Downloaded by guest on October 1, 2021 lh .Lee A. Alpha systems in nonequilibrium generation and spectra Fluctuation www.pnas.org/cgi/doi/10.1073/pnas.1701739114 of bath a in with plates (21). well solid particles between Brownian compare force self-propelling that the prop- of force scaling simulations extract fluctuation-induced recent we the spectrum, of unimodal erties narrow, approxima- a simple of making By tions objects. function between the a separation as nonmonotonic, the repulsion prediction of is and attraction spectrum key between fluctuation oscillate A can force it. the if within the that, fluctuations by is energy energy mathematically of of given partition the system, dependence by active determined the is in objects the passive on on tem exerted force the a 14–20). refs. of compute (e.g., inclusions to embedded microscopic system the (13). active on system particular focus nonequilibrium a studies flock- in many as realizable Therefore, such is behavior swarming and nontrivial ing and the (7–12), from self-propulsion analogue different qualitatively passive that behavior show phase particles complex induces Brownian sim- active and free theories of example, the ulations For and ground. sta- tenuous function general more partition on and energy, the convenient underlying places concepts, energy tistical of out-of- input of continuous variety wide scales. length the that different in across framework systems generation physical equilibrum convenient force that a describe question is physical could there fundamental whether A is (6). sys- sea arises and such stormy flow a scale, turbulent swimmers on large a ships in to the objects On between 2) interactions (3–5). arise (1, feature boundary tems might membranes soft a active interactions by on confined such proteins which from range in examples Bioinspired F effect Casimir this signal. the In nonequilibrium becomes spectrum. important noise fluctuation the that the sense, is within work encoded are our interactions of implication Maritime particles. Brownian key the active A of examples: simulations recent disparate and effect apparently oscillates Casimir two and separation. wall spectrum, examine of the function fluctuation We and a as the width attraction in and the repulsion peak between on the depends nonequilibrium of walls a position embedded by two nar- exerted on a force for system the that, spectrum, find medium, We unimodal controls generation. row, active wavenumber, force of the of function phenomenology of a the as spectrum energy fluctuation of elusive. partitioning the remains the the that Nonetheless, active show materials. these We active of artificial phenomenology inclu- may in general the forces exploited such on and be function, bath also regulate active often boundaries the active or by sions on exerted proteins forces passive from The boundaries. nonequilibrium by bath: as confined swimmers active microscopic viewed to an 2017) membranes 31, appropriately in January review are immersed for (received inclusions systems 2017 10, July biological approved and Many MA, Cambridge, University, Harvard Weitz, A. 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John and , fluctuations | ciematter active b,c,d,e,f,1 hsatcei NSDrc Submission. 1 Direct PNAS a is article This interest. of conflict no declare authors The paper. the wrote and data, analyzed a of scales length certain (28). fluid into by isotropic homogeneous turbulence” preferentially “active the energy create In pumping they medium. microswimmers, homogenous of otherwise example an nonmono- of into example, modes energy (for injecting particular continuously nontrivial by spectrum a fluctuation generate tonic) have to processes nonequilibrium potential that the is here point white key different with The between modes. energy associated of equipartition or usually to suspension corresponding are frame- noise Brownian (27), a a for noise provide that Johnson–Nyquist Equilibrium as can such behavior. fluctuations, spectra nonequilibrium thermal such understanding that for of showing work in properties interested general are we the here processes, kinetic scopic fre- the of key system. function one physical particular a is a be spectrum of signature general, fluctuation in wavenumber—this can, and quency noise magnitude intrinsic The of this freedom. view of of statistical degrees macroscopic to microscopic due the over intrinsic are averaging of fluctuations that feature is dynamic systems general uses physical A dynamics) particles’ (26). of the due scattering in means noise forces of natural random spectrum (the to A spectrum fluctuation spectrum. an the monitoring hand, fluctuation wavenumber- other the energy the via is dependent On system the (25). characterize fluctuation to equilibrium novel way alternative of that may out further, relation emerge and, dissipation modes shown (22–24), fluctuation been probed has the directly it of be suspension hand, breakdown a one the the from suspen- that On equilibrium a microswimmers? thermal distinguish active at we of grains can pollen How of question: sion the with begin We Force Fluctuation-Induced and Spectrum Fluctuation uhrcnrbtos ... .. n ...dsge eerh efre research, performed research, designed J.S.W. and D.V., A.A.L., contributions: Author lhleghradeu [email protected]. or [email protected], owo orsodnemyb drse.Eal [email protected], Email: addressed. be may correspondence whom To a atce,adse ih nteodrdl fteMaritime the of riddle old the effect. on Casimir light Brown- shed active and agree of particles, results simulations ian the Our dynamics separation. in molecular wall peak and recent of with the repulsion function a of between as position oscillates attraction force the and the and spectrum, width walls that fluctuation embedded the find two on on we system depends nonequilibrium spectrum, spectrum. a unimodal by fluctuation exerted narrow, nonmonotonic active a a in to For possible a lead only particular, can is In systems, which spectra. energy, fluctuation of is energy nonequipartition systems their nonequilibrium in physics. in generation biological encoded force and that statistical show in systems We challenge nonequilibrium significant a in is generation force Understanding Significance lhuhteflcuto pcrmcnb eie rmmicro- from derived be can spectrum fluctuation the Although f odcIsiuefrTertclPyis oa nttt of Institute Royal Physics, Theoretical for Institute Nordic b ahmtclIsiue nvriyo xod xodO26GG, OX2 Oxford Oxford, of University Institute, Mathematical d eateto ahmtc,Yl nvriy New University, Yale Mathematics, of Department NSEryEdition Early PNAS | f6 of 1

APPLIED PHISICAL SCIENCES The relation between fluctuation spectra and disjoining force driven to a nonequilibrium steady state via wind–wave inter- may be examined by generalizing the classic calculation of actions. We treat the one-dimensional case in which the wind Casimir (29). We consider an effectively one-dimensional sys- blows in a direction perpendicular to the plates (a simple model tem of two infinite, parallel plates separated by a L and of ships on the sea), and hence waves traveling parallel to the immersed in a nonequilibrium medium. We assume that the fluc- plates are negligible. Observations (38) show that the spec- tuations are manifested as waves and impart a radiative stress. trum G(k) is nonmonotonic (Fig. 1A). While various fits have We define the fluctuation spectrum as been proposed (38, 39), these are untested at large and small dE(k) wavenumber. Instead, we compute the force in Eq. 6 numeri- G(k) ≡ , [1] cally, approximating the spectrum by a spline through the mea- dk sured data points of Pierson and Moskowitz (38), and truncating where E(k) is the energy density of modes with wavenumber k. for wavembers beyond their measured ranges. Fig. 1B shows that Hence the radiation force per unit plate , δF , due to waves the resulting force is nonmonotonic and oscillatory as a function k k + δk k = |k| with wavenumber between and (where is of L: The force can be repulsive (Ffluct > 0) as well as attractive the magnitude of the wavevector), and with angle of incidence (Ffluct < 0). Physically, the origin of the attractive force is akin to between θ and θ + δθ, is the Casimir force between metal plates—the presence of walls δθ restricts the modes allowed in the interior, so that the energy δF = G(k)δk cos2θ . [2] 2π density outside the walls is greater than that inside. This attrac- One factor of cosine in Eq. 2 is due to projecting the momen- tive “Maritime Casimir” force has been observed since antiquity tum in the horizontal direction, the other factor of cosine is due (see, e.g., ref. 6, and references therein) and experimentally mea- to momentum being spread over an area larger than the cross- sured in a wavetank (40). However, the nonmonotonicity of the sectional length of the wave, and the factor of 2π accounts for the spectrum gives rise to an oscillatory force–displacement curve. force per unit angle (see, e.g., ref. 30 for a more detailed deriva- In particular, the force is repulsive when one of the allowed dis- tion of Eq. 2). For isotropic fluctuations, we can consider δθ as crete modes is close to the wavenumber at which the peak of the an infinitesimal quantity, and, upon integrating from θ = −π/2 spectral density occurs (Fig. 1C): Here the sum overestimates to π/2, we arrive at the in Eq. 6, and the outward force is greater than the inward force. Thus, the local maxima in the repulsive force are 1 δF = G(k)δk. [3] approximately located at 4 π Outside of the plates, any wavenumber is permitted, and so Ln ≈ n , [7] kmax Z ∞ 1 0 Fout = G(k)dk. [4] where G (kmax) = 0; the separation between the force peaks 4 0 is ∆L ≈ π/kmax. In a maritime context, our calculation implies However, the waves traveling perpendicular to and between the that, if the separation between ships is L > π/kmax, the repulsive plates are restricted to take only integer multiples of ∆k = fluctuation force will keep the ships away from each other. π/L, because the waves are reflected by each plate. The force To our knowledge, this prediction of a repulsive Maritime imparted by the waves to the inner surface of the plates is then Casimir force has yet to be verified experimentally. Clearly quan- ∞ titative measurement of this oscillatory hydrodynamic fluctuation 1 X F = G(m∆k) ∆k, [5] force in an uncontrolled in situ ocean environment influenced by in 4 m=1 intermittency would be challenging, although the controlled lab- in one dimension. Thus, the net disjoining force for a one- oratory framework used in pilot-wave hydrodynamics is ideally dimensional system is given by suited for direct experimental tests (e.g., ref. 41). We note that an ∞ oscillatory force has been observed in the acoustic analogue for 1 X 1 Z ∞ which a nonmonotonic fluctuation spectrum was produced (42, Ffluct = Fin − Fout = G(m∆k) ∆k − G(k) dk. 4 4 0 43). Moreover, one-dimensional filaments in a flowing 2D soap m=1 film are observed to oscillate in phase or out of phase depend- [6] ing on their relative separation (44), suggesting an oscillatory Note that Ffluct ≶ 0 for all plate separations L if the deriva- fluctuation-induced force; visualization of this instability reveals 0 tive G (k) ≶ 0 for all k: If a nonmonotonic force is observed, it the presence of waves and coherent fluctuations as the mecha- necessarily implies a nonmonotonic spectrum. Furthermore, in nism for force generation, which is the basis of our approach. higher dimensions, the continuous modes need to be integrated We would expect that the fluctuation-induced force vanishes to compute the force between the plates. when the fluid is at thermal equilibrium. To test this, we note that Clearly, the fluctuation spectrum G(k) is the crucial quantity a consequence of the equipartition theorem is that the energy in our framework, and can, in principle, be calculated for differ- spectrum for a 3D isotropic fluid at equilibrium is monotonic, ent systems. We note that previous theoretical approaches have and has the scaling (45) mostly focused on the stress tensor (31). For example, the effect 2 of shaking protocols on force generation has been investigated Geq(k) ∝ k . [8] 2 theoretically for soft (32) and granular (33) media. More gen- Noting that, in 3D, δk = δkx δky δkz /(4πk ), Eq. 6 becomes erally, nonequilibrium Casimir forces have been computed for ∞ ! reaction–diffusion models with a broken fluctuation–dissipation 1 Z ∞ Z ∞ X Z ∞ relation (34, 35), and spatial concentration (36) or thermal (37) Ffluct = dky dkz ∆kx − dkx = 0, [9] 4π 0 0 0 gradients. Moving beyond specific models, however, we argue m=1 that there are important generic features of fluctuation-induced where we have used the fact that the Riemann sum and inte- forces that can be fruitfully derived by considering the fluctuation gral agree exactly for a constant function. Checking this special spectrum and treating it as a phenomenological quantity. case confirms that our approach can, in certain circumstances, distinguish between equilibrium and nonequilibrium: In the Maritime continuum hydrodynamic setting, a nonzero fluctuation-induced We first illustrate the central result, Eq. 6, by applying it to the force implies nonequilibrium. We will comment on the UV classical hydrodynamic example of ocean surface waves that are divergence [divergence in G(k) as k → ∞] in Eq. 8 and on

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max 2 max πν n stehalf-width the is n ekwidth peak and G ofidthat find to , = L III O π  r ie by given are ) 3 0 .Nt htthe that Note ). eea Phe- General . | ν k ((ν/ p − . ν < . k −2G max fluct max |

0 0 = f6 of 3 [10] [15] [14] [12] [13] ν, < /G [11] ) 2 ), 2 )

APPLIED PHISICAL SCIENCES q  2 2 to oscillate between attractive and repulsive and has the asymp- ∞ G (n∆k) + q totic decay 1 X Z ∞ Fin = ∆k dq. [17] 2 4 q 2 π G0 1 n=1 0 2π (n∆k) + q 2 F ≈ − , [16] min 3ν L2 However, we can redefine which is the minimum (or maximal attractive) force. The inverse square decay is shown in Fig. 1B, Inset. Z ∞ G(pq 2 + k 2) h(k) ≡ dq [18] These predictions are borne out by the numerical results for p 2 2 0 2π q + k the Maritime Casimir effect discussed earlier (Fig. 1B), but, more importantly, form a phenomenological theory that can be applied as an effective 1D spectrum and substitute h(k) for G(k) in Eq. to systems where the fluctuation spectrum is not known a priori: 6. Performing the same asymptotic analysis as for the narrow- if force measurements are found to illustrate these scalings, then peak limit, the asymptotic scalings [12] and [14] are reproduced, we suggest that the underlying spectrum is likely to be narrow in quantitative agreement with simulations. (We note that the lin- and unimodal. (The scalings derived here are specialized to the ear scaling shown in Fig. 2B also implies that the width of the case of interactions between plates, which is a reasonable approx- peak scales linearly in L, as predicted by Eq. 14.) The ∼1/L2 imation for the interaction between objects when their separa- decay expected for large L is not observed in these data, as the tion is much less than their radii of curvature.) asymptotic approximations underlying Eq. 12 only break down for We can now revisit the case of classical fluids at equilibrium. L & Lthres ≈ 12σ, with σν = 0.2 estimated from the data. This Obviously, the divergence in Eq. 8 as k → ∞ is unphysical. This agreement between the data and our asymptotic framework sug- UV divergence is cured by noting that hydrodynamic fluctu- gests that the underlying spectrum for active Brownian systems is ations, as captured by the spectrum G(k), are suppressed at narrow and nonmonotonic. (For smaller values of the active self- the molecular length scale k ∼ 2π/σ where σ is the molecular propulsion force f simulated in ref. 21, the peaks are less pro- diameter. Therefore, our analysis (Eq. 11) predicts an oscillatory nounced and are obscured by numerical noise.) The slight dis- fluctuation-induced force with a period that is comparable to the crepancy with the linear fit at large n is likely due to the fact that molecular diameter. This is indeed observed in confined equilib- our asymptotic scaling only holds in the regime L  π/ν (note rium fluids (49), although, clearly, at the molecular length scale, that π/ν ≈ 15σ in Fig. 2B, and the linear fit deteriorates when our hydrodynamic description breaks down and other physical L & 7σ, confirming that the value of ν estimated from fitting phenomena, such as proximity induded layering, become rele- to the width and height of the force peak is at least of the cor- vant. Importantly, while the wavelength of the disjoin- rect order of magnitude). An additional source of the discrep- ing force in equilibrium fluids can be nanoscopic, of the order of ancy may be that the signal-to-noise ratio decreases for increasing the molecular scale, the oscillation wavelength in active nonequi- plate separation as the magnitude of the force becomes smaller. librium systems can be much larger than the size of the active Further analytical insights can be obtained by considering the particle, because the mechanism of force generation lies in a non- limit of no excluded volume interaction between particles in trivial partition of energy. which Ni et al. (21) observed that the disjoining is attrac- tive and decays monotonically with separation (similar results Force Generation with Active Brownian Particles have been obtained by Ray et al. (15) for run-and-tumble active Interestingly, our asymptotic results are in agreement with force matter particles). This observation can be explained within our generation in what one might consider to be the unrelated con- framework by noting that the self-propulsion of point particles text of self-propelled active Brownian particles. Ni et al. (21) sim- induces a Gaussian colored noise ζ(t) satisfying (50) ulated self-propelled Brownian hard spheres confined between 2 f 0 hard walls of length W and found an oscillatory decay in the hζ(t)i = 0, ζ(t)ζ(t 0) = e−2Dr |t−t |, [19] disjoining force (Fig. 2A). Although this system is 2D, our anal- 3 ysis can be generalized: In two dimensions, δk = δkx δky /(2πk), where f is the active self-propulsion force and Dr is the and hence rotational diffusion coefficient. In the frequency domain, the

ABC

Fig. 2. Comparison of our theory with the simulations of a 2D suspension of self-propelled Brownian spheres, confined between hard slabs, that interact via the Weeks–Chandler–Anderson potential (21). In A and B, the packing fraction in the bulk is ρσ2 = 0.4, where σ is the particle diameter, the wall 2 length is W = 10σ, and self-propulsion force f = 40kBT/σ.(A) The raw force–displacement curve for ρσ = 0.4 from ref. 21. (B) When replotted as suggested by our asymptotic predictions [12] and [14], these data suggest that the underlying fluctuation spectrum is unimodal and has a narrow peak, with 3 parameters G0 ≈ 4.8 × 10 and ν ≈ 0.2/σ. (As the peaks are spaced approximately σ apart, we assume kmax = π/σ, and G0 and ν are obtained from fits of Eq. 15 to the simulation data.) The positions of the stable (closed circles) and unstable (open circles) mechanical equilibria (when Ffluct = 0) are given by Leq, and the dotted lines are theoretical predictions (Eq. 15). Inset shows the force maxima in A ∝ 1/L and agrees with Eq. 12.(C) For ideal noninteracting 2 self-propelled point particles, the function Aσ/L (black dotted line; see Eq. 21) can be fitted (using A) to simulation data with Fσ /(WkBT) = 40 (A = 182) 2 and Fσ /(WkBT) = 20 (A = 31.6). Here W = 80σ.

4 of 6 | www.pnas.org/cgi/doi/10.1073/pnas.1701739114 Lee et al. Downloaded by guest on October 1, 2021 Downloaded by guest on October 1, 2021 reo ntedrcinprle otepae a eintegrated, be can plates the to yielding parallel the direction and the integral in freedom the between Eq. difference sum, Riemann the monotonic, now is aiiigwienie suigalna iprinrelation- linear a ship, Assuming noise. white Eq. maximizing of spectrum noise Lorentzian The e tal. et Lee of tor decay activity the A doubling scaling: further, this with Fig. consistent Indeed, is (21). al. et Ni 2C by observed as relation, displacement large for is and function correlation spectrum fluctuation iyfil ftegaua eimi lodrcl hw obe to shown den- directly the also (where is medium 54) granular (33, the medium of granular field shaken sity a inclu- including in systems, soft-matter sions other in forces reported been fluctuation-induced have long-range Fur- and here. shown oscillatory reported been framework thermore, force separation; has oscillatory plate viscosity geometry the of supports function effective Couette this oscillatory plane an the be be a to that also in (53) numerically note fluid can active We spectrum, an systems. unimodal of those general to also a applied for framework—is Eqs. derived background our framework, the asymptotic 12–15, into our of Therefore, enters spectrum (52). nonmonotonic that energy the spectrum numer- fluid, that fluid—the background show the not results but ical species, active the of tions where of class wide a the take For spectra (51) (52). fluctuation form fluid the analytical systems, a as turbulent” in well “active swimmers such as 51), active (28, of particles models active of description hydrodynamic of scale diameter. spectrum, length peak the a the indeed, to presence in diameter—and, rise the particle gives particular, injection—the 2A interactions energy In volume Fig. spectrum). excluded in inferred of cause seen the the of decay be tonicity oscillatory vol- must fitting the excluded self-propulsion the active between of of and coupling finite, interactions for estimate the seen ume evidently the only is particles, alter decay active force to oscillatory which, sufficient Since noise, parameter.) is sampling 2 and the Fig. arations by in caused seen be as to discrepancy slight n irsoi oe.Ti pcrmi arwypae when peaked narrowly is spectrum α/β This model. microscopic ing .BesaS 19)Amrtm nlg fteCsmreffect. Casimir the of analogy matter. maritime active A of (1996) trapping SL Acoustic Boersma (2016) JF 6. Brady J, Vermant R, Dier displacement De and SC, Takatori Shape (2016) L 5. Angelani MC, Marchetti R, Leonardo colloidal Di active M, Paoluzzi in compartmentalization 4. and control Shape (2015) al. proteins. et active M, of Spellings fluctuations force 3. by driven undulations pro- Membrane (2004) transmembrane N of Gov Activity 2. (1999) J Prost D, Levy P, Bassereau JB, Manneville 1. omntnceeg pcr r lofudi h continuum the in found also are spectra energy Nonmonotonic a Commun Nat particles. active by filled vesicles soft in fluctuations cells. Lett Rev Phys membranes. lipid of fluctuations 82:4356–4359. shape of magnification induces teins hw httedsonn rsueotie rmsimulations from obtained pressure disjoining the that shows  ω eynal h rdce atrof factor predicted the nearly very 5.6, rcNt cdSiUSA Sci Acad Natl Proc E 1/β 0 ∝ , ec,w xett e oooi force– monotonic a see to expect we Hence, L. and α, i.e., , k 93:268104. 7:10694. ent ht eas h pcrmo Eq. of spectrum the because that, note we , α sepcal infiata ag lt sep- plate large at significant especially is C, β k S smntncand monotonic is 6,  max (ω r osat htdpn nteunderly- the on depend that constants are S G lhuhEq. Although 1. = ) (ω 112:E4642–E4650. sapoiaeyteivreparticle inverse the approximately is , (k F ) = ) fluct 4D steFuirtasomo h time the of transform Fourier the is f 3 E r ∝ f nrae h rfco yafac- a by prefactor the increases 0 2 k − α 4D e f L −β 2 r 2 , 20 o,tedge of degree the Now, ∼1/L. 1 + k 22 ∝ c Rep Sci 2 eitsfo entropy- from deviates ω , 1/L atrstefluctua- the captures 2 adtenonmono- the (and . W eiv the believe (We 4. 6:34146. mJPhys J Am sosre and, observed is 64:539–541. hsRvLett Rev Phys [20] [21] [22] 20 1 utnn ,e l 21)Dnmclcutrn n hs eaaini upnin of suspensions in separation phase and clustering Dynamical (2013) al. Continuum et I, (2013) Buttinoni ME 11. Cates D, Marenduzzo RJ, Allen A, Tiribocchi J, Stenhammar 10. Ntoa cec onainadteOfieo aa eerhudrOCE- under Institution Research Oceanographic Naval Dynam- of Hole Office Fluid Woods the 1332750). the Geophysical and Foundation at 2015 Science Program the (National Society Study Royal and Summer a Award, ics 638-2013-9243, European Merit Grant the Research sup- Council by acknowledges Wolfson Research J.S.W. and Swedish D.V.). A.A.L.) (to from (to 637334 port Grant Schol- Fellowship Starting Fulbright Carrier Council F. Research Studentship, George Research and confined Council arship, Research in Sciences Physical experience they ACKNOWLEDGMENTS. forces the control geometries. that is actively designed (be speculation could to) natural settings (engineering) of a biological spectra in Indeed, swimmers fluctuation material. the intervening controlling controlling the actively walls, by bounding envisage forces the can of tuning (65)] one than properties rather [e.g., that, nature Casimir- the is suggested possibility long-range analysis intriguing to our an rise by Hence, 64). give (63, to behavior observed like a thermally such been a in have example, fluctuations system for temperature using, state; system steady pure can nonequilibrium fluid” a “active in an have constructed of we be form appear systems another the would Additionally, in here. correlations it studied time the (62), examine equilibrium to prudent time requires because generally, longer More particles. significantly reversal active can the wavelengths of systems size of with the out-of-equilibrium than out corollary, be force a exhibit must As scale monotonic). spectrum, length fluctuation is thermal molecular the wavelength the (because oscillation equilibrium case than force the a is larger it with much 49), system oscillations ref. hydrodynamic (e.g., force scale a that length that molecular so the scale, at molecular seen are the at a vanishes on fluids exerts charge theory. oscillating our test (active) also an may charge, that neighboring force the mea- as effect, settings, such Casimir electromagnetic biomimetic nonequilibrium and the of biological surements by var- motivated the this is Although and article (59–61). (26), interactions Casimir spectra of (56– measurements fluctuation calculations ied the microscopic of systems between measurements phe- nonequilibrium bridge 58), active the the and providing into driven by insight Brownian both crucial active of affords confined nomenology framework by Our generated forces particles. inter- the wind–water and by driven actions, is which systems: effect, consid- physical Casimir by Maritime nonequilibrium the approach disparate our seemingly two verified and ering and pressure spectrum, disjoining fluctuation the the between we relationship view, the top-down this computed adopting By drives energy. force of species for nonequipartition active a principle spectrum—the nonequilib- organizing fluctuation prepare the an to is that generation ways suggest of We plethora systems. a rium course, of are, There our of bed test Conclusion a as serve will systems of formalism. active measurements in on numerical forces particles or Casimir Experimental active (55). rotating monolayer and a our framework) with modes agreeing fluctuating qualitatively oscillatory, and inhomogeneous .Rde S aa F akrnA(03 tutr n yaiso phase- a of motile dynamics and in Structure (2013) balance A detailed Baskaran MF, without Hagan GS, transport Redner 9. Diffusive (2012) M Cates with particles 8. self-propelled of separation phase Athermal (2012) MC Marchetti Y, Fily 7. npriua,wieteflcuto pcrmo equilibrium of spectrum fluctuation the while particular, In efpoeldclodlparticles. colloidal particles. self-propelled Brownian active for kinetics separation 111:145702. phase of theory physics? fluid. colloidal active statistical separating need microbiology 042601. Does bacteria: alignment. no hsRvLett Rev Phys hswr a upre ya niern and Engineering an by supported was work This 108:235702. hsRvLett Rev Phys hsRvLett Rev Phys 110:055701. 110:238301. NSEryEdition Early PNAS e rgPhys Prog Rep hsRvLett Rev Phys G | ∼ f6 of 5 k 75: 2 ,

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