Motility-induced temperature difference in coexisting phases
MMotility-induced temperature difference in coexisting phases
Suvendu Mandal, ∗ Benno Liebchen,
1, 2, † and Hartmut L¨owen ‡ Institut f¨ur Theoretische Physik II: Weiche Materie,Heinrich-Heine-Universit¨at D¨usseldorf, D-40225 D¨usseldorf, Germany Theorie Weicher Materie, Fachbereich Physik, Technische Universit¨atDarmstadt, Hochschulstraße 12, 64289 Darmstadt, Germany (Dated: May 22, 2019)Unlike in thermodynamic equilibrium where coexisting phases always have the same tempera-ture, here we show that systems comprising “active” self-propelled particles can self-organize intotwo coexisting phases at different kinetic temperatures, which are separated from each other bya sharp and persistent temperature gradient. Contrasting previous studies which have focused onoverdamped descriptions of active particles, we show that a “hot-cold-coexistence” occurs if andonly if accounting for inertia, which is significant in a broad range of systems such as activateddusty plasmas, microflyers, whirling fruits or beetles at interfaces. Our results exemplify a routeto use active particles to create a self-sustained temperature gradient across coexisting phases, aphenomenon, which is fundamentally beyond equilibrium physics.
Introduction.–
In equilibrium systems, entropy maxi-mization (or free energy minimization) requires thermal,mechanical and chemical equilibrium among coexistingphases. Conversely, in nonequilibrium no fundamentallaw forbids different temperatures in coexisting phases,evoking the question if a specific mechanism exists whichcan generate such a difference. Such a mechanism mayappear counterintuitive, as heat-gradients, unless theyare sustained by a localized heat-source such as a starperforming nuclear fusion, usually cause processes oppos-ing them and driving the system towards thermal equi-librium (unless for ideal isolation): For example, a tem-perature difference in the air evokes a balancing wind,and air friction cools down a radiator once switched off.Here we report and systematically explore a surpris-ingly different scenario, where particles self-organize intocoexisting phases sustaining different temperatures. Thistwo temperature coexistence occurs spontaneously in auniform system and remarkably, there is no heat flux atsteady state, because the gradient in kinetic temperatureis balanced by a self-sustained, opposite density gradient.A “hot” and a “cold” phase are allowed to coexist in prin-ciple, as the system we consider comprises self-propelledmicroparticles which allow the system to bypass equilib-rium thermodynamics.By now, we know that such microparticles, often de-scribed as “active Brownian particles” [1–5], can self-organize into a liquid phase, coexisting with a gas-phase,even when interacting purely repulsively [6–17]. Coinedas “motility-induced phase separation”, or MIPS, thisphenomenon has advanced to a key paradigm in thephysics of self-propelled particles. When the micropar-ticles are overdamped, like microorganisms in a sol-vent [18] or active colloidal microswimmers [19–23], theyare equally fast in both phases. Hence, despite the pres-ence of active microparticles, liquid and gas as emergingfrom MIPS have identical kinetic temperatures, just likefor liquid-gas phase separation in equilibrium. (Note that MIPS involves a slow-down of particles in regions of highdensity [2, 6]; which occurs however only for the ’coarsegrained self-propulsion’, not for the actual velocity de-termining the kinetic temperature, as further discussedbelow.)When releasing the overdamped standard approxi-mation, as relevant e.g. for beetles at interfaces [24],whirling fruits [25] microflyers [26] or activated dustyplasmas [27], both the phase diagram and the proper-ties of the contained phases change dramatically, as weshow in this Letter. In particular, while MIPS gener-ally requires a sufficiently large self-propulsion speed v to occur, specifically for underdamped particles it breaksdown again if v is too large, i.e. MIPS is reentrant inthe presence of inertia [28]. This is because MIPS alsorequires particles to slow-down (regarding their directedmotion) in regions of high density [2]: such a slow-downoccurs instantaneously upon collisions of overdampedparticles, but in the presence of inertia, particles bounceback from collisions and do not slow down much beforeexperiencing subsequent collisions. Thus, at very large v , underdamped particles can exchange their kinetic en-ergies before slowing down much and MIPS breaks down.To see which physical mechanism controls the kinetictemperature difference (to be distinguished from the ef-fective temperature [29–32]) in coexisting phases, con-sider the collision of an active underdamped particle mov-ing with a fixed orientation towards an elastically reflect-ing wall. This problem is equivalent to a bouncing ballexperiencing friction and gravity (see Supplemental Ma-terial for details): while reaching a terminal speed ( v )when falling in free space, the ball continuously slowsdown, when reflected by a wall, even when the collisionsare elastic. Analogously, particles essentially move with v in the gas phase, where they rarely collide, but slowdown when entering the dense liquid phase, due to suc-cessive collisions with other particles (see Fig. 1). Noticethat inelastic collisions among the particles provide an al- a r X i v : . [ c ond - m a t . s o f t ] M a y persistence time τ p = 1 /D r mean time between collisions τ c = πσ/ (4 v ϕ )inertial time τ d = m/γ t TABLE I. Relevant time scales in active underdamped parti-cles. ternative, but mechanistically unrelated, route to achievea remarkable hot-cold coexistence, which has been dis-cussed for vibrated granular particles, where particlesdissipate energy due to inelastic collisions [33–35]. Incontrast, for the microparticles we consider, no inelasticcollisions are required: the emergence of coexisting tem-peratures is based on the interplay of activity and weakinertia.Our results exemplify a generic route to use activeparticles to create a self-sustained temperature gradientacross coexisting phases, a phenomenon, which is fun-damentally beyond equilibrium physics. This contraststhe overdamped standard case, which has been predomi-nantly explored in active matter physics so far and leadsto a dynamics which can be essentially mapped onto anequilibrium system at a coarse grained level [2, 6] yieldinga phase transition which is consistent with an equilibriumliquid-gas transition [36]. Thus, the existence of temper-ature differences in coexisting phases indicates a changeof the nature of MIPS, when releasing the overdampedstandard approximation: it changes from a liquid-gas liketransition to a new type of phase transition having nocounterpart in equilibrium. Accordingly, part of phe-nomenology of MIPS [6, 37–40], a key result in activematter physics, is even broader than anticipated previ-ously - but was curtained by the overdamped standardapproximation in previous studies.
Model.–
To demonstrate our results in detail, let usnow consider a generic model for active underdampedparticles in 2D, each having an internal drive, representedby an effective self-propulsion force F SP ,i = γ t v u ( θ i )where u ( θ i ) = (cos θ i , sin θ i ) is the direction of self-propulsion. The particles have identical diameters σ ,masses m and moments of inertia I . They interact viaan excluded-volume repulsive force F i (see SupplementalMaterial). Their velocities v i and orientations θ i evolveas m d v i d t = − γ t v i + F i + F SP ,i + (cid:112) k B T b γ t η i ( t ) ,I d θ i d t = − γ r d θ i d t + (cid:112) k B T b γ r ξ i ( t ) , (1)where η i , ξ i represent Gaussian white noise of zero-mean unit variance, T b is the effective bath temperatureand γ t , γ r are translational and rotational drag coeffi-cients, yielding diffusion coefficients D t,r = k B T b /γ t,r . gas (hot) dense (cold) FIG. 1. Scheme of the phase-separated state associated witha hot-cold coexistence in underdamped active particles. Par-ticles self-propel with the colored cap ahead (brown; greenishfor the tagged particle). Active particles move with ∼ v inthe gas phase, but can be an order of magnitude slower in thedense phase. To understand the behavior of active underdamped par-ticles, it is instructive to define three characteristic timescales (see table I): the persistence time τ p = 1 /D r , af-ter which the directed motion of active particles is ran-domized by rotational diffusion, the mean time betweencollisions for a given particle τ c = πσ/ (4 v ϕ ), where ϕ = N πσ / (4 L x L y ) is the area fraction, and the inertialtime scale τ d = m/γ t , characterizing the time a particleat rest needs to reach its terminal speed. (In principle,the moment of inertia I leads to an additional timescale( I/γ r ), but it turns out to be largely irrelevant to ourresults and is thus kept constant to I = 0 . (cid:15)τ p (seeSupplementary Material).Fixing the area fraction to a regime where MIPS canoccur ( ϕ = 0 . M = τ d /τ p , whichis a reduced mass measuring the impact of inertia, andthe P´eclet number Pe = v / ( D r σ ) ∝ τ p / ( τ c ϕ ) (seeSupplemental Material), measuring the strength of self-propulsion by comparing ballistic to a diffusive motion. Nonequilibrium phase diagram.–
To explore the im-pact of inertia on the collective behavior of active parti-cles, we first explore the phase-diagram using large-scalesimulations based on LAMMPS [41]. If M →
0, iner-tia plays no role and the particles are essentially over-damped. Accordingly, for M (cid:46) − , we recover theusual behavior: at fixed area fraction ϕ = 0 .
5, the par-ticles undergo MIPS [10, 12] when the P´eclet number islarge enough (Pe (cid:38) . ≤ M ≤ . FIG. 2. Nonequilibrium phase diagram at ϕ = 0 . L x × L y = 350 σ × σ ) at state points indicated in the phase diagram. Each simulation has been performed in a box of size L x × L y = 850 σ × σ , comprising N ≈ particles. Colors represent kinetic energies of individual particles in units of k B T .A hot-cold coexistence is visible in panel (e). Dashed lines in (c) show scaling predictions for the phase boundary between thehomogeneous and phase-separated state. (Fig. 2(d)). Thus, MIPS is reentrant for underdampedactive particles. Finally, when inertia is even stronger M (cid:38) .
08, MIPS does not occur at all. Overall, this leadsto the phase diagram shown in Fig. 2(c). The qualita-tive structure of this phase diagram can be understoodbased on simple scaling arguments. To see this, let usfirst remember how MIPS arises for overdamped parti-cles: consider a particle self-propelling towards a smalldense cluster of particles; when colliding, the particlestops and is blocked by the cluster, until rotational dif-fusion turns its self-propulsion direction away from thecluster on a timescale τ p = 1 /D r . When the time inbetween collisions τ c is smaller than τ p , the rate of par-ticles entering the cluster exceeds the leaving-rate andthe cluster rapidly grows [7, 9], later proceeding slowlytowards phase separation. This criterion explains the ex-istence of a (lower) critical P´eclet number. Since both τ c , τ p are mass-independent, we expect the lower criticalPe number also to be mass-independent: τ p (cid:38) τ c ⇒ Pe = const . (2)as approximately observed in Fig. 2(c). To understandthe upper critical Pe number, note that MIPS requiresa localized slow-down of particles to occur. Thus, atvery high collision rates (due to high Pe), underdampedparticles bounce back multiple times on the inertial timescale τ d , and can therefore not slow down locally. We,therefore, expect that MIPS occurs only if τ c (cid:38) τ d ⇒ Pe ∝ /m, (3)which yields the scaling law Pe ∼ /m shown as theupper dashed line in Fig. 2(c) and corresponds to oursimulation results. Temperature difference in coexisting phases.–
Let usnow explore the properties of the resulting liquid and the coexisting gas, in parameter regimes where MIPS takesplace. While in the overdamped case ( M → T eff ( x ) = m (cid:104) v ( x ) (cid:105) , which is thekinetic energy per particle, averaged along the lateralcoordinate. As shown in Fig. 3(a), T eff is uniform for M = 10 − , but develops a massively nonuniform shapewhen increasing M to 0 .
05 (see Supplementary MoviesS1 and S2, respectively). Fig. 3(c) quantifies the result-ing temperature difference, showing ( T gas − T dense ) /T dense as a function of M . Here, we see that the temperaturein the dilute phase can be almost two orders of magni-tude larger than in the dense phase. (Note that Fig. 3(c)shows that the temperature difference has a maximumat some M value before MIPS disappears, and then de-creases again; this is probably a consequence of the fact,that the collision rate in the gas phase increases in thecorresponding parameter domain, which cools the gas, aswe will see below.) This is further reflected by the veloc-ity distribution P ( v x ) in Fig. 3(b), showing a far-broaderdistribution for the gas phase than for the dense one, butonly if inertia is significant (see inset). Power-balance.–
To understand the temperature dif-ference quantitatively, we now derive a power-balanceequation. Multiplying the translational part of Eq. (1)by v , and averaging over all particles in a given phase,we obtain FIG. 3. (a) Spatial profiles of the effective temperature T eff ( x ) + 2 . ϕ ( x ) (dashed lines)for different reduced masses M . (b) Steady-state distributions of particle velocities v x for moderate inertia M = 5 × − .Solid lines are fits to the Maxwell-Boltzmann distribution P ( v x ) = (cid:112) m/ (2 πT eff ) exp[ − mv x / (2 T eff )], where T eff is the kinetictemperature. Inset: P ( v x ) for vanishing inertia M = 10 − . (c) The relative temperature and area fraction difference betweenthe two phases as a function of inertia. Other parameters: Pe = 100, ϕ = 0 . m d (cid:104) v ( t ) (cid:105) d t = − γ t (cid:104) v ( t ) (cid:105) + (cid:104) v ( t ) · F ( t ) (cid:105) + (cid:104) v ( t ) · F SP ( t ) (cid:105) + (cid:112) k B T b γ t (cid:104) v ( t ) · η ( t ) (cid:105) . (4)Here, the left hand side equals the time derivative of theeffective temperature ∂T eff /∂t ; γ t (cid:104) v ( t ) (cid:105) = 2 T eff /τ d de-scribes the energy dissipation rate due to Stokes dragand (cid:104) v ( t ) · F ( t ) (cid:105) represents the dissipated power dueto interactions among the particles, which is negligi-ble here since particle collisions are elastic, see Supple-mentary Fig. S4. The third-term (cid:104) v ( t ) · F SP ( t ) (cid:105) rep-resents the self-propulsion power. The last-term is re-lated to the bath temperature by the following relation √ k B T b γ t (cid:104) v ( t ) · η ( t ) (cid:105) = 2 k B T b γ t /m = 2 k B T b /τ d , whichis identical in the gas and in the dense phase. Plug-ging these expressions into Eq. (4), and and using that ∂T eff /∂t = 0 in each phase individually in steady state,we obtain T gas − T dense = τ d (cid:2) (cid:104) v · F SP (cid:105) gas − (cid:104) v · F SP (cid:105) dense (cid:3) . (5)Therefore, if and only if τ d (cid:54) = 0, self-propulsion can cre-ate a temperature difference in coexisting phases. Since τ d = 0, in overdamped particles, both phases have thesame kinetic temperature. In contrast, for underdampedparticles we have τ d (cid:54) = 0. The contributions of the indi-vidual terms to the power balance is visualized in Sup-plementary Fig. S4, revealing that the self-propulsionpower is much higher in the gas phase than in the densephase and dominates the kinetic temperature (ratherthan diffusion as for overdamped particles). To see, whythe self-propulsion power is different in the gas phasecompared to the dense phase, we explore the distribu-tion of the particle effective speeds v eff = v · u in bothphases; here (cid:104) v · F SP (cid:105) = γ t v (cid:104) v eff (cid:105) . Thus, Figure 4 showsthat the average effective speed in the gas phase is v ,whereas negative speed values are rare, showing thatparticles in the gas phase rarely move against their self-propulsion direction (Fig. 1, left panel). This suggests FIG. 4. Probability distributions of effective speeds in thegas phase as well as in the dense phase. Other parameters:Pe = 100, ϕ = 0 . that (cid:104) v · F SP (cid:105) gas ∼ γ t v . In contrast, in the dense phase,the effective particle speed is almost symmetrically dis-tributed around 0, which results from the fact that par-ticles have no space to move and bounce back after eachcollision; thus, they move against their self-propulsion di-rection about half of the time (Fig. 1, right panel), whichimplies (cid:104) v · F SP (cid:105) dense ∼ Conclusion.–
Unlike equilibrium systems, self-drivenactive particles can self-organize into a liquid and a co-existing gas phase at different temperatures. This resultexemplifies a route to use self-driven particles to create aself-sustained temperature gradient, which might serve,in principle, as a novel paradigm to create isolating lay-ers at the microscale, e.g. to keep bodies at differenttemperatures.On a more fundamental level, our results show thatmotility-induced phase separation, one of the best ex-plored phenomenon in active matter research, is funda-mentally different from a liquid-gas phase separation –an insight which has been curtained by the focus onoverdamped particles so far. As a consequence, the phe-nomenology of motility-induced phase separation is evenricher than anticipated previously - it can, in particular,lead to phenomena at the macroscale which are funda-mentally beyond equilibrium physics.For future studies, it would also be interesting to studythe effect of inertia on anisotropic active particles [42–45] where translational and rotational motions are cou-pled. Specifically for such particles, ref. [46] has recentlyobserved (but hardly analyzed) the occurrence of differ-ent kinetic energies in coexisting phases, suggesting thatthe present findings survive for particles of nonsphericalshape.An interesting challenge would also be to derive a mi-croscopic theory for motility-induced phase separation inunderdamped particles to predict the joint temperatureand density profiles across the interface between the twocoexisting states [47]. Such an approach needs to bedesigned for non-isothermal situations as considered re-cently in Enskog kinetic theories [48, 49] or in dynamicaldensity functional theory [50, 51].We thank Christian Scholz and Alexei Ivlev for fruit-ful discussions. This work is supported by the GermanResearch Foundation (Grant No. LO 418/23-1) ∗ [email protected] † [email protected] ‡ [email protected][1] P. Romanczuk, M. B¨ar, W. Ebeling, B. Lindner, andL. Schimansky-Geier, “Active brownian particles,” Eur.Phys. J. Spec. Top. , 1 (2012).[2] M. E. Cates and J. Tailleur, “Motility-induced phaseseparation,” Annu. Rev. Condens. Matter Phys. , 219(2015).[3] R. Ni, M. A. C. Stuart, and M. Dijkstra, “Pushing theglass transition towards random close packing using self-propelled hard spheres,” Nat. Commun. , 2704 (2013).[4] C. Kurzthaler, C. Devailly, J. Arlt, T. Franosch, W. C. K.Poon, V. A. Martinez, and A. T. Brown, “Probingthe spatiotemporal dynamics of catalytic janus particleswith single-particle tracking and differential dynamic mi-croscopy,” Phys. Rev. Lett. , 078001 (2018).[5] R. G. Winkler, A. Wysocki, and G. Gompper, “Virialpressure in systems of spherical active brownian parti-cles,” Soft Matter , 6680–6691 (2015).[6] J. Tailleur and M. E. Cates, “Statistical mechanics ofinteracting run-and-tumble bacteria,” Phys. Rev. Lett. , 218103 (2008).[7] Y. Fily and M. C. Marchetti, “Athermal phase separationof self-propelled particles with no alignment,” Phys. Rev.Lett. , 235702 (2012).[8] A. Patch, D. M. Sussman, D. Yllanes, and M. C.Marchetti, “Curvature-dependent tension and tangentialflows at the interface of motility-induced phases,” SoftMatter , 7435–7445 (2018).[9] I. Buttinoni, J. Bialk´e, F. K¨ummel, H. L¨owen,C. Bechinger, and T. Speck, “Dynamical clustering andphase separation in suspensions of self-propelled colloidalparticles,” Phys. Rev. Lett. , 238301 (2013).[10] J. Stenhammar, A. Tiribocchi, R. J. Allen, D. Maren-duzzo, and M. E. Cates, “Continuum theory of phaseseparation kinetics for active brownian particles,” Phys.Rev. Lett. , 145702 (2013). [11] J. Stenhammar, D. Marenduzzo, R. J. Allen, and M. E.Cates, “Phase behaviour of active brownian particles:the role of dimensionality,” Soft Matter , 1489–1499(2014).[12] G. S. Redner, M. F. Hagan, and A. Baskaran, “Struc-ture and dynamics of a phase-separating active colloidalfluid,” Phys. Rev. Lett. , 055701 (2013).[13] P. Digregorio, D. Levis, A. Suma, L. F. Cugliandolo,G. Gonnella, and I. Pagonabarraga, “Full phase dia-gram of active brownian disks: From melting to motility-induced phase separation,” Phys. Rev. Lett. , 098003(2018).[14] Z. Mokhtari, T. Aspelmeier, and A. Zippelius, “Collec-tive rotations of active particles interacting with obsta-cles,” Europhys. Lett. , 14001 (2017).[15] A. P. Solon, J. Stenhammar, M. E. Cates, Y. Kafri, andJ. Tailleur, “Generalized thermodynamics of motility-induced phase separation: phase equilibria, laplace pres-sure, and change of ensembles,” New J. Phys. , 075001(2018).[16] D. Levis, J. Codina, and I. Pagonabarraga, “Activebrownian equation of state: metastability and phase co-existence,” Soft Matter , 8113–8119 (2017).[17] J. T. Siebert, F. Dittrich, F. Schmid, K. Binder,T. Speck, and P. Virnau, “Critical behavior of activebrownian particles,” Phys. Rev. E , 030601(R) (2018).[18] J. Elgeti, R. G. Winkler, and G. Gompper, “Physicsof microswimmers—single particle motion and collectivebehavior: a review,” Rep. Prog. Phys. , 056601 (2015).[19] C. Bechinger, R. Di Leonardo, H. L¨owen, C. Reichhardt,G. Volpe, and G. Volpe, “Active particles in complexand crowded environments,” Rev. Mod. Phys. , 045006(2016).[20] H.-W. Huang, F. E. Uslu, P. Katsamba, E. Lauga,M. Sakar, and B. Nelson, “Adaptive locomotion of arti-ficial microswimmers,” Sci. Adv. , eaau1532 (2019).[21] A. Aubret, S. Ramananarivo, and J. Palacci, “Eppur simuove, and yet it moves: Patchy (phoretic) swimmers,”Curr. Opin. Colloid Interface Sci. , 81–89 (2017).[22] A. Aubret and J. Palacci, “Diffusiophoretic design of self-spinning microgears from colloidal microswimmers,” SoftMatter , 9577–9588 (2018).[23] C. Maggi, M. Paoluzzi, L. Angelani, and R. Di Leonardo,“Memory-less response and violation of the fluctuation-dissipation theorem in colloids suspended in an activebath,” Scientific reports , 17588 (2017).[24] H. Mukundarajan, T. C. Bardon, D. H. Kim, andM. Prakash, “Surface tension dominates insect flight onfluid interfaces,” J. Exp. Biol. , 752–766 (2016).[25] J. Rabault, R. A. Fauli, and A. Carlson, “Curving to fly:Synthetic adaptation unveils optimal flight performanceof whirling fruits,” Phys. Rev. Lett. , 024501 (2019).[26] C. Scholz, S. Jahanshahi, A. Ldov, and H. L¨owen, “In-ertial delay of self-propelled particles,” Nat. Commun. ,5156 (2018).[27] G. E. Morfill and A. V. Ivlev, “Complex plasmas: An in-terdisciplinary research field,” Rev. Mod. Phys. , 1353–1404 (2009).[28] A. Suma, G. Gonnella, D. Marenduzzo, and E. Or-landini, “Motility-induced phase separation in an activedumbbell fluid,” EPL , 56004 (2014).[29] L. F. Cugliandolo, “The effective temperature,” J. Phys.A , 483001 (2011).[30] C. Nardini, E. Fodor, E. Tjhung, F. van Wijland, J. Tailleur, and M. E. Cates, “Entropy production infield theories without time-reversal symmetry: Quanti-fying the non-equilibrium character of active matter,”Phys. Rev. X , 021007 (2017).[31] D. Levis and L. Berthier, “From single-particle to col-lective effective temperatures in an active fluid of self-propelled particles,” EPL , 60006 (2015).[32] Z. Preisler and M. Dijkstra, “Configurational entropy andeffective temperature in systems of active brownian par-ticles,” Soft Matter , 6043–6048 (2016).[33] Y. Komatsu and H. Tanaka, “Roles of energy dissipationin a liquid-solid transition of out-of-equilibrium systems,”Phys. Rev. X , 031025 (2015).[34] K. Roeller, J. P. D. Clewett, R. M. Bowley, S. Herming-haus, and M. R. Swift, “Liquid-gas phase separation inconfined vibrated dry granular matter,” Phys. Rev. Lett. , 048002 (2011).[35] T. Schindler and S. C. Kapfer, “Nonequilibrium steadystates, coexistence, and criticality in driven quasi-two-dimensional granular matter,” Phys. Rev. E , 022902(2019).[36] D. Levis, J. Codina, and I. Pagonabarraga, “Activebrownian equation of state: metastability and phase co-existence,” Soft Matter , 8113 (2017).[37] P. Krinninger, M. Schmidt, and J. M. Brader, “Nonequi-librium phase behavior from minimization of free powerdissipation,” Phys. Rev. Lett. , 208003 (2016).[38] A. P. Solon, J. Stenhammar, M. E. Cates, Y. Kafri, andJ. Tailleur, “Generalized thermodynamics of phase equi-libria in scalar active matter,” Phys. Rev. E , 020602(2018).[39] T. Speck, J. Bialk´e, A. M. Menzel, and H. L¨owen, “Ef-fective cahn-hilliard equation for the phase separation ofactive brownian particles,” Phys. Rev. Lett. , 218304(2014).[40] A. Wysocki, R. G. Winkler, and G. Gompper, “Co-operative motion of active brownian spheres in three-dimensional dense suspensions,” EPL , 48004 (2014).[41] S. Plimpton, “Fast parallel algorithms for short-rangemolecular dynamics,” J. Comput. Phys. , 1–19(1995).[42] N. H. P. Nguyen, D. Klotsa, M. Engel, and S. C. Glotzer,“Emergent collective phenomena in a mixture of hard shapes through active rotation,” Phys. Rev. Lett. ,075701 (2014).[43] S. Farhadi, S. Machaca, J. Aird, B. O. T. Maldonado,S. Davis, P. E. Arratia, and D. J. Durian, “Dynamicsand thermodynamics of air-driven active spinners,” SoftMatter , 5588–5594 (2018).[44] G. Kokot, S. Das, R. G. Winkler, G. Gompper, I. S.Aranson, and A. Snezhko, “Active turbulence in a gasof self-assembled spinners,” Proc. Natl. Acad. Sci. ,12870–12875 (2017).[45] A. Aubret, M. Youssef, S. Sacanna, and J. Palacci, “Tar-geted assembly and synchronization of self-spinning mi-crogears,” Nat. Phys , 1114 (2018).[46] I. Petrelli, P. Digregorio, L. F. Cugliandolo, G. Gonnella,and A. Suma, “Active dumbbells: Dynamics and mor-phology in the coexisting region,” Eur. Phys. J. E ,128 (2018).[47] K. Miyazaki, Y. Nakayama, and H. Matsuyama, “En-tropy anomaly and linear irreversible thermodynamics,”Phys. Rev. E , 022101 (2018).[48] J. J. Brey, V. Buz´on, P. Maynar, and M. I. Garc´ıa deSoria, “Hydrodynamics for a model of a confined quasi-two-dimensional granular gas,” Phys. Rev. E , 052201(2015).[49] V. Garz´o, R. Brito, and R. Soto, “Enskog kinetic theoryfor a model of a confined quasi-two-dimensional granularfluid,” Phys. Rev. E , 052904 (2018).[50] R. Wittkowski, H. L¨owen, and H. R. Brand, “Extendeddynamical density functional theory for colloidal mix-tures with temperature gradients,” J. Chem. Phys. ,224904 (2012).[51] J. G. Anero, P. Espa˜nol, and P. Tarazona, “Functionalthermo-dynamics: A generalization of dynamic densityfunctional theory to non-isothermal situations,” J. Chem.Phys. , 034106 (2013).[52] J. D. Weeks, D. Chandler, and H. C. Andersen, “Role ofrepulsive forces in determining the equilibrium structureof simple liquids,” J. Chem. Phys. , 5237–5247 (1971).[53] Q.-L. Lei, M. P. Ciamarra, and R. Ni, “Nonequilibriumstrongly hyperuniform fluids of circle active particles withlarge local density fluctuations,” Sci. Adv. , eaau7423(2019).[54] C. Rycroft, “Voro++: A three-dimensional voronoi celllibrary in c++,” Chaos , 041111 (2009). SUPPLEMENTARY MATERIALSIMULATIONS
Simulations are performed with a slightly modified version of LAMMPS [41], which integrates the equations ofmotion given in Eq. (1) using the Euler method. The conservative force on particle i from particle j is F i = −∇ i u ( r ij ),which results from a purely repulsive WCA potential [52]: u ( r ij ) = (cid:40) (cid:15) (cid:2) ( σr ij ) − ( σr ij ) (cid:3) + (cid:15), r ij /σ ≤ / , r ij /σ> / where (cid:15) = k B T is the interaction strength, and r ij is the distance between particles i and j . The equations of motionare integrated with a time step δt = 10 − τ p . Recent experiments [26] on microflyers reveal that diffusion coefficients( D r and D t ) and friction coefficients ( γ r and γ t ) are not related by the Stokes-Einstein relation. Thus, for simplicity,we choose γ t = γ r /σ as shown, e.g., in Ref. [53].In order to clarify the importance of the moment of inertia I , we have performed simulations with two differentmoments of inertia I = 0 . (cid:15)τ p (Fig. 2(c) in the main text) and I = 0 . (cid:15)τ p (Supplementary Fig. 5). These twofigures display qualitatively similar results, which implies that we are close to overdamped rotational dynamics, where I = 0. MIPS No MIPSNo MIPS
FIG. 5. Nonequilibrium phase diagram same as Fig. 2(c), but now for I = 0 . (cid:15)τ p . BOUNCING BALL PICTURE
To develop an intuition for the emergence of temperature differences let us exploit a simple formal analogy: thedynamics of an active particle with fixed orientation, which is elastically reflected by a fixed obstacle, is identical tothe dynamics of an elastically bouncing ball under the influence of gravity (representing self-propulsion) and Stokesdrag (Fig. 6(a)). To characterize the bouncing dynamics, we show the vertical position y ( t ) as a function of time t in Fig. 6. For vanishing drag, γ t = 0, energy is conserved and the ball bounces periodically without slowing down(Fig. 6(b)). However, when experiencing drag, the ball, initially at rest, accelerates due to gravity to a velocity whichcannot exceed v before hitting the fixed obstacle (Fig. 6(c) and inset). The ball bounces back elastically, preservingits speed upon the collision, but now ascends against the gravitational force to a turning point below the startingposition. From here, the ball accelerates towards the obstacle again, but has less space to accelerate this time. Thus,each time the ball hits the obstacle, it is slower. The same slow-down mechanism applies to a particle entering thedense phase and encountering a series of collisions, each time bouncing back, against its self-propulsion direction, and FIG. 6. (a) A ball with fixed orientation bounces back elastically from a fixed obstacle (blue particle). Typical trajectories(and velocities in insets) of a bouncing ball for (b) vanishing drag γ t = 0 , (c) finite drag γ t (cid:54) = 0 (underdamped), and (d) infinitedrag γ t = ∞ (overdamped). (e) A typical trajectory of a ball when it encounters an inelastic collision rather than drag. having less space to accelerate. This is in stark contrast to the behavior in the gas phase, where collisions are rareand particles have enough time to reach their terminal speed v in between collisions. Thus, the active gas is much’hotter’ than the active liquid. The behavior of an overdamped bouncing ball is yet different (Fig. 6(d) and inset):this ball reaches its terminal speed instantaneously; when hitting the obstacle, it does not bounce back, and does notmove any further, apart from translational diffusion. Here, while directed motion immediately stops when hitting theobstacle, the actual velocity of the particle hardly changes: This is because the instantaneous speed of overdampedparticles is dominated by the diffusive micromotion, not by self-propulsion. Consequently, overdamped particles areequally fast in the gas and in the liquid, yielding identical temperatures in both phases – as in equilibrium. Finally, tocontrast the present slow-down mechanism, crucially based on self-propulsion, from the scenario in vibrated granularparticles, let us emphasize that the latter corresponds to a ball experiencing inelastic collisions, i.e. to a case wherekinetic energy is drained from the system upon a collision. NONEQUILIBRIUM PHASE DIAGRAM
To construct the phase diagram, an elongated box with periodic boundary conditions is used. Simulations wererun up to 10 τ p in order to reach the steady state. To characterize the phase-separation, we measure the distribution P ( ϕ loc ) of the local free-area ϕ loc of active underdamped particles using the Voronoi tessellation method [54] (seeSupplementary Fig. 7). Once the free-area distribution is bimodal, we identify it as a phase-separated state. AREA FRACTION DIFFERENCE IN COEXISTING PHASES
In the phase-separated state, we can measure the local area fractions in the two different phases by dividing thesimulation box into slabs of width (cid:39) . σ . We find that the area fraction profiles (dashed lines) in Fig. 3(a) in themain text are similar to ABPs and can be fitted to a hyperbolic tangent function ϕ ( x ) = 12 ( ϕ dense + ϕ gas ) −
12 ( ϕ dense − ϕ gas ) tanh (cid:104) x − x ) w (cid:105) , (6)where x and w are the location and width of the gas-liquid interface. We extract the corresponding area fractions ofthe gas phase ϕ gas and the dense phase ϕ dense by fitting each side of the interface using Eq. (6). In Fig. 3(c) (in themain text), we plot the relative area fraction difference ( ϕ dense − ϕ gas ) /ϕ gas in coexisting phases by varying inertiawhile keeping the P´eclet number fixed at Pe = 100. Notably we find that the area fraction difference between the twophases is 10 times higher than the gas phase and interfacial width w (cid:39) σ . As we move from phase-separated to ahomogeneous state with increasing inertia M at fixed Pe, the relative area fraction decreases monotonically towards acritical inertia M ≈ .
08. This behavior is similar to the first-order-phase transition, but occurs in a non-equilibriumsetup. Most importantly, the control parameter is inertia M instead of the thermodynamic temperature. FIG. 7. Local free-area distributions for various P´eclet numbers at fixed inertia M = 5 × − and global area fraction ϕ = 0 . LEGENDS TO MOVIES
In all movies ϕ = 0 . L x × L y = 850 σ × σ , N = 10 , Pe = 100, while reduced mass M and reduced temperature T ∗ = T eff /k B T are provided for each movie. The simulation time t is measured in units of τ p .1. Movie S1:
Underdamped active particles with M = 10 − . Here, coexisting phases have the same temperatureas shown by the colors, just like in equilibrium physics.02. Movie S2:
Underdamped active particles with M = 0 ..