Arrested States in Persistent Active Matter: Gelation without Attraction
Carl Merrigan, Kabir Ramola, Rakesh Chatterjee, Nimrod Segall, Yair Shokef, Bulbul Chakraborty
AArrested States in Persistent Active Matter: Gelation without Attraction
Carl Merrigan, Kabir Ramola, Rakesh Chatterjee, Nimrod Segall, Yair Shokef, and Bulbul Chakraborty Martin Fisher School of Physics, Brandeis University, Waltham, MA 02454, USA Centre for Interdisciplinary Sciences, Tata Institute of Fundamental Research, Hyderabad 500107, India School of Mechanical Engineering and Sackler Center for ComputationalMolecular and Materials Science, Tel Aviv University, Tel Aviv 69978, Israel
We explore phase separation and kinetic arrest in a model active colloidal system consisting ofself-propelled, hard-core particles with non-convex shapes. The passive limit of the model, namelycross-shaped particles on a square lattice, exhibits a first order transition from a fluid phase to asolid phase with increasing density. Quenches into the two-phase coexistence region exhibit an agingregime. The non-convex shape of the particles eases jamming in the passive system and leads tostrong inhibition of rotations of the active particles. Using numerical simulations and analyticalmodeling, we quantify the non-equilibrium phase behavior as a function of density and activity. Ifwe view activity as the analog of attraction strength, the phase diagram exhibits strong similaritiesto that of attractive colloids, exhibiting both aging, glassy states and gel-like arrested states. Thetwo types of dynamically arrested states, glasses and gels, are distinguished by the appearance ofdensity heterogenities in the latter. In the infinitely persistent limit, we show that a coarse-grainedmodel based on the asymmetric exclusion process quantitatively predicts the density profiles of thegel states. The predictions remain qualitatively valid for finite rotation rates. Using these results,we classify the activity-driven phases and identify the boundaries separating them.
I. INTRODUCTION
Active matter, constituted of particles that convertambient energy to directed motion, has emerged as animportant class of non-equilibrium systems with exam-ples ranging from bacterial suspensions to synthetic col-loids. Being driven out of equilibrium at microscopicscales, the collective dynamics of these systems are farricher [1–7] than thermal systems, which are bound byfluctuation-dissipation relations.A particular collective behavior that has been widelystudied is motility-induced phase separation (MIPS) [8–13]. MIPS, a kinetic phenomenon, is striking in its sim-ilarity to equilibrium phase separation such as in pas-sive colloids with attractive interactions [14]. The non-equilibrium phases and transitions between them whilehaving exact analogs in equilibrium systems [8–13] ex-hibit anomalous fluctuations that can be traced back totheir non-equilibrium nature. The universality of MIPShas led to the proposition that activity mimics attrac-tion [10]. Under certain conditions, the non-Brownianrandom walks representing active-particle dynamics canbe mapped onto systems with detailed balance [15].In addition to phase separation, passive colloids ex-hibit dynamically arrested phases in the form of glassesand gels. These two types of disordered, amorphoussolids have distinct structural and dynamical signatures.The glass transition occurs in both repulsive and at-tractive colloids at packing fractions close to randomclose packing and is structurally homogeneous on largelength scales. Gelation in attractive colloids leads tostrongly heterogeneous states with fluid-like regions co-existing with an arrested, percolated, dense phase [14].Active analogs of the glass transition [16–24] and jam-ming [25, 26] have been explored extensively in activeBrownian particles (ABPs) interacting via repulsive po- tentials. A review of the emergent behavior of active par-ticles in crowded environments appears in [27]. Recentwork on an extreme limit of ABPs with long persistencetime of their self-propulsion direction has revealed fluc-tuations in the dense limit that are qualitatively differentfrom those at short persistence times [28]. In this limitof long, yet finite persistence times, clustering and het-erogeneous dynamics analogous to passive gels has beenobserved [29] lending further credence to the idea that ac-tivity can act as an effective attractive interaction. Sim-ilarly, soft disks with infinitely persistent active motionalong quenched random directions also display a transi-tion to an absorbing jammed phase above a critical den-sity [30]. Other studies have focussed on the variation ofpersistence times in such systems using the static fluidstructure as well as non-equilibrium velocity correlations[19–22]. The activity-induced change in the effective at-traction in these systems depends on the microscopic de-tails of the particles as well as their persistence times,and therefore activity may enhance or suppress glassydynamics.In this paper, we explore the two well-knownparadigms of dynamical arrest in passive colloids, gela-tion and glass formation, in a lattice model of ABPswith purely repulsive interactions but with a non-convexshape that can interlock and hinder rotations. Using nu-merical simulations and a coarse-grained model based ona mapping to an asymmetric simple exclusion process(ASEP) [33], we classify the activity-induced phases andconstruct a non-equilibrium phase diagram. A novel fea-ture of this phase diagram is the appearance of a phasewith coexisting voids and solids separated by an inter-face wetted by an active fluid. The coarse-grained modelquantitatively predicts the width of this interface, whichcombined with conservation laws leads to predictions ofthe non-equilibrium phase boundaries separating the ar- a r X i v : . [ c ond - m a t . s o f t ] M a y FIG. 1. (a)-(f) Snapshots of the system obtained at the end of simulations runs, t max = 2 × . The color-bar representsthe stationary time for each particle on log scale, log ( τ i ( t max )) (see Section III). Unoccupied lattice sites are colored white.Particles that have not taken a single step through the duration of the simulation are colored darkest. The phases we find byvarying the density ρ and activity ∆ v are (a) steady state passive fluid, (b) passive fluid-solid/aging glass states, (c) finite-activity states resembling the aging regime of the passive system, (d)-(f) active phase with void-solid coexistence (see SectionV). There is a progression from majority non-arrested states (d), to majority arrested states (e)-(f) with increasing density.Panel (g) displays a color map of the fraction of arrested states (see Section IV). Locations of the snapshots shown in (a)-(f)are marked on the phase diagram with pink circles. ρ rcp in the figure denotes the random-close-packing density of hard crosses:the maximal density that can be reached via the RSAD process [31]. The morphology exhibited by the snapshots at highdensities (e)-(f) are reminiscent of high-density gels in attractive colloids, while (c)-(d) resemble the gel-bubble states at lowdensities [32]. rested states. The main result, summarized in Fig. 1,is that activity triggers arrest into a percolated phaseof immobile particles akin to a gel in attractive colloids.At high densities, the transition is from an aging glass,whereas at low densities it is from an active fluid. Toour knowledge, this model provides the first realizationof an activity-induced transition from a repulsive glass toa gel.As described in detail below, we study an active latticegas model [11, 34] of hard-core, cross-shaped particles ona square lattice, which is also referred to as the N3 modelsince each cross prevents the occupation of the first, sec-ond and third neighbors of its central square, as shown inFig. 2. In the passive limit, this is the simplest lattice-gasthat exhibits a finite-density first-order transition froma fluid phase to a sublattice-ordered phase with tenfoldsymmetry [35]. The sublattice-ordered states can be fur-ther grouped into right-handed and left-handed chiral or-der. In continuum, experiments have demonstrated theemergence of long-range chiral order in crystals of cross-shaped particles [36]. As the density is quenched intothe two-phase coexistence region [31, 37], one observes acrossover from a simple fluid to a slowly coarsening or ag- ing regime in which concentrated immobile clusters withlocal crystalline order emerge. There is evidence for theexistence of a glass transition [38] in this passive systemin the form of diverging timescales and the appearanceof dynamical heterogeneities [39, 40].This paper is organized as follows. Section II describesour model and the simulation methodology. In SectionIII, we quantify the spatially heterogeneous dynamicsthat is visible in the snapshots shown in Fig. 1. Next,in Section IV, we classify the states into two categories,arrested and non-arrested, based on measurements ofthe mean-squared-displacements (MSDs) of the particles.The MSD measurements also distinguish between aging,glassy states, and gel-like arrested states. This classifi-cation is used to construct the phase diagram shown inFig. 1 (g). In Section V, we present a coarse-grainedmodel of the dynamics that leads to a prediction of thedensity profile. We compare these results to the densityprofiles obtained in our numerical simulations in SectionVI, and construct a non-equilibrium phase diagram thatdelineates states based on the density profiles of the ar-rested states. This classification connects the dynamicalsignatures of arrest shown in Section IV to phase sepa- FIG. 2. (a) Nearest neighbor labels and hopping rates for ouractive hard-cross model on a square lattice. The particles canperform thermal moves in any of the four directions with arate v (black arrows), and active moves with a rate v + ∆ v along the direction of their orientation (red arrow). (b) Thecross at the central site (green) is prevented from rotating bythe presence of neighboring crosses occupying the fourth andfifth nearest-neighbor sites. ration. Lastly, in Section VII, we consider the effects offinite persistence times. The appendices provide furtherdetails of the passive, glassy dynamics, and discuss finitesize effects. II. MODEL AND SIMULATIONS
We study a model of hard-core particles on the squarelattice with exclusion up to third nearest-neighbors [35,38, 41–43]. Each particle can be represented as a hard,cross-shaped object occupying five lattice sites, see Fig. 2.The highest possible density of ρ = 0 . ρ fluid ≈ .
16 with a crystalline solid at ρ solid ≈ .
19 [35, 38, 41–43]. There are 10 distinct sublattice orderings possible forthe crystalline packings. The competition between thesephases leads to frustration at high densities, and indeedthis model displays a glass transition at higher densities[31, 39, 40]. By adding activity we can, therefore, studyactive analogs of the glass transition and gelation in pas-sive colloids [14, 32, 44].In the active generalization of the model [45], each par-ticle is assigned an active direction which can point alongany of the four lattice directions (cid:0) , π , π, π (cid:1) . The par-ticles perform active Brownian walks on the lattice witha rate v + ∆ v along the active direction, where v is the“thermal” hopping rate along each of the four lattice di-rections. Each particle can change its active direction by ± π with a rotation rate D R . The thermal diffusion co-efficient, D T = a v , where a is the lattice spacing, is setto unity in our simulations. Since there is no energy scalein this model, the only role of temperature is to set themagnitude of the diffusion coefficient, which simply fixes the unit of time. The expected self-propulsion velocityfor a single cross in the dilute limit is v p = a ∆ v [45],and hence the translational Peclet number is given by P e t = v p aD T = ∆ vv . Since we fix v = 1, we use ∆ v torepresent P e t .The active dynamics we prescribe for hard crosses areidentical to those implemented in simulations of MIPSfor a simple-exclusion lattice gas model [11]: squares on asquare lattice, which do not exhibit an equilibrium phasetransition or glassy dynamics. Further, we consider rota-tion of the active direction and require a rigid rotation ofthe whole cross. Consequently, rotations are disallowedfor crosses that have neighboring crosses which occupyeither the fourth or fifth nearest-neighbor site, see Fig. 2(b). We note that unlike simulations of continuum activedynamics, D T cannot be set to zero because of a kinetictrap which only exists for random walks on a lattice [11].This is especially true for non-rotating active particles:without thermal moves to free them, non-rotating ac-tive particles become immediately trapped upon colli-sion [46, 47].We use a continuous time, rejection-free, kinetic MonteCarlo algorithm to implement the active dynamics [48–50]. All allowed events in the system are assigned a rate,and the relative weight of each rate determines the proba-bility for the event to occur. Time proceeds by randomlyselecting an event, and then advancing the clock by an in-terval − log( r ) / R , where r is a uniform random variable,and R is the sum of all non-zero rates for the allowedevents at a give time t . The time increments after eachevent are exponentially distributed with mean (cid:104) ∆ t (cid:105) = R .This algorithm is especially efficient for simulations atlarge densities, where most moves are disallowed by theexcluded volume constraint.The initial states of the system are prepared using aRandom Sequential Adsorption and Diffusion (RSAD)process [37] which can generate disordered packings upto a maximum density of ρ rcp = 0 . ... [38], corre-sponding to the random-close-packing density for hardcrosses. We study a range of global densities between ρ = 0 .
10 and ρ = 0 .
17, and a range of activity values ∆ v from 0 to 1. Note that even though ∆ v is a dimension-less activity, it may take values larger than unity [45].Thesystem domain is a two-dimensional square box of linearlength L , periodic boundary conditions, and a fixed totalnumber of particles N = ρL . Unless otherwise stated,we present results for L = 450. The longest simulationtime is set to t = 2 × , which is much larger than the α -relaxation time at ρ = 0 . τ α ≈
100 (see AppendixA).In this work we focus primarily on the limit D R → D R > D R leads to qualitativelysimilar results as the infinite persistence-time limit, in-cluding overall global arrest due to percolating gel-likestructures, as shown in Section VII. III. DYNAMICAL HETEROGENEITY ANDACTIVITY-INDUCED AGING
Our measure of dynamical heterogeneity is based onthe definition of “stationary times” for each cross. Thestationary time τ i ( t ) at the observation time, t , is definedto be the time that cross i has spent at its currently occu-pied site, (cid:126)r i , namely τ i ( t ) = t − t i , where t i is the time atwhich particle i arrived at (cid:126)r i . Distributions of τ i ( t ) pro-vide a quantitative measure of the spatial heterogeneityof the dynamics, and is closely related to the distributionof persistence times used to analyze the glass transitionin kinetically constrained models [51–53]. In the agingregime and at large activities, the distributions of τ i de-pend explicitly on time, therefore, spatial configurationsof τ i ( t max ) are used to construct the color bar in Fig. 1.The distributions of τ i ( t max ) are shown in Fig. 3.We first discuss the nature of dynamical hetero-geneities in the passive system. The adsorption processof the RSAD protocol generates, at time t = 0, a uni-form configuration with density ρ in which crystallineorder is minimized. In the passive system, the diffu-sion of the crosses at t > ρ ≤ . P (log τ i ( t max )) has a single peak with anincreasingly broad tail at large τ i as ρ → . ρ > . τ i ( t max ) (cid:39) t max appearing within a background of parti-cles with 10 < τ i ( t max ) < (Fig. 1 (b)). Bimodal dis-tributions of persistence times have been used to identifydynamical heterogeneities in several glass-forming kinet-ically constrained models [51, 54]. We show in AppendixA that standard measures such as the self-intermediate -4 -3 -2 -1 -4 -3 -2 -1 -4 -3 -2 -1 -4 -3 -2 -1 FIG. 3. Distributions of stationary times, P (log ( τ i ( t max )).In the passive system (∆ v = 0), a signature of the crossoverfrom the steady state passive liquid to an aging glass regimeis the appearance of a bimodal distribution, indicating twopopulations of particles with mobilities that differ by severalorders of magnitude. This bimodal distribution appears atlower densities when persistent active motion is introduced.The different figures represent (a) Passive crosses ∆ v = 0. (b)The transition to an“active glass” with increasing density ata small activity ∆ v = 0 . ρ = 0 .
14 and (d) ρ = 0 .
16 with increasing activity. scattering function also indicate the onset of aging at ρ ≈ . P (log τ i ( t max )) can develop abimodal structure at densities lower than 0 . ρ ≥ .
16, thestates (Fig. 1 (c)) resemble the passive aging fluid (Fig.1 (b)) with growing clusters of immobile, solid-like re-gions suspended in a fluid. Increasing the activity in thisdensity regime leads to states with percolated clusters ofimmobile particles as seen in Figs. 1 (f). At densitieslower than this regime of activity-induced aging, the ap-pearance of a bimodal distribution in P (log τ i ( t max ))with increasing activity is accompanied by a clear spatial separation of the particles into “voids” and dense regionsaccommodating the most immobile particles (Fig 1 (d)-(e)). In this regime, we observe large variations in thefinal structures from one simulation run to another.The variance of the stationary times, Q ( t ) = (cid:104) ( τ i ( t ) −(cid:104) τ i ( t ) (cid:105) ) (cid:105) , provides a global measure of the time evolu-tion of spatial heterogeneity in our dynamics. Fig. 4shows the ensemble averaged time series, (cid:104) Q ( t ) (cid:105) , of Q ( t )at different densities and activities. In the passive sys- = 0.1550 = 0.1600 = 0.1625 = 0.1650 = 0.1675 = 0.1700 = 0.1400 = 0.1500 = 0.1600 = 0.1625 = 0.1650 = 0.1700 FIG. 4. Time series of the ensemble averaged stationary timevariance, (cid:104) Q ( t ) (cid:105) , for the same data sets as in Fig. 3. (a)Passive system: (cid:104) Q ( t ) (cid:105) increases with time for ρ ≥ . v = 0 . (cid:104) Q ( t ) (cid:105) increases with time at densities ρ ≥ . ρ =0 . (cid:104) Q ( t ) (cid:105) increases with time for ∆ v ≥ . ρ =0 . (cid:104) Q ( t ) (cid:105) increases with time for ∆ v ≥ . τ active , is marked by black arrows inpanels (b)-(d). tem, (cid:104) Q ( t ) (cid:105) rapidly reaches a small steady state valuein the non-aging regime of densities (Fig. 4 (a)). Boththe equilibration time and the magnitude of the steady-state dynamical heterogeneity grow with density, untilfor ρ > . (cid:104) Q ( t ) (cid:105) growsindefinitely. Activity drives strong growth of (cid:104) Q ( t ) (cid:105) atlower densities. However, as shown in Figs. 4 (b)-(d),there is a delay time, τ active , before which the system dy-namics match the passive system. To quantify this delaytime, we define τ active to be the time at which the deriva-tive of the stationary-time variance becomes greater thana small threshold, d (cid:104) Q ( t ) (cid:105) /dt > (cid:15) . For any activity thederiviative is positive for t >
0, but it may remain smallfor some time: we find (cid:15) = 10 gives a robust signa-ture for the approximate time when the growth rate of (cid:104) Q ( t ) (cid:105) first becomes significantly different from zero, seeFig. 4. We have checked that the choice of (cid:15) does notsignificantly change our results other than providing ascale. This delay time increases with decreasing activityor density. At very weak activities, therefore, we observethe passive behavior of (cid:104) Q ( t ) (cid:105) since τ active increases be-yond our maximum simulation time. For instance, at ρ = 0 . v ≥ . Q ( t ), whichis the analog of the zero-wavevector, four-point suscepti-bility, χ ( q = 0 , t ) [56], resembles that observed at smallwave vectors in a model of chemical gelation [57]. Astriking feature of the arrested and non-arrested statesat high activities is the appearance of “voids”. We showbelow that the appearance of these voids is a manifesta-tion of an extreme form of MIPS for these non-rotatingparticles that can be understood from ASEP dynamics.Physical gels arise from arrested phase separation [14],the gel-like states in our active lattice gas similarly seemto arise from arrested MIPS. IV. COLLECTIVE ARREST, PRESENCE OF“ABSORBING STATES”
In this section, we explore the appearance of arrestedstates through measurements of the MSD of individualparticles. The MSD of all particles ( i = 1 , ..., N ) in thesystem is defined as R ( t ) = 1 N i = N (cid:88) i =1 | (cid:126)r i ( t ) − (cid:126)r i (0) | . (1)In order to determine if states are arrested, we study thelog derivative of this MSD for each system, defined as γ ( t ) = d log R ( t ) d log t , (2)which has been used for studying dynamics in a model ofgelation [44]. This observable is an instantaneous expo-nent that indicates whether the MSD is diffusive ( γ = 1),ballistic ( γ = 2), or arrested ( γ = 0). If there is perco-lation of a dynamically arrested phase, then γ ( t ) → R ∼ t to a flat plateau R ∼ t once the walker reaches thewalls.Fig. 5 shows MSD and γ ( t ) measurements for indi-vidual runs at large and intermediate activity values,∆ v = 0 .
20 and ∆ v = 0 .
07. We can construct a mean-field model for the behavior of γ ( t ) in the non-arrestedstates by using known results about the dynamics of asingle non-rotating active tracer moving in a backgroundof passive particles with density ρ [45]. For a non-rotatingactive tracer moving on a lattice at ρ = 0, the MSD isgiven by ∆ r i ( t ) = D (4+∆ v ) t + D ∆ v t [11]. This formmay be generalized to higher densities by introducing adensity-dependent diffusion coefficient D ( ρ ), giving∆ r i ( t ) = D ( ρ )(4 + ∆ v ) t + D ( ρ ) ∆ v t . (3)Note that this requires a measurement of D ( ρ ) for thepassive lattice gas. For a single active tracer, there-fore, γ ( t ) displays a smooth crossover from 1 to 2 ata characteristic timescale, which we can estimate from γ ( t ∗ ) = 3 /
2, yielding t ∗ = (4 + ∆ v ) / (∆ v D ( ρ )). Thiscrossover time increases as the density is increased or ac-tivity is decreased. This is the same trend as exhibitedby τ active . Thus, any non-trivial collective behavior aris-ing from the activity appears once the ballistic motiontakes over.We define arrested runs to be those for which γ ( t max ) < .
5. For strong activities all runs fall into the arrestedclass, whereas at lower values of ∆ v only a finite fractionof runs become arrested. When considering the ensembleof possible dynamical trajectories, we observe two differ-ent types of activity-driven behavior: arrested states forwhich γ ( t ) → γ ( t ) fluctuating around a value of 2. It is clear fromFig. 5 that the differentiation between arrested and non-arrested runs emerges only at times longer than t ∗ .The behavior of γ ( t ) offers the clearest contrast be-tween activity-induced arrest, as seen in our model, andthe attraction-induced gelation seen in passive colloids.In passive colloids, diffusing particles become arrested ei-ther to indicate a glass or gel transition. In contrast, asseen from Fig. 5, it is the persistent motion, indicatedby the ballistic behavior, that leads to arrest in the ac-tive hard crosses. The coarse-grained model for densityinhomogeneity that we present in the next section is con-sistent with this picture.In passive systems, low-density gels can exhibit sub-diffusive behavior at intermediate times but γ ( t ) asymp-totes to unity since there are always particles that havefinite mobility and can diffuse [44]. In fact, the MSDexhibits only weak signatures of gelation in passive col-loids [14, 58, 59]. The behavior we observe is akin to col-loids trapped in a porous environment [60]. Since the ar-rested states (Figs. 1 (e)-(f)) are characterized by a per-colating network of immobile particles with no gaps be-tween them, all the active crosses are effectively trappedand thus γ ( t ) → γ ( t ) → ρ, ∆ v ). Between 10 to 35 runs were conductedat each set of parameter values, and the color bar indi-cates the fraction of those runs for which γ ( t max ) < . ρ > . , ∆ v < . FIG. 5. Mean-squared displacements (MSD) R ( t ) (insets)and their logarithmic derivatives γ ( t ) (main panels) at (a) ρ = 0 . , ∆ v = 0 .
2, (b) ρ = 0 . , ∆ v = 0 .
07, (c) ρ =0 . , ∆ v = 0 .
07, and (d) ρ = 0 . , ∆ v = 0 .
05. Blue curvesshow “arrested” states ( γ < . (cid:104) γ ( t ) (cid:105) is shown inblack, and the mean-field prediction for a single active traceris shown in green. The single tracer prediction crosses overfrom diffusive ( γ = 1) to ballistic ( γ = 2) at a characteristictime t ∗ (black arrows), which we estimate from γ ( t ∗ ) = 3 / come arrested within our simulation time since the time t ∗ needed for the active particles to exhibit ballistic mo-tion is dramatically slowed down by both the small valueof the diffusion coefficient D ( ρ ), as well as the very weakactivity ∆ v . Consequently, the states in this range ofdensities resemble those found in the passive system. V. COARSE GRAINED DENSITY PROFILESAND HYDROSTATIC LENGTHSCALE
Since dynamic differentiation emerges between differ-ent realizations of the simulations over a range of activityand density values, a further question arises about howthese different classes of states differ structurally at longtimes. For our system of non-rotating, infinitely persis-tent active crosses, the morphology of the arrested states(see Figs. 1 and 6) depict voids coexisting with a solid-like ( ρ solid ≈ . I n c r e a s i n g a c ti v it y Increasing density
FIG. 6. Contour maps of the coarse-grained density profile of arrested states at the longest simulation time t max = 2 × . Weused a coarse-graining box size with area L / lattice sites. The colorbar represents the localdensity which varies from 0 ≤ ρ ≤ .
2. The arrows represent the gradient of the density field, the lowest density point for eachfigure has been shifted to the center of the frame (making use of the periodic boundary conditions). Rows show fixed activity∆ v = 0 . , . , .
05, and columns show densities increasing left to right ρ = 0 . , . , . , .
16. Density fluctuations for atypical non-arrested active liquid state ( ρ = 0 . , ∆ v = 0 . v . tive flux towards the interface causes an increasing den-sity in the vicinity of the solid, giving rise to a diffusivecurrent away from the solid. In order to model this pro-cess, we coarse grain the system to construct a spatiallyvarying density field ρ ( x, y ). We consider a 1D sectionof the system perpendicular to an interface (oriented inthe y -direction for convenience) between an active fluidand the solid, giving rise to a linear density profile ρ ( x ).The exclusion due to particles in adjacent rows as well astheir lateral diffusion give rise to correlations, which weignore for large enough coarse graining blocks. This pref-erence for biased motion perpendicular to the interfacecan also be modeled using the well-known ASEP modelwhich incorporates both the hard-core exclusion alongwith diffusion and biased motion. The steady state den-sity profile in the arrested states can then be derived froma hydrodynamic treatment of the ASEP [33], as we showbelow.The ASEP is a paradigmatic model where many ex-act statements can be made regarding the coarse grained dynamics in a non-equilibrium system. We can there-fore, through this mapping, write equations for the coarsegrained densities appearing at late times in our activecrosses system. In steady state, the density ρ ( x ) is in-dependent of time, and there is no net particle currentbetween the different coarse grained blocks. There aretwo components to this current determined by the den-sity profile ρ ( x ): (1) A diffusive (or thermal) current J T arising due to the spatial variations in density, which tolowest order is J T = − D ∂ρ ( x ) ∂x , (4)(2) An active current J A proportional to the density ofparticles. However, if neighboring blocks are at high den-sities, this current decreases due to exclusion. Once againto lowest order we have J A = α ∆ vρ ( x ) ( ρ solid − ρ ( x )) , (5) ARRESTED SOLIDALVOID VOIDALACTIVE LIQUID ACTIVE LIQUIDARRESTED SOLID ρ solid ρ ( x ) x Lρ ( x ) x L b ) a ) ρ solid l void l solid ξ FIG. 7. One dimensional linear density profiles illustratingthe non-equilibrium phase classification based on the observedconfigurations and local density distributions. (a) A densesolid region along with an active liquid interface which fillsthe remaining space available in the system (see, for example,bottom row in Fig. 6). (b) Solid region, bordered by a narrowactive liquid (AL) interface of total width ξ . The remainingarea in the system is left empty of particles, creating a void(see, for example, top two rows in Fig. 6). where α is an as yet undetermined proportionality con-stant. These are essentially the mean field currents inASEP [64].In steady state, the net currents are zero. Hence, com-bining Eqs. (4) and (5), we obtain: ∂ρ ( x ) ∂x = α ∆ vD ρ ( x ) ( ρ solid − ρ ( x )) . (6)The only two homogeneous solutions to this equation arevoids with ρ = 0 and the solid state with ρ = ρ solid .Eq. (6) is the logistic equation in space, which generatessigmoidal solutions. The saturation values represent thesolid and void regions, whereas the decaying part rep-resents the wetting active fluid. At large distances thissolution decays as exp[ − ( α ∆ v/D ) x ], implying a wettinglengthscale ξ = Dα ∆ v . (7)We note that this “hydrostatic lengthscale” diverges inthe limit of zero activity. This divergence as the activityis decreased is shown in Fig. 8 along with the theoreticalprediction from Eq. (7) showing near perfect agreementwith the ∆ v − decay. Note that Eq. (7) does not involvethe global density of the system, and the correlationsbetween rows in our two-dimensional lattice can providenon-trivial corrections to the derived behavior for largerdensities. -2 -1 FIG. 8. The divergence of the lengthscale of the “wettingactive liquid” as the activity is decreased along with the the-oretical prediction in Eq. (7). The data displays a good agree-ment with the ∆ v − decay. The dashed black line shows thebest fit ξ = v . VI. NON-EQUILIBRIUM PHASE DIAGRAM
As seen in Fig. 6, two types of profiles are observed inthe arrested states: coexistence of an “active liquid” withsolid regions (bottom row), and a solid network, punctu-ated by voids that have a characteristic size, and an ac-tive liquid interface separating the two. In addition, thereare non-arrested, active liquid states with density fluctu-ations of amplitude much smaller than the solid density.In the previous section, we showed that our theory cor-rectly predicts the variation of the width of the inter-face. In this section, we extend our analysis to constructa non-equilibrium phase diagram of the arrested states,and provide a theory for the emergent length scales char-acterizing the voids and the solid regions.We can derive phase boundaries between the threetypes of states by considering the conditions that mustbe satisfied at a given density and activity to create eachof these configurations. For the arrested states at strongactivities, the dense immobile solid ( ρ solid ≈ .
19) is bor-dered by an active liquid interface that can be fit by alinear profile (as an approximation to the sigmoidal so-lutions of Eq. (6)) with slope m = dρdx and width ξ , suchthat ξm = ( ρ solid − ρ void ) = 0 .
19. We use these lineardensity profiles (as shown in Fig. 7) for all states, includ-ing the active liquid, in the computation of the phasediagram. The total length of the system is fixed at L ,and the total number of particles in the system is con-served as N = ρL .The first condition needed to create a solid region alongwith an active liquid interface is that there must beenough mass available in the system to populate boththese regions (as in Fig. 7 (a)). The total area under thetrapezoid must conserve the total number of particles insystem at a given global density ρ , yielding N = ρL = ( ρ solid )( ξ + l solid ) . (8)Since the interface width is determined by the activity(Eq. (7)), the active liquid region can only contain a fixedactivity-dependent mass, and consequently the width ofthe solid region depends on both the density and theactivity as l solid = ρρ solid L − Dα ∆ v . (9)The solid first appears at the point where l solid = 0, yield-ing the equation for the solid-active liquid phase bound-ary ∆ v = DαL ρ solid ρ . (10)Below this activity value, the interface width is largerthan L/
2, therefore we expect the active liquid phase tocontain all the mass in the system.So far we have only imposed conservation of mass onthe system. To determine when voids first appear, wemust also consider whether the shape of the trapezoidalsolid-active liquid profile can fit within the total spaceof the system. Since none of the regions overlap, theirlengths sum to the system size L . We therefore have (seeFig. 7 (b)) L = l solid + l void + 2 ξ. (11)Since the thickness of the solid region is constrained byconservation of mass, Eq. (9), we can solve for how muchspace remains available for the void region, l void = (cid:18) − ρρ solid (cid:19) L − Dα ∆ v . (12)To determine when a void region may first appear, wefind the point l void = 0. This yields the second phaseboundary between states containing voids and thosewithout voids ∆ v = DαL − ρ/ρ solid . (13)Below this activity the active liquid is confined within aspace smaller than its preferred width ξ .Phase boundaries based on the 1D linear profile anal-ysis, presented above, are shown in Fig. 9 (a). In Fig. 9(b) we show that a numerical classification of the threetypes of non-equilibrium phases agrees qualitatively withthe 1D theory. It is straightforward to extend the treat-ment developed in this section to a derivation of the phaseboundaries based on 2D density profiles. We find that thequalitative features of the resulting phase diagram do notchange as compared to the 1D case studied here. Thestates in the phase identified as the solid-active liquid, -2 -1 -2 -1 r c p = . Solid-Active Liquid-VoidActive-LiquidSolid-Active Liquid v = 0.00
FIG. 9. (a) Classifications of the activity-induced phases,based on linear density profiles. The solid lines indicate thetheoretically predicted boundary between the active liquidand solid + active liquid regions in Eq. (10), and the ac-tivity beyond which void regions open up (predicted by Eq.(13)). We have used the observed value Dα = 10, ρ solid = 0 . L = 450. (b) Phase diagramobtained from the numerically sampled phase space of the ac-tive lattice gas, including the passive and the active states.Each point is classified as either pure active liquid, solid +active liquid, or solid + active liquid along with void regions.Open symbols denote the transition from the passive liquidto the aging glassy regime. These classifications are basedon ensemble-average density profiles. The topology of thisphase diagram is the same as that observed in attractive col-loids [14, 32, 65]. The black region denotes densities largerthan ρ solid = 0 .
19. Note that the predicted phase diagramshown in (a) is valid only at finite activities and is expectedto accurately capture the physics only at large enough activi-ties where activity-induced correlations overwhelm the corre-lations intrinsic to the passive hard crosses at ρ ≈ ρ rcp . Thisis because, as discussed in the text, the coarse-grained modelincludes only simple exclusion, not the extended N3 exclusion. based on the density profiles, are dynamically the ag-ing glassy states shown in Fig. 1. The arrested, gel-likestates, shown in Fig. 1 are in the solid-active liquid-voidregion of Fig. 9.In Appendix C we show that the variation of the phase0 = 0.1300 = 0.1400 = 0.1500 = 0.1600 = 0.1300 = 0.1400 = 0.1500 = 0.1600 = 0.1300 = 0.1400 = 0.1500 = 0.1600 = 0.1300 = 0.1400 = 0.1500 = 0.1600 FIG. 10. Comparison of the local density distributions forpersistent ( D R = 0) and finite rotation rate ( D R = 0 . v = 0 .
5, the zero density peak cor-responding to the empty void regions at D R = 0 (a) is re-placed by a well-defined low density peak at D R = 0 .
005 (b).For the smaller activity value, the non-rotating crosses showphase separation into solid and void-like regions, with an in-tervening active liquid region (c). In contrast, at D R = 0 . boundaries with system size is consistent with the predic-tions of the theory derived above. In the infinite systemsize limit, the ASEP analysis implies that voids can beaccommodated at any density and activity, as the phaseboundaries become lines with infinite slopes at ρ = 0 and ρ = ρ solid . This feature is a consequence of the simpleexclusion process, which cannot describe the non-trivialcorrelations in the passive, hard-cross system (c. f. Fig.9 (b)). We are currently exploring avenues for incorpo-rating these effects, possibly through the construction ofa large-deviation function that incorporates the physicsin of void-solid coexistence of persistent hard crosses, en-capsulated in the ASEP model, and the glass transitionphysics of passive hard crosses. VII. EFFECT OF FINITE ROTATION RATE
In this section we discuss the effect of a finite butsmall rotation rate on the arrested phase separationobserved for non-rotating hard crosses ( D R = 0) de-rived above. We perform simulations in two regimes D R = 0 . , ∆ v = 0 .
1, and D R = 0 . , ∆ v = 0 .
5. Itshould be noted that D R is the attempted rotation rate.The non-convex shape of the hard-crosses leads to strongrotational locking that leads to a smaller effective rota-tion rate, which also decreases as the density increases.We find that at large activities ∆ v ≥ .
5, a percolated,globally arrested solid network, similar to the ones ob-served at D R = 0, is able to form in the same regime -2 -1 FIG. 11. Phase diagram at D R = 0 .
005 showing the binodalconstructed from the peaks in ensemble density distributionswith triangles marking the high-density peak and circles thelow-density one. Crosses indicate phase space points (∆ v, ρ )at which the system is in a homogeneous fluid state with asingle peak at the global density in the density distribution.Colors indicate the global density ρ . Density distributionswere averaged over 5 runs for each sampled phase space point. of densities as shown in Fig. 1. However, there arestructural differences between the arrested states at zeroand non-zero D R . Specifically, the voids observed at D R = 0 are replaced by a low-density, gaseous fluid at D R = 0 .
005 as seen in Fig. 20. The fluid phase stillexhibits an inhomogeneous density profile with the high-est densities occurring at the interface with the solid. Inaddition, a finite rotation rate seems to create a round-ing of the facets separating the solid from the fluid. Atlower activities, the fluid density becomes homogeneous,as seen in Fig. 21. The morphology of these states re-semble those observed in the active-aging regime (Fig. 1(c)) rather than the percolating arrested solid observedat D R = 0, seen in Fig. 21 (a).The ensemble-averaged density distributions shown inFig. 10, demonstrate that at D R = 0 . , the “void”peak disappears. In addition, there is a clear signatureof two-phase coexistence between a fluid and a solid, withlittle dependence of the peak positions on the global den-sity [66]. With increasing activity, the peak marking thefluid density shifts significantly to lower densities whereasthe solid peak stays roughly pinned ρ ≈ .
19. Since theensemble-averaged density distributions do not dependon the global density at a fixed activity (Fig. 21), a bin-odal line can be constructed from the positions of thepeaks, as shown in Fig. 11. Interestingly, this phase di-agram looks different from that of spherical ABPs [66],and remarkably similar to the one observed in actively-driven dumbbell shaped particles, which are also non-convex [12].1We can analyze the stability of the D R = 0 phase di-agram to small but non-zero D R . On the lattice, sincethe particles can only have discrete orientations, we cancompute the mean free path of the particles in the per-sistent direction. Since the density of rotation events intime have a gap distribution D R exp( − D R t ), and the av-erage time between succesful rotation events scales 1 /D R [45], the average length traveled without a rotation move,is given by ξ R ∼ (∆ v )(1 /D R ). In order for the rota-tional diffusion to affect the motion of the particles, theremust be a sufficient time for the particles to move be-fore reaching the dense regions and become a part of thewetting fluid. Therefore, when the average length trav-eled in the persistence direction is comparable to the sizeof the voids l void , the phases appearing will no longercorrespond to the zero rotation limit. This provides acrossover value D ∗ R = ∆ v/l void , above which the rota-tional diffusion takes on significant effect. Since the sizeof the voids can be estimated from Eq. (12), we can es-timate the crossover rotation rate below which voids areexpected to appear in the system1 D ∗ R = (cid:18) − ρρ solid (cid:19) L ∆ v − Dα ∆ v . (14)For the values of ∆ v used in the simulations, and usingthe values D/α = 10 and L = 450 deduced from the data,we estimate the crossover value for density ρ = 0 . D R = 0 .
005 for ∆ v = 0 .
1, and D R = 0 .
009 for ∆ v = 0 . D R decreases with ∆ v , weexpect that as the activity is decreased, the qualitativefeatures change from those observed in arrested states atzero rotation, and begin to resemble a MIPS between alow and high density fluid, as is observed in Fig. 21.To summarize, we find that the most significant differ-ence between hard crosses with D R = 0 and D R = 0 . VIII. DISCUSSION
In this paper we have studied a lattice gas of activeparticles with a non-convex shape that leads to stronginhibition of rotations. We showed that unlike the usualMIPS observed in active particles with large rotationaldiffusion, highly persistent particles are able to arrest thephase separation leading to states that are characterizedstructurally by a void-solid coexistence with a fluid ofrelatively mobile particles “wetting” the void-solid inter-face. The voids have a characteristic size that dependson density and activity in contrast to coarsening, as theywould if the phase separation was not arrested. Dynami-cally, the glassy dynamics characterizing the aging regimeof the passive system transitions to complete arrest in thephase-separated regime. In addition, activity enlarges the density range of the aging regime. We also show thatadding moderate rates of rotational diffusion does notchange this picture qualitatively.By appealing to the dynamics of a single active tracerin a passive background of hard crosses [45], we showedthat the dynamic differentiation between arrested andnon-arrested states appears at times longer than thatneeded for the tracer dynamics to cross over from dif-fusive to ballistic. Thus, the gel-like arrest, in contrastto the glassy caging dynamics of the passive system,is driven by activity. The morphology of the arrestedstates, however, closely resembles gelation or arrestedphase separation in passive colloids with attractive in-teractions [32, 65]. Our analysis shows a clear crossoverfrom the passive glassy dynamics to gel-like arrest at highdensities [65].We used the fact that our persistently active dynam-ics effectively causes one-dimensional motion against aninterface, to map the late-time dynamics in the arrestedstates onto the Asymmetric Exclusion Process (ASEP).This microscopic mapping of the long time behavior ofthe system to a well-known lattice model allowed us toinvoke well-established coarse graining procedures whichwe used to describe the non-trivial collective behaviourobserved in our system. Building on this understand-ing, we used the predicted length scale to map out anon-equilibrium phase diagram that predicts non-trivialphases, a non-trivial topology, as well as non-trivial finitesize scaling. While the position of the phase boundariesare not in quantitative agreement with our numerical re-sults, the topology matches the numerical results overmost of the phase space. The ASEP-based predictionsfail at low activity and high densities because it takesinto account only simple exclusion, and thus does notincorporate the shape-induced frustration and glassy dy-namics in the passive system at high enough densities.In this paper we make concrete predictions about thephase behavior and the morphology of arrested stateswhen rotation of the active direction is strongly inhib-ited. This should occur naturally in collections of activeentities with non-convex shapes [67]. Numerical simula-tions have explored MIPS [12] and glassy dynamics [56] inactive dumbbells. In these studies, the mechanism of de-correlation of the active direction is, however, related tothermal diffusion [56] and not independently controlled.It would be interesting to explore dynamical arrest insimulations of rigid non-convex shapes where the rota-tion rate is affected by rotational locking. Experimentalinvestigation of phase separation and arrest in collectionsof active colloidal particles with non-convex shapes offerpossibilities of testing our theoretical predictions. We arenot aware of any such experimental investigation, how-ever, extending studies such as the Brownian dynamics ofhard crosses [36] to include self propulsion seem feasible.2
ACKNOWLEDGMENTS
We thank Mustansir Barma, Pinaki Chaudhuri, Chan-dan Dasgupta, Madan Rao, Abhishek Dhar, Eli Eisen-berg and Andras Libal for helpful discussions. The workof CM and BC has been supported by the Brandeis MR-SEC (nsf-dmr 1420382). CM was hosted as graduatefellow at KITP, with support from the National ScienceFoundation under Grant No. NSF PHY-1748958 andthe Heising-Simons Foundation. This work was also par-tially supported by the Israel Science Foundation GrantNo. 968/16 and by a grant from the United States-IsraelBinational Science Foundation.
Appendix A: Dynamics of Passive Hard Crosses
In this appendix, we provide a brief description of ouranalysis of slow dynamics and the “glass transition” inthe passive hard cross system (∆ v = 0) [31, 39, 40]. Thisanalysis was performed to establish a baseline for thedynamics in the absence of activity. The initial stateswere created through a quench into the two-phase coex-istence region using the Random Sequential Adsorptionand Diffusion (RSAD) protocol [37]. The hard-crosseswere then evolved according to the dynamics describedin Section II in the main text, and two standard measureswere used to probe the glassy dynamics [68, 69]: (i) theself-intermediate scattering function (ISF) and the (ii)mean-squared-displacement (MSD) of individual crosses.The slowest relaxation occurs at the wave-vector withmagnitude q at which the static structure factor has apeak. For N particles, the ISF is defined as F q ( t w , t w + ∆ t ) = 1 N N (cid:88) j =1 exp( i(cid:126)q · ( (cid:126)r j ( t w + ∆ t ) − (cid:126)r j ( t w ))) , (A1)where the wave vector (cid:126)q has magnitude q . The MSD isdefined as: R ( t w , t w + ∆ t ) = 1 N N (cid:88) j =1 | (cid:126)r j ( t w + ∆ t ) − (cid:126)r j ( t w ) | . (A2)In a supercooled liquid, both the ISF and MSD shouldrespect time-translational invariance, and should be in-dependent of the waiting time, t w . From Figs. 12 and13, this expectation is met for densities ρ < . t w from 200,000 to 800,000, to a stretched exponential form[70, 71], F q (∆ t ) ∝ exp( − (∆ t/τ α ) β ), as shown in Fig. 14(a). The τ α extracted from this fit increases by two or-ders of magnitude over the density range 0 .
12 to 0 . -2 -1 -2 -1 FIG. 12. Mean-squared displacements (MSD) of passive hardcrosses starting from: (a) the initial state produced by theRSAD protocol, t w = 0, and (b) t w = 200 , t w , for densities ρ ≥ . Within our simulation time window of t max = 2 × ,passive hard crosses do not reach a time-translationallyinvariant state for densities ≥ . t w , at these densities. We have es-tablished that in the aging regime the ISF exhibits a t/t w scaling, as shown in Fig. 15. Scalings of this form arecharacteristic of the aging regime in glassy systems [72–74]. Appendix B: Ensemble Averaged DensityDistributions
The standard evidence for MIPS is based on measure-ments of ensemble-averaged density distributions. In thisappendix, we present numerical results for these distri-butions over the full range of densities and activities.These ensemble averages include both arrested and non-arrested states.3 FIG. 13. Self-intermediate scattering functions (ISF) of pas-sive hard crosses starting from: (a) the initial state producedby the RSAD, t w = 0, and (b) t w = 200 , t w , for densities ρ ≥ . As in standard equilibrium phase separation, MIPSphase boundaries are drawn based on coexisting peaks inthe density distributions, whose positions do not dependon the global density but which trade intensity as theglobal density is varied [66]. In our system, this expecta-tion is met at low activities (Fig. 16) where we observe acontinuation of the two-phase coexistence characteristicof the passive hard-crosses.At higher activities, as seen in Fig. 17, the density dis-tribution shows more structure. The ensemble averageddensity distributions for the passive hard crosses exhibita fluid-solid coexistence between ρ ≈ .
16 and 0 .
19. Inthe presence of activity, the fluid peak at ρ ≈ .
16 be-comes broader and ultimately develops a peak at verylow densities, characteristic of the voids seen in the ar-rested states. Before the void peak emerges clearly, inthe solid-active liquid regime of the phase diagram, thepositions of these low-density peaks are observed to de-pend both on activity and global density (c. f. Fig. 17(c)-(d)). This observation is consistent with the fact that = 0.1000 = 0.1100 = 0.1200 = 0.1300 = 0.1400 = 0.1450 = 0.1500 = 0.1525 = 0.1550 = 0.1575 = 0.1600 = 0.1610 = 0.1625 FIG. 14. (a) Stretched exponential fits (black lines) to theISFs at densities ρ ≤ . ρ = 0 . β , decreases from 0 . ρ = 0 . . ρ = 0 .
16. (b) τ α extracted from the stretchedexponential fits. the phase separation is “arrested”. As we have shown us-ing the ASEP-based coarse-grained theory, the arrest ofthis phase separation leads to a wetting layer of width ξ within which the active liquid is confined. The densityof particles in this active layer is a function of both ac-tivity and global density, as observed from the snapshotspresented in the bottom row of Fig. 6. Appendix C: Finite-Size Dependence of PhaseBoundaries
The coarse-grained theory, based on ASEP that wehave used to construct the non-equilibrium phase dia-gram [32] of arrested states at D R = 0, predicts that thephase boundaries depend explicitly on the system size L .In the infinite-system size limit, voids can form at anydensity and activity, according to this theory. We haveperformed limited studies of the size dependence of thephenomena we observe with the sole intent of checking4 -2 -1 FIG. 15. Waiting time ( t w ) dependence of the long time decayof the ISF at ρ = 0 . , . , . t w , and as shown in theinset, depends only on the ratio tt w . whether the results of numerical simulations agree qual-itatively with the predictions of the theory.In Fig 18 we compare ensemble averaged density distri-butions for system sizes L = 225 , , and L = 600 at fourdifferent values of ∆ v at ρ = 0 . L . For example, at ∆ v = 0 . L = 225 , L = 600. Similarly, at∆ v = 0 .
05, the active-liquid solid phase is not observedat L = 225.As mentioned above, the reason for the strong finitesize effects is the ability of the system at D R = 0 to formvoids at any activity and density. This feature is vali-dated in the simulations, as seen in Fig 18, which showsregions that appear as low-density fluctuations in smallsystems transition to voids at larger L . These effectsare in turn a consequence of the emergence of a singlelength scale, ξ , which depends only on ∆ v , characteriz-ing the density variation. When the system size is largerthan ξ , voids open up to ensure mass conservation. Con-versely, squeezing the system down to sizes smaller than -1
2 = 0.1300 = 0.1400 = 0.1500 = 0.1550 = 0.1600 = 0.1625 = 0.1650 = 0.1700 -1
2 = 0.1400 = 0.1500 = 0.1550 = 0.1600 = 0.1625 = 0.1650 = 0.1700 -1
2 = 0.1400 = 0.1500 = 0.1550 = 0.1600 = 0.1625 = 0.1650 -1
2 = 0.1550 = 0.1600 = 0.1625 = 0.1650 -1
2 = 0.1550 = 0.1600 = 0.1625 = 0.1650 = 0.1700 -1
2 = 0.1550 = 0.1600 = 0.1625 = 0.1650 = 0.1675 = 0.1700
FIG. 16. Ensemble averaged local density distributions forsmaller activity values showing the transition from the uni-form liquid state to coexistence of solid-regions with liquidregions. The liquid peaks shows large low-density fluctua-tions for the largest activity values. For the passive system∆ v = 0 . ρ > . v >
0, this coexistence region expands toreach lower densities. ξ forces the interfaces to overlap. The ensemble averageddensity distributions reflect this overlap through the de-velopment of a broad tail on the low density side of thesolid peak in the the regime of solid-active liquid coex-istence in finite systems. Since this phase disappears inthe thermodynamic limit, it cannot be characterized bythermodynamic measures such as binodals derived fromthe peaks of the density distributions. As seen in Fig.17, the shape of the distribution that interpolates be-tween the void and the solid peaks depends on the globaldensity and the system size (Fig. 18).5 -1
3 = 0.1000 = 0.1100 = 0.1200 = 0.1300 = 0.1400 = 0.1500 = 0.1550 = 0.1600 = 0.1650 = 0.1700 -1
3 = 0.1000 = 0.1100 = 0.1200 = 0.1300 = 0.1400 = 0.1500 = 0.1550 = 0.1600 = 0.1625 = 0.1650 -1
3 = 0.1100 = 0.1200 = 0.1300 = 0.1400 = 0.1500 = 0.1550 = 0.1600 = 0.1625 -1
3 = 0.1100 = 0.1200 = 0.1300 = 0.1400 = 0.1500 = 0.1550 = 0.1600 = 0.1625 = 0.1650 = 0.1700 -1
3 = 0.1200 = 0.1300 = 0.1400 = 0.1500 = 0.1550 = 0.1600 = 0.1625 = 0.1650 = 0.1700 -1
3 = 0.1200 = 0.1300 = 0.1400 = 0.1500 = 0.1550 = 0.1600 = 0.1625 = 0.1650 = 0.1700
FIG. 17. Ensemble average distributions of the local density(measured in square boxes with side length 45 lattice sites, forconfigurations at the simulation time t max = 2 × ) com-pared for different activities ∆ v and different overall globaldensities ρ . For large enough activity ∆ v ≥ .
2, there areonly two clear peaks, a peak for the solid phase with den-sity ρ solid ≈ .
19 and a peak for the empty void regions ρ void = 0. As the activity is reduced, a liquid-like peak near ρ ≈ . − .
15 appears. At small enough activity, the clearvoid peak disappears, and only large low density fluctuationsof the liquid peak remain, suggesting that void regions havebeen filled in by the growing activity liquid interface. Notethese distributions are ensemble averages over both arrestedand non-arrested states. FIG. 18. Finite-size dependence of the local density distri-butions for several different activities at ρ = 0 .
15 and linearsystem sizes L = 225 , , v = 0 .
03 becomes a bimodal activeliquid + solid coexistence for L = 600. Similarly, at largeractivity, for instance ∆ v = 0 .
07, increasing the system sizeopens up enough space for empty void regions to form, so thatthe active liquid + solid coexistence at L = 225 transformsinto an active liquid + solid + void state at L = 600. Theresults qualitatively support the predicted finite size depen-dence of the boundaries between the three non-equilibriumphase types, active liquid, solid-active liquid, and solid-activeliquid-void. FIG. 19. Snapshots illustrating the finite-size dependence ofthe final non-equilibrium state reached, for ρ = 0 .
15 and L =225 , , v = 0 .
03, increasing thesystem size transforms the final state from an active liquid inthe smaller systems with L = 225 , L = 600. For larger activity∆ v = 0 .
07, increasing the system size allows room for voidsto open up, so that the active liquid + solid state at L = 225clearly becomes an active liquid + solid + void state at L =600. The black scale bars indicate a length of 100 lattice sites. D R =
005 (e)-(h). Color bars denote the stationary times of each cross, log ( τ i ). Forthis large activity value ∆ v = 0 .
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