldof what is sometimes called “dry active matter” [3]. Similar equations are, e.g., obtained inthe Toner-Tu continuum treatment [29] of polar active systems. In that case the alignment oforientations leads to nonlinear terms of the polarization p in Eq. (16) and thus to dynamicalcollective behavior.

4 Large-scale behavior and phase separation

4.1 Linear stabilityClearly, the homogeneous density profile (ρ = ρ and p = 0) is a solution of the dynamicalequations (15) and (16). On closer inspection, however, one finds that these equations exhibitan instability, i.e., a small perturbation of the homogeneous suspension does not decay anymorebut grows. Initially, the growth∼ eσ(q)t of a perturbation with wave vector q is exponential withrate

σ(q) = −1

2Dr −Deq

2 +1

2

√D2

r − 4αβq2 ≈ −(De +

αβ

Dr

)q2 − (αβ)2

D3rq4 +O(q6) (17)

with expansion coefficients

α ≡ v(ρ) = v0 − ρζ, β ≡ v02− ρζ. (18)

D1.8 T. Speck

λ

λ

λ = 0

ampl

itude

ampl

itude

stable

unstable��

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⇠p"

⇠ "2

(b)

instabilityline

critical point

vc

density ρ

spee

d v 0

Fig. 3: Linear stability analysis. (a) Shown is the dispersion relation Eq. (17) for two speedsbelow (σ1 > 0) and above (σ1 < 0) a critical speed vc. Also indicated is how the maximum ofthe dispersion relation scales with the small parameter ε. (b) Schematic “phase diagram” in the(ρ, v0) plane. The shaded area indicates the linearly instable region bounded by the instabilityline of critical speeds vc. The smallest speed on this line defines the critical point (ρ∗, v∗).

This function is plotted in Fig. 3(a). If σ(q) < 0 then the homogeneous profile is linearly stablesince all perturbations decay. A range of wave vectors become instable when the coefficientof the quadratic term becomes positive, i.e., when DeDr + αβ < 0 becomes negative. We arethus concerned with a large-scale instability since σ(q) is positive for small q corresponding toperturbations on large scales, cf. Fig. 3(a). This is at least consistent with the assumptions thatwe have made in deriving the effective hydrodynamic equations. The fact that σ(0) = 0 reflectsthe conservation of density. For a given state point (given global density ρ and ζ) the criticalspeed vc corresponds to the speed for which DeDr + αβ = 0 exactly vanishes, i.e. the onset ofthe instability. There is a smallest speed v∗ ≡ 4

√DeDr for which the instability might occur,

hence for v0 < v∗ the suspension is always stable. In the plane (ρ, v0) we thus obtain a line ofcritical speeds, the instability line, see Fig. 3(b).

4.2 Weakly non-linear analysisThe hydrodynamic equations of motion can be further reduced to yield a dynamical equationfor the density alone. The technique employed is called weakly non-linear analysis and is wellknown from the study of pattern formation [30]. In the following we try to convey the gist andoutline of the calculation without going into detail (for the full account, see Ref. 27).The basic idea is to consider the wave vector q0 with the largest growth rate σ(q0), which willthus dominate the morphology right after the onset of the instability. Using as a small expansionparameter the deviation from the critical speed

ε =v0 − vc

vc, (19)

one easily finds that q0 ∼√ε and further σ0 ≡ σ(q0) ≈ −σ1εq20 ∼ ε2 (with yet another

coefficient σ1). It is more convenient to use a non-negative ε as the small expansion parameterand let the sign of σ1 indicate whether we are in the linearly stable (σ1 > 0) or linearly unstable(σ1 < 0) regime.


We have thus found the typical length scale 1/q0 and the typical time scale 1/σ0 for the evolutionof the dominant mode. The density is now studied on the coarser scale of this dominant mode,ρ(r, t) = ρ(r/q0, t/σ0), with dimensionless length r and time t. Switching to this scale anddropping the tilde amounts to

∇ 7→√ε∇, ∂t 7→ ε2∂t. (20)

In the following we use dimensionless quantities with τr the unit of time and√D0/Dr the unit

of length. One now expands the deviation from the global density ρ in the small parameter ε andcollects terms of the same order. This leads to a hierarchy of equations, which to lowest orderreproduces the linear stability criterion. On the next order we find a closed, effective dynamicalequation for the density deviation alone.Actually, there are two choices for the expansion. The first is ρ = ρ+ ε1/2c+ . . . , which leadsto the dynamical equation

∂tc = σ1∇2c+ ζ2c∇ · (c2∇c)−D2e∇4c, (21)

where ζc is the value of the force imbalance coefficient exactly at the onset of the instability.One outcome of the calculation is that this equation only holds in the vicinity of v∗. To lift thisrestriction, we need to expand ρ = ρ + εc + . . . and repeat the weakly non-linear analysis,which now leads to another dynamical equation

∂tc = σ1∇2c− 2g∇ · (c∇c)−D2e∇4c (22)

with new coefficientg ≡ 1

2ζc(αc + βc) =

1

4ζc

√v2c − v2∗ > 0. (23)

Again, only quantities evaluated at the onset enter this expression. A posteriori this justifies ourassumption of a spatially uniform force imbalance coefficient ζ corresponding to the homoge-neous profile right at the loss of linear stability.

4.3 Effective free energyThe final step is to realize that both equations (21) and (22) can be rewritten

∂tc = ∇2 δF

δc(24)

employing a potential F [c]. This is a functional of the density deviation (our order parameter).These equations are thus formally equivalent to the well-studied Cahn-Hilliard equation [31],and extending this analogy we can identify

F [c] =

∫dr

[1

2D2

e |∇c|2 + f(c)

], f(c) =

a

2(c− c0)2 +

κ

4(c− c0)4 (25)

as an effective free energy. Plugging these expressions into Eq. (24) and comparing with thedynamical equations derived in the previous section, we find

a = σ1 −g2

3κand c0 =

g

3κ. (26)

D1.10 T. Speck

The coefficient κ cannot be determined this way except for the point vc = v∗, at which it readsκ(v∗) = 1

3ζ2∗ . The reason is the missing ∇c3 term in Eq. (22), which in turn is due to the

scale separation between the particle size and the length over which density fluctuations arecoarse-grained.Eq. (25) has the form of the Ginzburg-Landau equation (1), where De sets the effective interac-tion range. We can thus construct, at least qualitatively, the phase diagram for ABPs exploitingthe extensive knowledge for passive suspensions and fluids. For simplicity, we consider thebulk free energy density close to the critical point (we have restored the density as variable)

f(ρ) =1

2σ1ε(ρ− ρ)2 +

1

12ζ2c (ρ− ρ)4, (27)

which is plotted in Fig. 4(b) for some v0 > v∗. The inflection points f ′′(ρ) = 0 define aline called “spinodal” in the phase diagram Fig. 4(a), which corresponds to the loss of linearstability and is thus identical with the instability line of Sec. 4.1. A second line, the “binodal”,

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Fig. 4: Phase behavior. (a) Spinodal (dashed) and binodal (solid) lines in the (ρ, v0)-plane.Indicated is the behavior for global density ρ = 1: Increasing the propulsion speed v0 (verticalarrow), the homogeneous profile loses linear stability when reaching the spinodal at vc. Forany quenched v0, the coexisting densities are predicted by the points on the binodal (horizontalarrows for vc). (b) The corresponding (tilted) bulk free energy density f(ρ). The continuationof the dashed line from the upper panel indicates the inflection point for ρ = 1. (c) Coexistingdensities for a model suspension of (nearly) hard disks. Shown is the data from three differentnumerical studies with the same repulsive pair potential but varying ratios of effective propul-sion force to potential strength. The dashed lines are fits with (�) indicating the extrapolatedposition of the critical point. (d) Binodal and critical point of Bialke et al. as a function of areafraction. Also shown is the relative size of the dense domain after an instantaneous quench.(From Ref. [27])


is obtained from the double tangent construction. For Eq. (27) it is simply the two minima,more generally it will be the points with identical slope (which equals the chemical potential).The corresponding densities are the densities of the coexisting phases since these have the samechemical potential. The resulting phase diagram is shown in Fig. 4(a). It has a lower criticalpoint at which spinodal and binodal meet. This phase diagram is equivalent to that of a passivefluid with attractive interactions, cf. Fig. 1(d), where now the speed takes on the role of aninverse temperature.

5 Numerical resultsAfter having constructed the qualitative phase diagram we finally turn to computer simulationsof ABPs. The arguably best studied version in two dimensions employs the repulsive Weeks-Chandler-Andersen potential [32]

u(r) =

{4ε[(σ/r)12 − (σ/r)6] + ε (r < 21/6σ)

0 (r > 21/6σ)(28)

for the pairwise interactions with potential strength ε and length scale σ. The results for thecoexisting densities are shown in Fig. 4(c), where we employ the rotational Peclet numberPe ≡ 3v0/(σDr). Complete phase diagrams and coexisting densities so far have been reportedin three studies: (i) Redner et al. [15] use ε = kBT for N = 15, 000 disks and vary the speedPe. This means that at higher propulsion speeds particles will overlap more strongly. (ii) Bialkeet al. [16, 17] use ε = 100kBT for N = 10, 000 disks and also vary the propulsion speed. Notethat Dr = 3D0/a

2 with a = 21/6σ. (iii) Solon et al. [18] effectively vary the temperature bykeeping the ratio between the effective propulsion force and the potential strength constant withPe = 24βε. The number of disks is N = 20, 000. Common to all three studies is that belowPe . 60 fluctuations are so strong that a reliable determination of coexisting densities is notpossible anymore. The most likely explanation is that these fluctuations are due to the presenceof the critical point and a growing length scale. The estimated location of this critical point isalso indicated in Fig. 4(c).Finally, in Fig. 4(d) coexisting densities (converted to area fractions φ ≡ ρa2π/4) are overlaidby results from instantaneous quenches [16]. Here the suspension (v0 = 0) is equilibrated ata given global density and then quenched to the final speed v0, after which it is relaxed to thesteady state. The size of the dense domain is measured and also shown. One clearly discernsthe gap between dilute coexisting density and the onset of instantaneous clustering, which canbe identified with the loss of linear stability. The state points in between are metastable, whichimplies that a critical nucleus has to appear before the suspension can reach the stable phaseseparated state. Indeed, nucleation behavior and hysteresis close to the instability line havebeen observed in simulations [16].

6 ConclusionsTo summarize, we have outlined how to systematically treat the large-scale behavior of ac-tive Brownian particles following Refs. 16, 26, 27. Starting from the microscopic many-bodySmoluchowski equation, the dynamical equation for the one-point density is derived. This is amean-field description, in which every particle moves in an effective environment characterized

D1.12 T. Speck

by the pair distribution function. The input to the theory is static two-point correlations in formof the force imbalance coefficient ζ defined in Eq. (11). The next step is then to derive a dy-namical equation for the density of active particles alone, which is accomplished by a weaklynon-linear analysis. Rewriting this dynamical equation leads to a potential function, which canbe interpreted as an effective free energy by association with the phase separation of passivecolloidal particles.The effective free energy takes on the bimodal shape of a Ginzburg-Landau free energy. Hence,the large-scale behavior of active Brownian particles can been mapped onto that of an equilib-rium system with effective attractions, where the speed v0 takes the role of an inverse tempera-ture. At this point one might wonder whether active Brownian particles are entirely accountedfor by a mapping to an equilibrium system with effective attractions. To this end one shouldstress that the results derived here have been obtained from an expansion using a small parame-ter ε. These results hold in a region where the relevant scale is much larger than the persistencelength `p = v0τr. Already to next order a description solely in terms of the density ceases to bevalid. However, recently it has been suggested that such a mapping can be even extended to thelevel of an effective, isotropic equilibrium pair potential that reproduces the non-equilibriumbehavior of ABPs [33]. This remains to be tested more thoroughly.Acknowledgments. I thank Hartmut Lowen, Andreas Menzel, and Julian Bialke for an in-spiring collaboration. Financial support by the DFG is gratefully acknowledged within priorityprogram SPP 1726 under grant number SP 1382/3-1.


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