PHYSICAL REVIEW X 9, 041026 (2019)...dynamical bias. The explicit form of the bias is inspired by the results of Sec. II connecting dissipation to many-body interactions. Using analytical

How Dissipation Constrains Fluctuations in Nonequilibrium Liquids: Diffusion,Structure, and Biased Interactions

Laura Tociu,1,2 Étienne Fodor,3 Takahiro Nemoto,4 and Suriyanarayanan Vaikuntanathan1,21James Franck Institute, University of Chicago, Chicago, Illinois 60637, USA

2Department of Chemistry, University of Chicago, Chicago, Illinois 60637, USA3DAMTP, Centre for Mathematical Sciences, University of Cambridge, Wilberforce Road,

Cambridge CB3 0WA, United Kingdom4Philippe Meyer Institute for Theoretical Physics, Physics Department, École Normale Supérieure and

PSL Research University, 24, rue Lhomond, 75231 Paris Cedex 05, France

(Received 15 October 2018; revised manuscript received 15 September 2019; published 5 November 2019)

The dynamics and structure of nonequilibrium liquids, driven by nonconservative forces which can beeither external or internal, generically hold the signature of the net dissipation of energy in the thermostat.Yet, disentangling precisely how dissipation changes collective effects remains challenging in many-bodysystems due to the complex interplay between driving and particle interactions. First, we combine explicitcoarse-graining and stochastic calculus to obtain simple relations between diffusion, density correlations,and dissipation in nonequilibrium liquids. Based on these results, we consider large-deviation biasedensembles where trajectories mimic the effect of an external drive. The choice of the biasing function isinformed by the connection between dissipation and structure derived in the first part. Using analytical andcomputational techniques, we show that biasing trajectories effectively renormalizes interactions in acontrolled manner, thus providing intuition on how driving forces can lead to spatial organization andcollective dynamics. Altogether, our results show how tuning dissipation provides a route to alter thestructure and dynamics of liquids and soft materials.

DOI: 10.1103/PhysRevX.9.041026 Subject Areas: Chemical Physics, Soft Matter,Statistical Physics

I. INTRODUCTION

Nonequilibrium forces can drive novel and specificpathways to modulate phase transitions and self-assemblyin materials. The close connection between the net dis-sipation of energy, powered by these forces, internaltransport, and spatial organization is especially apparentin living systems [1–4]. As an example, the flagella motorsof Escherichia coli exhibit a unique phenomenologycombining ultrasensitive response, adaptation, and motorrestructuring as a function of the applied load [5–7].Moreover, in vivo studies of the cellular cytoskeleton, aswell as in vitro experiments on reconstituted systems, alsoshow that motor-induced forces control a large variety offunctionality in the cell [8–12].To elucidate the role of nonequilibrium forces in materi-

als, it is crucial to examine how dissipation affects theemerging dynamics and structure. While equilibrium

features arewell established, progress in controlling systemswith sustained dissipation is hampered by a lack of generalprinciples [13–19]. In this context, minimal models of activeand driven systems provide analytically and numericallytractable test beds to investigate the interplay betweendissipation and material properties far from equilibrium[20–24]. They illustrate, for instance, how nonequilibriumdriving can induce phase transitions and excite novelcollective responses in soft media [15,21,25–27]. Recenttheoretical work proposes extending equilibrium conceptsto active media, such as the definition of pressure [14,28], torationalize their phenomenology [29,30]. Others strive toobtain stationary properties of active matter through per-turbation close to equilibrium [16,31,32], inspired by otherapproaches on driven systems [33–36].To investigate how dissipation controls emerging behav-

ior, yet another approach focuses on introducing a bias indynamical ensembles. Using large-deviation techniques,trajectories are conditioned to promote atypical realizationsof the dynamics [37,38]. Such techniques are used, forinstance, to investigate the role of dynamical hetero-geneities in glassy systems [39–45] and soliton solutionsin high-dimensional chaotic chains [46,47]. More recently,it has been shown that changing dissipation, through adynamical bias, strongly affects the internal transport and

Published by the American Physical Society under the terms ofthe Creative Commons Attribution 4.0 International license.Further distribution of this work must maintain attribution tothe author(s) and the published article’s title, journal citation,and DOI.

PHYSICAL REVIEW X 9, 041026 (2019)

2160-3308=19=9(4)=041026(21) 041026-1 Published by the American Physical Society

https://crossmark.crossref.org/dialog/?doi=10.1103/PhysRevX.9.041026&domain=pdf&date_stamp=2019-11-05https://doi.org/10.1103/PhysRevX.9.041026https://doi.org/10.1103/PhysRevX.9.041026https://doi.org/10.1103/PhysRevX.9.041026https://doi.org/10.1103/PhysRevX.9.041026https://creativecommons.org/licenses/by/4.0/https://creativecommons.org/licenses/by/4.0/

the density fluctuations of nonequilibrium liquids [48,49],thus confirming that controlling dissipation is indeed afruitful route to tailoring material properties. In spite ofthese advances, anticipating the emergent dynamics andstructure of biased nonequilibrium systems is still chal-lenging in the presence of many-body interactions [38,50],so that precise control has remained elusive so far in thiscontext. Consequently, any generic principle rationalizingspatial organization in terms of dissipation is still lacking.In this paper, we explore how dissipation affects the

dynamics and structure of many-body diffusive systems.First, we consider in Sec. II two types of assemblies ofBrownian particles: one in which only a subset is driven byan external force and one in which a subset of the particlesexperience an internal active force. We first focus oninstances where the fraction of driven particles is less thanthe fraction of undriven particles, so that driven andundriven particles are, respectively, referred to as tracerand bath particles. Using the diffusion coefficient of atagged tracer particle and the density correlations betweentracer and bath particles, we connect dissipation to liquidproperties. In contrast to Ref. [23], our prediction fordiffusion follows from a systematic coarse graining withexplicit dependence in terms of microscopic details[51–53].Next, importantly, we put forward a generic relation

between density correlations and dissipation valid for anarbitrary driving force: This relation is our first main result.We demonstrate that this result holds both for fluids inwhich a fraction of the particles are driven by a fixedexternal drive and for fluids in which either a fraction of theliquid or the entire liquid is driven by an internal noise,analogous to the driving used in model active mattersystems. This result opens the door to estimating dissipa-tion directly from the liquid structure, in contrast toprevious approaches based either on perturbing the system[54–58] or on analyzing trajectories and currents in phasespace [3,59–62]. We illustrate this result with numericalsimulations for which dissipation is quantified by thedeviation from equilibrium tracer-bath correlations.Using these results as a basis, we also show how variousaspects of the pair correlation function of a nonequilibriumliquid are effectively constrained by the energy dissipation.Altogether, this set of results clarifies how nonequilibriumforces affect the transport and structure of the liquid, thusshowing how liquid properties can be modified at the costof energy dissipation.Motivated by these results, and to provide concrete

intuition for how particular configurations can be stabilizedby nonequilibrium forces, we next investigate in Sec. III theemerging structure of Brownian particles subject to adynamical bias. The explicit form of the bias is inspiredby the results of Sec. II connecting dissipation to many-body interactions. Using analytical calculations andnumerical simulations based on the cloning algorithm

[46,63–68], we show that biased sampling trajectoriescan be used to renormalize any specific interparticleinteraction in a multicomponent liquid. The rare noisefluctuations sampled with dynamical bias effectively drivethe system away from typical behavior [38–45,50]. Suchnoise realizations can then serve as proxies of how tocontrol the dynamics by applying an external force withcomplex protocols. We also illustrate the generality of ourideas by considering an assembly of aligning self-propelledparticles [69]. Specifically, we show how biased energyflows can renormalize interactions between particles andstabilize a flocking transition. Overall, our results lay thegroundwork for precise control of the emerging structureand collective dynamics in many-body diffusive nonequi-librium systems.

II. DISSIPATION AND LIQUID PROPERTIES

In this section, we provide a series of relations betweenenergy dissipation and liquid properties in nonequilibriumliquids. Specifically, we consider interacting Brownianparticles where a specific set of particles Ω is driven bya nonconservative force Fd;i:

γ_ri ¼ δi∈ΩFd;i −∇iXj

vðri − rjÞ þ ξi; ð1Þ

where δi∈Ω ¼ 1 if i ∈ Ω and δi∈Ω ¼ 0 otherwise. Thedriven particles belonging to the set Ω are referred to astracers and others as bath particles. The fluctuating term ξiis a zero-mean Gaussian white noise with correlationshξiαðtÞξjβð0Þi ¼ 2γTδijδαβδðtÞ, where γ and T, respec-tively, denote the damping coefficient and the bath temper-ature, with the Boltzmann constant set to unity (kB ¼ 1).

A. Deterministic vs active drive

In what follows, we consider two types of drive: (i) anexternal force following the same deterministic protocol forall driven particles and (ii) an internal force given by a noiseterm independent for each driven particle. Building onrecent work [21,23], we take for drive (i) a time-periodicprotocol given in two dimensions by

FdðtÞ ¼ f½sinðωtÞêx þ cosðωtÞêy�; ð2Þ

where f and ω are, respectively, the amplitude and thefrequency of the drive, so that the drive persistence readsτ ¼ 2π=ω. The relative strength of the drive is given by thePéclet number Pe ¼ σf=T, where σ is the typical particlesize [21,23]. In the absence of interactions (v ¼ 0), theaverage position of driven tracers follows a periodic orbit,describing a circle in two dimensions. In contrast, drive(ii) corresponds to a random self-propulsion as is oftenconsidered in active liquids [70–72]. Specifically, we use aset of zero-mean Gaussian noises with correlations

TOCIU, FODOR, NEMOTO, and VAIKUNTANATHAN PHYS. REV. X 9, 041026 (2019)

041026-2

hFd;iαðtÞFd;jβð0Þi ¼ δijδαβf2

de−jtj=τ; ð3Þ

where d is the spatial dimension. The parameters f and τ,respectively, control the amplitude and the persistence offluctuations.Interestingly, the active force with correlations (3) can be

obtained from a generalized version of the deterministicforce (2) where each particle i is now subjected to anindependent drive. The period of the orbit is determined bya series of n oscillators with identical frequencies for allparticles yet independent amplitudes:

Fd;iðtÞ ¼fffiffiffiffiffiffind

pXna¼1

½Aai cosðωatÞ þ Bai sinðωatÞ�: ð4Þ

The essential ingredient of the mapping into active force isto implement disorder in the drive, which is done by takingthe oscillator amplitudes as uncorrelated zero-meanGaussian variables with unit variance:

hAaiαAbjβid ¼ δabδijδαβ ¼ hBaiαBbjβid; ð5Þ

where h·id denotes an average over the disorder. It followsthat Fd;i is also a Gaussian process with zero mean andcorrelations given by

hFd;iαðtÞFd;jβð0Þid ¼ δijδαβf2

nd

Xna¼1

cosðωatÞ: ð6Þ

In the limit of a large number of oscillators (n ≫ 1), weexpress these correlations in terms of the density of drivingfrequencies ϕ as

hFd;iαðtÞFd;jβð0Þid ¼n≫1

δijδαβf2

d

Zϕðω0Þeiω0jtj dω

0

2π: ð7Þ

This expression establishes that, in the limit of manyoscillators, the deterministic drive (4) with disorderedamplitude is equivalent to a noise term with spectrum ϕ.In particular, by choosing ϕðω0Þ ¼ 2τ=½1þ ðω0τÞ2�, thedrive correlations (7) reproduce exactly the ones of therandom force in Eq. (3).To illustrate the relevance of this mapping, we simulate

numerically the many-body dynamics (1) where everyparticle is subjected to a disordered drive of the form(4). We use the potential vðrÞ ¼ v0ð1 − jrj=σÞ2Θðσ − jrjÞ,where Θ denotes the Heaviside step function, which setspurely repulsive interactions. To implement numerically thedisorder in driving, it is sufficient to sample the amplitudesfAai;Baig and frequencies fωig at the initial time. In theregime of high persistence τ and large average density ρ0,we observe the spontaneous formation of clusters up to acomplete formation at a large time; see Fig. 1. Thisformation is analogous to the motility-induced phase

separation commonly reported in standard models of activeparticles [13,26]. Interestingly, it appears in our case evenin the absence of fluctuations (T ¼ 0), namely, for a purelydeterministic set of equations.In short, we thus demonstrate that the disordered drive

alone reproduces the emerging physics of active systems.This important result bridges the gap between two mainclasses of nonequilibrium liquids, where the driving forcestems from either a deterministic protocol or a randomnoise. In what follows, we obtain analytic and numericalresults for both drives to illustrate the broad applicability ofour framework, ranging from systems driven by determin-istic fields to active matter systems.

B. Dissipation controls tracer diffusion

To connect tracer diffusion with dissipation, we firstdescribe the dynamics of undriven particles in terms of acoarse-grained variable. Using standard techniques, thedynamics of the density field ρðr; tÞ ¼ Pi∉Ω δ½r − riðtÞ�can be written as a nonlinear Langevin equation [51]. In theregime of weak interactions, the density fluctuationsδρðr; tÞ ¼ ρðr; tÞ − ρ0 around the average density ρ0 areGaussian and captured by the following Hamiltonian[53,74,75]:

H ¼ T2

ZδρðrÞKðr − r0Þδρðr0Þdrdr0

þZ X

i∈Ωvðr − riÞρðrÞdr; ð8Þ

where KðrÞ ¼ δðrÞ=ρ0 þ vðrÞ=T. Note that density fluc-tuations remain generally Gaussian even for a homo-geneous active liquid [70]. The conserved densitydynamics reads

FIG. 1. Snapshot of particles subjected to a disordered drive. Aphase separation emerges which is analogous to the motility-induced phase separation of active particles [13,26]. Simulationdetails are in the Appendix A and the movie in Ref. [73].

HOW DISSIPATION CONSTRAINS FLUCTUATIONS IN … PHYS. REV. X 9, 041026 (2019)

041026-3

∂δρðr; tÞ∂t ¼ DG∇

2

ZKðr − r0Þδρðr0; tÞdr0

þ 1γG

∇2Xi∈Ω

v½riðtÞ − rjðtÞ� þ∇ · Λðr; tÞ; ð9Þ

where DG ¼ ρ0T=γ and γG ¼ γ=ρ0 are, respectively, thefield diffusion coefficient and the field damping coefficient.The term Λ is a zero-mean Gaussian white noise withcorrelations hΛαðr;tÞΛβðr0;t0Þi¼2DGδαβδðr−r0Þδðt−t0Þ.Owing to the linearity of the density dynamics (9),

it can be readily written in Fourier space δρðq; tÞ ¼Rρðr; tÞe−iq·rdr as

∂δρðq; tÞ∂t ¼ −jqj

2DGKðqÞδρðq; tÞ

− jqj2 vðqÞγG

Xj∈Ω

e−iq·rjðtÞ þ iq · Λðq; tÞ; ð10Þ

so that the field dynamics can be directly solved as

δρðq; tÞ ¼Z

t

−∞dse−DGjqj2KðqÞðt−sÞ

×

�iq · Λðq; sÞ − jqj2 vðqÞ

γG

Xj∈Ω

e−iq·rjðsÞ�: ð11Þ

Considering the limit of dilute driven tracers, whereinteractions among them are negligible, their dynamicsreads

γ _rj ¼ Fd þ ξj −Zqiqvð−qÞeiq·rjðtÞδρðq; tÞ; ð12Þ

withRq ¼

Rdq=ð2πÞd. As a result, Eqs. (11) and (12)

provide closed time-evolution equations for tracers only. Itshould be valid only for weak interactions a priori, yetprevious works show that it remains qualitatively relevanteven beyond this regime in practice [76–78]. Indeed,Gaussian field theories for density fluctuations provide avery good description of simple liquids [74].To characterize the transport properties of the liquid in

the presence of driving forces, our first goal is to obtain anexplicit expression, in terms of microscopic details, for thetracer diffusion coefficient:

D ¼ limt→∞

1

2dth½hriðtÞi − riðtÞ�2i: ð13Þ

We aim to explore connections between D and dissipation,which is defined from stochastic thermodynamics as thepower of the forces exerted by all tracers on solvent:J ¼ Pih_ri · ðγ _ri − ξiÞi, where · denotes a Stratonovichproduct [79,80]. Dissipation is directly related to entropyproduction, as a measure of irreversibility, both when the

drive is deterministic [79,80] and when it is a correlatednoise [81–84]. Substituting the dynamics (1), the dissipa-tion coincides with the power of driving forces:J ¼ Pi∈Ωh_ri · Fd;ii. Besides, replacing _ri by its expres-sion in Eq. (1) and using the fact that ξi and Fd;i areuncorrelated, we deduce that the dissipation can be furtherseparated into free-motion and interaction contributions asJ ¼ Nf2=γ − _w, where the rate of work reads

_w ¼ 1γ

Xi∈Ω;j

hFd;i ·∇ivðri − rjÞi: ð14Þ

Given that _w is the only nontrivial contribution to dis-sipation, connecting diffusion and dissipation simplyamounts to expressing D in terms of _w.Deriving transport coefficients in nonequilibrium many-

body systems, whose collective effects result from thecomplex interplay between interaction and driving forces,is a notoriously difficult task [85–89]. We set up a properperturbation scheme by scaling the pair potential v with asmall dimensionless parameter h ≪ 1 which controls thecoupling between tracer and bath. In Appendix B, weobtain some explicit expressions for D and _w to quadraticorder in h and in the scaled driving amplitude Pe.First, we discuss the case of the deterministic drive (2),

and we focus on the limits of small and large drivingfrequency, respectively, ωτr ≪ 1 and ωτr ≫ 1, where therelaxation timescale τr ¼ ðDG=σ2ÞKðjqj ¼ 1=σÞ is set bydensity diffusion over the tracer size σ. First, at highfrequencies ωτr ≫ 1, the rate of work per particle _w=N andthe deviation from equilibrium diffusion D −Deq, whereDeq is the diffusion coefficient for Pe ¼ 0, are given,respectively, by

_wN

¼�hPeω

�2

·ðT=σÞ2dγ3

Zqjqj4jvðqÞj2 1þ ρ0KðqÞ

KðqÞ ;

D −Deq ¼�hPeω

�2

·T=σ2

dγ3

Zq

jqj2jvðqÞj2KðqÞ½1þ ρ0KðqÞ�

: ð15Þ

In the opposite limit of low frequencies ωτr ≪ 1, we get

_wN

¼ ðhPeÞ2

dγσ2

Zq

jvðqÞj2KðqÞ½1þ ρ0KðqÞ�

;

D −Deq ¼5ðhPeÞ2dγTσ2

Zq

jvðqÞj2jqj2KðqÞ½1þ ρ0KðqÞ�3

: ð16Þ

Both _w=N andD −Deq are now independent of the drivingfrequency ω. As a result, our perturbation theory shows thatthe scalings of _w andD −Deq are identical, in terms both ofthe drive amplitude Pe and of its frequency ω, in asymptoticfrequency regimes. Note that the scaled rate of workγ _w=ðNf2Þ coincides with the reduced equilibrium diffusion


041026-4

γDeq=T − 1 to this order [52,53], as expected from linearresponse.The case of the active drive with correlations (3) follows

by using the mapping between the disordered drive andactive forcing in Sec. II A. In practice, we first derive thediffusion coefficient D and the rate of work _w for thedriving force (4) at fixed disorder, as a straightforwardgeneralization of the deterministic driving case, and wethen average over the disorder. At small persistence τ ≪ τr,we get

_wN

¼ τTðhPeÞ2

dðσγÞ2Zq

jqj2jvðqÞj2KðqÞ ;

D −Deq ¼3τðhPeÞ2dðσγÞ2

Zq

jvðqÞj2KðqÞ½1þ ρ0KðqÞ�2

: ð17Þ

In contrast, the large persistence limit τ ≫ τr yields thesame results as for the low-frequencies regime of deter-ministic drive, namely, the expressions (16). Indeed, theforce Fd;i has a constant direction in such a limit, and thedifference between deterministic and active drives, which,respectively, correspond to independent or similar direc-tions for each tracer, is irrelevant in the limit of dilutetracers.When the size a of the bath particles is significantly

smaller than the tracer size σ ≫ a, which amounts to settingdifferent pair potentials v for bath-bath and for bath-tracerinteractions, one can safely neglect the variation of KðqÞ inEqs. (15)–(17), so thatKðqÞ ≃ Kðjqj ¼ 1=aÞ. Then, in bothregimes ωτr ≫ 1 (τ ≪ τr) and ωτr ≪ 1 (τ ≫ τr), therenormalization of the diffusion coefficient D −Deq canbe simply written in terms of the rate of work per particle_w=N for Pe ≪ 1 as

D −Deqσ2

∼_w

NT: ð18Þ

Thus, the excess rate at which tracers move over their ownsize compared to equilibrium, set by the lhs of Eq. (18), iscontrolled by the rate at which work is applied on tracers bynonequilibrium forces, set by the rhs of Eq. (18). Theproportionality factor depends on the details of interactionsand of density fluctuations. Interestingly, this result is validfor both deterministic and active drives. It corroboratesnumerical observations obtained previously in a systemwhere composition-dependent diffusion constants can leadto phase transitions [23].

C. Dissipation sets density correlations

We now explore how dissipation relates to static densitycorrelations of the liquid. To this end, we treat undrivenbath particles without any approximation in what follows,instead of relying on the Gaussian density field theory forδρ as in Sec. II B, and we consider an arbitrary set of

driving forces Fd;i. In equilibrium, the liquid structure canbe derived from a hierarchy of equations for densitycorrelations, whose explicit form reflects the steady-statecondition on the many-body distribution function [90]. Inour settings, steady-state conditions should now providemodified equations for density correlations, which canpotentially make apparent the connection with dissipation.This observation motivates us to consider the average

rate at which the potential U ¼ Pi∈Ω;j vðri − rjÞ changes,which can be written using Itô calculus as

γh _Ui ¼Xi∈Ω;j

h½γð_ri − _rjÞ þ 2T∇i� ·∇ivðri − rjÞi: ð19Þ

Substituting the dynamics (1) and using hξi · ∇ivi ¼ 0within the Itô convention, we get

γh _Ui ¼Xi∈Ω;j

ð1þ δj∈ΩÞhFd;i ·∇ivðri − rjÞi

þX

i∈Ω;j;kh½∇ivðri − rjÞ� ·∇k½vðri − rkÞ− vðrj − rkÞ�i

þXi∈Ω;j

2Th∇2i vðri − rjÞi: ð20Þ

In the first line of Eq. (20), we recognize the rate of work _was defined in Eq. (14) and the term γ _wact ¼

Pfi;jg∈ΩhFd;i ·

∇ivðri − rjÞi which quantifies the contribution of inter-actions among driven particles to dissipation. The lattervanishes exactly when the drive is identical for all particles,since

Pfi;jg∈Ω∇ivðri − rjÞ ¼ 0 by symmetry, and it can be

neglected for an active drive when the fraction of drivenparticles is small. Then, using the steady-state conditionh _Ui ¼ 0, we deduce

_wþ _wact ¼2ρ0γ

ZgðrÞf½∇vðrÞ�2 − T∇2vðrÞgdr

þ ρ20

γ

ZZ½g3aðr; r0Þ þ g3bðr; r0Þ�

× ½∇vðrÞ� · ½∇vðr0Þ�drdr0; ð21Þwhere

gðrÞ¼ 1N

X0i∈Ω;j

hδðr−riþrjÞi;

g3aðr;r0Þ¼1

N2X0i∈Ω;j;k

hδðr−riþrjÞδðr0−riþrkÞi;

g3bðr;r0Þ¼1

N2X0i∈Ω;j;k

hδðr−riþrjÞδðr0−rjþrkÞi; ð22Þ

andP0

denotes a sum without the overlap of indices: i ≠ j,k ≠ i, and k ≠ j. The power balance (21), valid for an


041026-5

arbitrary driving, either deterministic or active, is our firstmain result. Importantly, it holds for generic interactionsand for any number of driven particles, namely, not only inthe limit of dilute tracers, in contrast with the results inSec. II B.In practice, it reflects how density correlations adapt to

the presence of nonequilibrium forces. For a vanishing rateof work ( _w ¼ 0 ¼ _wact), one recovers the first order of theequilibrium Yvon-Born-Green (YBG) hierarchy, in itsintegral form, for two-component fluids [90]. At a finiterate of work ( _w ≠ 0), the relation between the two-bodycorrelation g and the three-body terms fg3a; g3bg is nowimplicitly constrained by dissipation. A direct implicationis that the rate of work can now be inferred simply bymeasuring static density correlations, provided that thepairwise interaction potential is known, for a given drivenliquid. Importantly, such an approach does not require anyinvasive methods based on comparing fluctuations andresponse [54–58], and it does not rely on a detailed analysisof particle trajectories [60,61] or currents in phase space[59,62], whose experimental implementation can requireelaborate techniques [3,4].However, the power balance (21) is not straightforward

to test, either numerically or experimentally, due to thethree-body correlations. In equilibrium, where tracer andbath particles are indistinguishable, we get g3a ¼ g3b.Assuming that this result remains approximately valid inthe driven case for a small fraction of tracers, the rate ofwork can simply be written in terms of the force exerted ona tracer Fi ¼ −

Pj ∇ivðri − rjÞ as

_w ≃2

γ

Xi∈Ω

½hF2i i þ Th∇i · Fii�: ð23Þ

To probe the validity of this result, we simulate thedynamics (1) where 10% of particles are subject to thedriving force, considering either the deterministic periodicdrive (2) or the active noise drive (3). Interactions areset by the Weeks-Chandler-Andersen potential vðrÞ ¼4v0½ðσ=jrjÞ12 − ðσ=jrjÞ6�Θð21=6σ − jrjÞ [91]. Our measure-ments in Fig. 2 show that Eq. (23) is indeed a goodapproximation at small Pe and small τ, namely, when thedrive only weakly perturbs the liquid. The discrepancy ishigher for the active case compared with the deterministicone, since _wact ¼ 0 in the latter without any approximation.In contrast to previous approaches [3,54,92], which rely onprospecting the whole system, our results demonstrate thatthe rate of work can actually be evaluated with only a smallerror by considering solely forces acting on tracer: Thecontribution of forces on other particles is negligible for asmall fraction of driven tracers.To evaluate further the change in liquid structure induced

by dissipation, we measure the deviation from equilibriumpair correlations g − geq due to the driving forces (leftcolumn in Fig. 3). In particular, inspired by the two-body

contribution in the power balance (21), we focus on theobservable I ¼ ½ð∇vÞ2 − T∇2v�ðg − geqÞ. At a given τ,scaling I by Pe2 reveals that all curves almost collapse intoa master curve for our numerical range Pe ∈ ½12; 36�, asreported in the middle column in Fig. 3. In practice,particles overlap more for a stronger drive, so that g departsfrom zero at a smaller interparticle distance. To correct forthis aspect, we introduce a shift of the curves IðjrjÞ asjrj → jrj þ a, where aðPeÞ is a fitting parameter. Given thatthe rate of work also scales like Pe2, it suggests theexistence of an underlying relation between

RIðrÞdr

and _w. In practice, a linear fitting provides a satisfactory

(a)

(b)

FIG. 2. Parametric plot of the rate of work _w=T and of thestatistics of bath-tracer forces

Pi∈Ω½hF2i i þ Th∇i · Fii�=ðγTÞ

when 10% of particles are driven by either (a) a deterministicforce or (b) an active force. The solid line with slope 2 refers tothe approximate relation (23). The satisfying agreement withnumerical data indicates that the rate of work can be estimated byonly measuring bath-tracer forces. The simulations are performedwith N ¼ 4500 particles using the procedure described inAppendix A. Parameters: Pe ¼ 12 (hexagons), 18 (square),24 (triangle), 30 (circle), and 36 (diamond); (a) τT=ðγσ2Þ ¼2 × 10−1 (black), 3 × 10−1 (brown), 4 × 10−1 (red), and 5 × 10−1

(orange); (b) τT=ðγσ2Þ ¼ 2 × 10−2 (black), 3 × 10−2 (brown),4 × 10−2 (red), and 5 × 10−2 (orange).


041026-6

(a) (b) (c)

(d) (e) (f)

(g) (h) (i)

(j) (k) (l)

FIG. 3. Connecting dissipation and structure for a liquid where 10% of particles are driven by either deterministic or active forces.(Left) Bath-tracer density correlation g as a function of interparticle distance r=σ. The blue solid line corresponds to the equilibriumcorrelation function geq for Pe ¼ 0. (Middle) Deviation from equilibrium correlations I ¼ ½ð∇vÞ2 − T∇2v�ðg − geqÞ scaled by Pe2 as afunction of ðrþ aÞ=σ, where aðPeÞ is a fitting parameter. The data almost collapse into a master curve for each row, namely, at a given τ.(Right) Parametric plot of the rate of work _w=T and the integrated deviation from equilibrium correlations ρ0

RIðrÞdr=ðγTÞ showing a

linear relation. The black solid line with slope α is the best linear fit, and the marker colors refer to the same Péclet values as in theleft and right columns. Simulation details are in Appendix A. Parameters: fτT=ðγσ2Þ; αg ¼ f2 × 10−1; 1.50g (a)–(c), f5 × 10−1; 1.41g(d)–(f), f2 × 10−2; 1.38g (g)–(i), and f5 × 10−2; 1.33g (j)–(l).


041026-7

agreement between them, as shown in the right column inFig. 3:

_w ¼ αρ0γ

Z½gðrÞ − geqðrÞ�f½∇vðrÞ�2 − T∇2vðrÞgdr; ð24Þ

where α is a fitting parameter independent of the Pécletnumber. This empirical relation demonstrates that, in thelimit of dilute tracers, the rate of work can actually bedirectly estimated by comparing driven and equilibriumpair correlations for both deterministic and active drives.Comparing Eqs. (21) and (24), we deduce the followingintegral relation between density correlations:

Z½ð2 − αÞgðrÞ þ αgeqðrÞ�fT∇2vðrÞ − ½∇vðrÞ�2gdr

¼ ρ0ZZ

½g3aðr; r0Þ þ g3bðr; r0Þ�½∇vðrÞ� · ½∇vðr0Þ�drdr0:ð25Þ

Interestingly, it is reminiscent again of the connectionbetween density correlations provided by the YBG hier-archy at equilibrium [90]. Similarly, the relation (25)amounts to a constraint on density correlations, now validfor nonequilibrium liquids, which could guide the search

for explicit predictions on the emerging structure.Importantly, it does not rely on any equilibrium mapping,in contrast to previous works [93–95], since it remains validfor non-negligible dissipation.The power balance (21) can actually be extended to the

case where all particles in the liquid are driven as

_w ¼ ρ0γ

ZgðrÞf½∇vðrÞ�2 − T∇2vðrÞgdr;

þ ρ20

γ

ZZg3ðr; r0Þ½∇vðrÞ� · ½∇vðr0Þ�drdr0; ð26Þ

where g and g3 now refer, respectively, to the two-body andthree-body density correlations among all particles. Thisextension leads to an exact relation between the rate ofwork and the forces applied to particles Fi as

_w ¼ 1γ

Xi

½hF2i i þ Th∇i · Fii�; ð27Þ

which differs from the relation (23) for driven tracers by anoverall factor of 2. A result analogous to Eq. (27) was foundpreviously for a deterministic drive [36,96]. The maindifference is that Eq. (27) features interaction forces Fi onlyin the rhs, thus allowing one to evaluate the rate of work

(b) (c)

(d) (e) (f)

FIG. 4. Connecting dissipation and structure for a liquid where 100% of particles are driven by an active force. (Left) Densitycorrelation g as a function of interparticle distance r=σ. The blue solid line corresponds to the equilibrium correlation function geq forPe ¼ 0. (Middle) Deviation from equilibrium correlations I ¼ ½ð∇vÞ2 − T∇2v�ðg − geqÞ scaled by Pe2 as a function of ðrþ aÞ=σ,where aðPeÞ is a fitting parameter. The data almost collapse into a master curve for each row, namely, at a given τ. (Right) Parametricplot of the rate of work _w=T and the integrated deviation from equilibrium correlations ρ0

RIðrÞdr=ð2γTÞ showing a linear relation. The

black solid line with slope α is the best linear fit, and the marker colors refer to the same Péclet values as in the left and right columns.Simulation details are in Appendix A. Parameters: fτT=ðγσ2Þ; αg ¼ f2 × 10−2; 1.22g (a)–(c) and f5 × 10−2; 1.14g (d)–(f).


041026-8

without any prior knowledge on the driving force. Besides,it is valid for both deterministic and active drives.Moreover, conducting the same analysis of density corre-lations as for driven tracers, I exhibits again a scaling withPe2, as reported in Fig. 4. We show that _w and

RIðrÞdr are

also linearly related. Introducing the linear coefficient asαρ0=ð2γÞ is consistent with substituting Eq. (25) intoEq. (26), where g3;a þ g3;b is now replaced by 2g3.Hence, it demonstrates that the rate of work is alsoaccessible from the nonequilibrium deviation of paircorrelations in fully driven liquids.Overall, the results of this section illustrate how dis-

sipation affects the transport and structural properties ofdriven liquids, measured in terms of the diffusion coef-ficient and density fluctuations. These findings motivate thefollowing question: Can nonequilibrium forces be tuned toreliably stabilize target configurations? To explore thisquestion, we rely in what follows on the framework oflarge-deviation theory. In practice, our strategy amounts tobiasing trajectories in terms of dissipation, related to many-body interactions by Eq. (21), to mimic the effect of anexternal drive. Following this route, our analytical andnumerical results provide some concrete intuition for howinteractions in a multicomponent system can be control-lably renormalized by nonequilibrium forces. Hence, wedemonstrate the ability to nucleate structures different fromthose characteristic of the equilibrium Boltzmann distribu-tion to help guide self-assembly [97] and collective motion[25] far from equilibrium. These results further illustrate theinterplay between energy dissipation and organization innonequilibrium many-body settings.

III. INTERACTIONS IN BIASED ENSEMBLES

To investigate how target structures and dynamics can bepromoted by means of a dynamical bias, we begin byconsidering a system of interacting Brownian particleswithout any driving force:

γ_ri ¼ −∇iXj

vðri − rjÞ þ ξi; ð28Þ

where the statistics of the noise term ξi is the same as theone in Eq. (1). The rate of work _w defined in Eq. (14) iszero because of the absence of driving. In Sec. II, to obtaina nonzero rate of energy flow through the system, weconsider an explicit driving force Fd;i, and we explore itseffects on the transport and structural properties of theliquid. In practice, different types of driving can lead to thesame dissipation. In this section, using the framework ofthe large-deviation theory, we take an alternative approachwhere the dynamics is now conditioned by enforcing arequired energy flow without any explicit driving. Thus,exploring how the system adapts to this requirementprovides a new insight into the relation between dissipation

and organization in driven systems, which is distinct fromyet complements the approach in Sec. II.To this end, we focus on the subset of noise realizations

that are conditioned on a nonzero rate of work. Inparticular, these realizations no longer have a zero average,so that one can redefine the noise term in Eq. (28) as ξi →ξi þ Faux;i by introducing an auxiliary force Faux;i [38,50].Hence, the stochastic dynamics given by Eq. (28) withadded force Faux;i provides an explicit case which ensures anonzero energy flow rate. In practice, this dynamics can bedrastically different from the original one, thus opening thedoor to stabilizing unexpected structure and to promotingnovel collective effects. Interestingly, such a dynamics canactually be regarded as the optimal strategy to effectivelyenforce a target condition on the rate of work [98].Formally, to study the dynamics conditioned by dis-

sipation, we bias the probability of trajectories. This biasingis done by introducing an exponential weighting factorexp½κ R t0 EðsÞds�, where E is the observable which con-ditions the dynamics, e.g., energy flow rate, and κ is aconjugate field. In practice, the relative importance ofbiasing in the dynamics is controlled by κ, which, in turn,controls the average value of E [37]. Before deriving thecentral results of this section, namely, relations betweenbiased energy flow rates and organization, we first intro-duce a simple example in which the connection betweenauxiliary forces and an exponentially biased ensemble canbe clearly seen.

A. Dynamical bias and external forces

To introduce pedagogically our methods, we first showhow biasing trajectories can lead to effectively introducinga driving force. Inspired by the role of dissipation inemerging liquid properties, as discussed in Sec. II, webias the equilibrium dynamics (28) with the sum of thedissipation and the rate of work, scaled by T, that would beproduced by applying a constant force Fd to a subset Ω ofparticles:

E ¼ 1γT

Xi∈Ω

Fd · ½γ _ri þ∇iV�; ð29Þ

where V ¼ ð1=2ÞPi;j vðri − rjÞ. The path probability P ∼exp½−Pi R t0 AiðsÞds� corresponding to this biased ensem-ble is obtained with standard methods [99,100]:

Ai¼1

4γT½γ_riþ∇iV�2− 1

2γ∇2i V− κγT δi∈ΩFd · ½γ _riþ∇iV�;

ð30Þ

where the first two terms correspond to the unbiaseddynamics (28) and the third one to the bias in Eq. (29).It can also be written as


041026-9

Ai ¼1

4γT½γ_ri − 2κδi∈ΩFd þ∇iV�2 − 1

2γ∇2i V − κ

2

γTδi∈ΩF2d:

ð31Þ

As a result, given that the last term in Eq. (31) can beabsorbed in a normalization factor, we deduce that thetrajectories biased by Eq. (29) can be generated, at leadingorder, in a physical dynamics where the external force 2κFdis applied to every particle in Ω. In particular, it does notfeature any long-range interactions which are usually foundin auxiliary dynamics [101].

B. Dynamical bias and modified interactions

To go beyond the case of applying a constant force, wenow seek for a dynamical bias which regulates particleinteractions in a controlled manner. In particular, weexamine cases where the control parameters κij are specificto particle pairs fi; jg, so that the biasing factor in pathprobability now reads exp½Pi;j κij R t0 EijðsÞds�. Now, ourchoice for the biasing function Eij is informed by theconnection between the rate of work and many-bodyinteractions in driven liquids, as detailed in Sec. II C.Specifically, we observe that the power balance (21) for adeterministic drive ( _wact ¼ 0) can be written as _w ¼−P

i∈Ω;jhLvðri − rjÞi in terms of the evolution operatorof the equilibrium dynamics (28) defined by γL ¼P

i½T∇i −∇iV� · ∇i. This observation motivates us toconsider the following bias:

Eij ¼1

4TLvðri − rjÞ: ð32Þ

In the unbiased ensembles in Sec. II C, hEiji provides ameasure of the rate at which driving forces pump energyinto or extract energy from the specific interaction betweenthe ith and jth particles. Here, instead of driving the systemwith a specific driving force, trajectories are driven byatypical realizations of the noise generated by biasedsampling.To explore how this bias modifies interactions, we first

employ a derivation different from the path integralapproach in Sec. III A. Based on the procedure inRefs. [38,50], the auxiliary physical dynamics, whichhas the same statistical properties as in the biased ensemble,can be constructed by solving the eigenvalue equation

�Lþ

Xi;j

κijEij

�Gðfrkg; κÞ ¼ λðκÞGðfrkg; κÞ; ð33Þ

where the eigenvalue λ, parametrized by κij, is the scaledcumulant-generating function appropriate to Eij. The aux-iliary dynamics is then defined by replacing the interactionpotential in Eq. (28) by the following auxiliary potential:

Ṽ ¼ 12

Xi;j

vðri − rjÞ − 2T lnG: ð34Þ

In practice, computing G is a highly nontrivial procedurefor many-body systems. The explicit solutions consideredso far concern either exclusion processes [66,102,103] orparticle-based diffusive systems restricted to small noiseregimes [104,105] and noninteracting cases in some spe-cific potentials [106–108].In our case, a simple expression can be obtained for the

auxiliary potential Ṽi by solving Eq. (33) perturbatively atsmall bias parameter κ. Specifically, we expand

λðκÞ ¼Xij

κijhEiji þOðκ2Þ;

Gðfrkg; κÞ ¼ Gð0Þ þXij

κijGð1Þij ðfrkgÞ þOðκ2Þ; ð35Þ

where G0 is the uniform eigenvector associated with thezero eigenvalue. Given that hEiji ¼ 0 in the steady state,which follows from the vanishing current condition in theunbiased dynamics (h _vi ¼ 0), the leading nontrivial orderof Eq. (33) reads

Xij

κij½LGð1Þij þ Gð0ÞEij� þOðκ2Þ ¼ 0: ð36Þ

Substituting the explicit expressions for the biasing func-

tion in Eq. (32), we then deduce that 4TGð1Þij ¼−G0vðri−rjÞis a solution of the eigenvalue problem to the order of κ.The auxiliary potential follows as

Ṽ ¼ 12

Xi;j

ð1þ κijÞvðri − rjÞ þOðκ2Þ: ð37Þ

Therefore, biasing with Eq. (32) amounts to changing thestrength of particle interaction by a factor κij specificallyfor any pair fi; jg, which is the main result of this section.While energy flows are sustained by explicit nonequili-

brium forces in Sec. II, we now maintain a nonzero averagefor Eij by a biased sampling of trajectories. The corre-sponding noise realizations can be thought of as an externalprotocol, which leads to modifying the energy landscapesampled by the biased system as given in Eq. (37). Notethat the tuning interaction strength between targeted pairs isqualitatively consistent with the effect of external driving.Indeed, phase separation in mixtures of driven and undrivenparticles, reported both experimentally and numerically,can be rationalized in terms of an effective decrease ofspecific interactions between these particles [21,23].Moreover, the techniques in Sec. III A allow one to

anticipate the trajectories generated at higher order whennow biasing with Eq. (32). To this end, we consider theensemble where the first-order dynamics, given by the


041026-10

potential (37), is biased with exp½R t0 εðsÞds� defined interms of

ε ¼ 14γT

Xk

�Xi;j

κij∇kv½riðsÞ − rjðsÞ��2

: ð38Þ

As detailed in Appendix C, this ensemble is equivalent tobiasing the original dynamics (28) with Eq. (32). Thus, theeffect of higher-order bias on trajectories amounts tomaximizing the squared forces in the integrand ofEq. (38), which effectively tends to cluster particles forboth signs of κij.Finally, the decomposition between the first-order aux-

iliary potential and higher-order symmetric bias can beextended to a generic class of biases of the form TEij ¼LAðri − rjÞ for an arbitrary observable A: The correspond-ing first-order auxiliary potential V þ 2Pi;j κijAðri − rjÞis now complemented with the higher-order bias (38),where A replaces v. Such a bias is reminiscent of, yet

qualitatively different from, the escape rate used to promotedynamical heterogeneity in glassy systems [41,109]. In ourcase, clustering is favored for both positive and negativebias parameters κij. In particular, this result is in contrastwith the emergence of a hyperuniform phase, where large-scale fluctuations are suppressed, reported when biasingsome hydrodynamic theories of diffusive systems [110].

C. Numerical sampling of biased structures

To illustrate the potential of our bias to control liquidproperties, we focus in what follows on the specific caseκij ¼ κδi∈Ωδj∉Ω, where all pairs between a subset Ω andother particles are biased with the same strength κ. Here, theset Ω could, for instance, refer to some tracer particlesimmersed in the liquid, to connect with the settingsin Sec. II.To confirm numerically the validity of our approach, we

first probe the range of the first-order auxiliary dynamics

(a) (b)

(d)(c)

FIG. 5. (a),(b) Average biasing observableP

i∈Ω;j∉ΩhEijiκ ¼P

i∈Ω;j∉ΩhLvðri − rjÞiκ=T as a function of bias parameter κ, where Land v, respectively, denote the evolution operator and the pair potential of the equilibrium dynamics (28). Results from the first-orderauxiliary dynamics (solid lines) and from a direct sampling of the biased ensemble (circles) coincide for a finite range of κ. (c),(d) Biaseddensity correlation gκ as a function of interparticle distance r=σ obtained from auxiliary dynamics (solid lines) and direct sampling(dotted lines). At leading order, our dynamical bias effectively renormalizes the potential v by a factor κ for specific pairs of particlesfi ∈ Ω; j ∉ Ωg, in satisfying agreement with direct sampling. This renormalization illustrates the control of liquid structure at small κand weak interactions. Simulation details are in Appendix A.


041026-11

where interactions are predicted to be simply renormalized.We compare measurements of

Pi∈Ω;j∉ΩhEijiκ, where h·iκ

denotes an average in the biased ensemble, obtained fromsimulations with the renormalized potentials in Eq. (37)and from a direct sampling of the biased ensemble. Thelatter is implemented with a cloning algorithm whichregularly selects and multiplies rare realizations for effi-cient sampling [46,63–68]. For convenience, interactionsare now given by the soft-core potential vðrÞ ¼v0 expf−1=½1 − ðjrj=σÞ2�gΘðσ − jrjÞ. For weak inter-actions (v0 ¼ 4T), we observe a very satisfying agreementbetween the two measurements for a finite range of κ, asreported in Fig. 5(a), which supports the validity of ourperturbation up to an interaction change between −20%and þ40%. The range of validity decreases as v0=Tincreases, as shown in Fig. 5(b), and we expect a similartrend when also increasing the number of biased pairs.To explore further the features of this biased ensemble,

we now compare the density correlations of biased pairsgκðrÞ ∼

Pi∈Ω;j∉Ωhδðr − ri þ rjÞiκ obtained from both

direct sampling and first-order auxiliary dynamics. Forκ ¼ �0.1, we observe that the structural modificationinduced by the bias becomes more dramatic as v0=Tincreases. The agreement between the cloning and auxiliarydynamics is good for the whole curve when v0=T ¼ 4,whereas a deviation appears beyond r ≃ σ whenv0=T ¼ 12, as shown in Figs. 5(c) and 5(d). In both cases,the region of particle overlap r < σ is well reproduced.These results corroborate the ability of the first-orderauxiliary dynamics to capture interaction changes as asimple renormalization of the potential strength. In con-trast, the tendency for particles to cluster, manifest numeri-cally in the increased peak value at r ≃ σ, is a higher-ordereffect missed by this auxiliary dynamics when v0=T ¼ 12.

Yet, note that the peak value is comparable for κ ¼ �0.1, inagreement with Eq. (38) being symmetric in κ. Altogether,these results demonstrate that our bias modulates the liquidstructure in a controlled manner for a small bias and weakinteractions as predicted by Eq. (37).Finally, we probe numerically the effect of a large bias

(jκj > 1) using direct sampling, to explore configurationssignificantly distinct from the one of the equilibriumdynamics (28). The particles spontaneously tend to clusterfor both positive and negative κ, as shown in Fig. 6 andmovies in Ref. [73]. This result confirms the propensity oftrajectories to maximize interaction forces at a high bias, ascaptured by Eq. (38). Importantly, the shape of clustersdiffers depending on the sign of κ: A micellelike structurefeaturing the particles in Ω at the core (blue) surrounded byothers (red) appears for κ ¼ −3, whereas clusters have arandom composition for κ ¼ 3. Again, this result agreeswith the renormalized interactions being either increased(κ > 0) or decreased (κ < 0). In practice, the interactionstrength changes sign when κ < −1 according to Eq. (37),so that the red-blue pairs are effectively attractive forκ ¼ −3. To optimize the overall energy, the most favorableconfiguration then consists in maximizing (minimizing) theoverlap of particles in Ω (not in Ω), which, in turn,stabilizes a cluster of blues surrounded by reds. In general,two types of configuration should generically be stabilizedfor a given interaction potential v, depending on the sign ofthe bias. Overall, this result establishes a reliable proof ofprinciple for the design of tailored self-assembled struc-tures with our specific choice of biased ensembles.

D. Bias-induced collective motion

As a final illustration of how collective effects canbe controlled by dynamical bias, we consider a model of

FIG. 6. Configurations obtained from a direct sampling of the biased ensemble where the pair interactions between red and blueparticles are selectively modified. In the unbiased dynamics (κ ¼ 0), interactions are purely repulsive with a soft core which has a similarstrength for all particles, either red or blue, so that the system is homogeneous. The dynamical bias promotes clustering for both signs ofκ, yet it changes interaction selectively for either sign. The repulsion is increased between red and blue particles for κ ¼ 3, and theirinteractions become effectively attractive for κ ¼ −3. As a result, the clusters which emerge spontaneously have different structures:either a random composition of mixed reds and blues (κ ¼ 3) or a micellelike structure with a blue core (κ ¼ −3). This result illustrateshow biasing specific pairs leads to supervised spatial organization. Simulation details are in Appendix A and movies in Ref. [73].


041026-12

self-propelled particles where interactions are now medi-ated only via the angular dynamics [69]:

_ri ¼ V0uðθiÞ; _θi ¼ μrXj

T ðθj − θi; ri − rjÞ þ ηiðtÞ;

ð39Þ

where V0 denotes the self-propulsion velocity, uðθÞ ¼ðcos θ; sin θÞ is the unit vector, and μr is the rotationalmobility. The term ηi is a zero-mean Gaussian white noisewith correlations hηiðtÞηjð0Þi ¼ 2DrδijδðtÞ given in termsof the rotational diffusion coefficient Dr. To promotealignment between neighboring particles, we choose thepairwise torque as T ðθ; rÞ ¼ Θðσ − jrjÞ sin θ=ðπσ2Þ. Thisdynamics was originally introduced as a generalization ofthe Vicsek model to continuous time [25]. Thus, it exhibitsa transition between a isotropic state for small density ρ0and large noiseDr and a polar state for large density ρ0 andsmall noise Dr. In practice, the linear stability analysis of

the corresponding hydrodynamic equations predicts thetransition to occur when 2Dr ¼ μrρ0 [69]. In whatfollows, our aim is to show that such a transition canalso be mediated by tuning interactions with a dynami-cal bias.To this end, we take the biasing factor in path proba-

bility as exp½κ R t0 EθðsÞds�, where the biasing observable Eθreads

Eθ ¼ −1

2

Xi;j

� ∂∂θi þ

μrDr

Xk

T ðθi − θk; ri − rkÞ�

× T ðθi − θj; ri − rjÞ: ð40ÞThe average value hEθi is proportional to ðd=dtÞ

Pi;j ×

hcosðθi − θjÞΘðσ − jri − rjjÞi, which vanishes in thesteady state. Then, following the procedure detailed inSec. III B, we deduce that biasing the dynamics (39) withEθ amounts to considering renormalized interactions of theform

Isotropic state Polar state

= - 0.2 = 0 = 0.2

(e)

(a) (b) (c)

(d) (e) (f)

FIG. 7. Configurations obtained from a direct sampling of the biased ensemble for aligning self-propelled particles. The color coderefers to the orientation of particles. In the unbiased dynamics (κ ¼ 0), we observe isotropic and polar states, respectively, at large noise(Dr > D�r) and small noise (Dr < D�r ). Here, the critical noise isD�r ¼ 8, and we take the noise valuesDr ¼ f7; 9g for, respectively, thepolar and isotropic regimes. The dynamical bias leads to renormalizing interactions in a controlled manner, which effectively changesthe transition threshold asD�r → D�rð1þ κÞ at leading order. As a result, one can stabilize either isotropic or polar states, respectively, forκ < 0 and κ > 0, thus illustrating the ability to trigger or inhibit collective effects in nonequilibrium systems. Simulation details are inAppendix A and movies in Ref. [73].


041026-13

T̃ ¼ ð1þ κÞT þOðκ2Þ: ð41Þ

Thus, by promoting a nonzero average for Eθ, aligninginteractions can be tuned in a controlled manner at firstorder in κ. A higher-order bias leads to maximizing thesquared torque, as presented in Appendix C.We test this prediction numerically using a direct

sampling of biased trajectories. We consider values offDr; μr; ρ0g above and below the threshold D�r ¼ μrρ0=2,where the system exhibits either isotropic or polar states inthe unbiased dynamics (κ ¼ 0), as shown in Fig. 7.Specifically, when the original system is isotropic(Dr > D�r), we observe a transition to polar for κ > 0,and, conversely, when it is polar (Dr < D�r), a transition toisotropic for κ < 0. This observation confirms our result(41), where the bias amounts to changing the angularmobility as μ̃r ¼ ð1þ κÞμr þOðκ2Þ for weak κ, so that thelinear instability is either triggered or suppressed by solelytuning κ, all other parameters being held the same. Thus,these results demonstrate how biasing the dynamics with anappropriate observable leads to control of the emergence ofspontaneous organization, with potential interest for othernonequilibrium dynamics.

IV. CONCLUSION

Developing techniques to characterize and control thebehavior of systems operating far from equilibrium remainsa central and outstanding problem. Despite the apparentlycomplex interplay between internal dissipation and emerg-ing properties, we demonstrate that tracer diffusion anddensity correlations can simply be connected to dissipationin driven liquids. We also construct a mapping betweendeterministic and active drives for a specific active mattermodel, thus showing how our approach can potentially beextended to a broad class of systems. Importantly, ourresults open promising perspectives to evaluate dissipationsimply from the structure of the system. Inspired by recentworks [18,111], one could also introduce a map ofdissipation, directly related to the statistics of interactionforces, to resolve spatially where energy is released in thethermostat. Though the corresponding integrated mapwould not cover the total dissipation, it would alreadyprovide insightful information about locations of low andhigh dissipation with respect to a constant background setby the squared driving amplitude.In practice, monitoring dissipation with a well-defined

parameter remains an open challenge for many-bodysystems. To this end, biased ensembles enable one tospecify the statistics of dissipation by introducing anadditional control parameter, analogously to the changefrom a microcanonical to a canonical ensemble in equi-librium thermodynamics [38,50], which is done by select-ing rare noise realizations which drive the system awayfrom typical behavior, without introducing any drivingforce. Pioneering works were focused on favoring

dynamical heterogeneities, without affecting the structure,of kinetically constrained models [39–43]. Yet, more recentstudies show the potential to also modify density correla-tions in diffusive systems [48,49,112].Using these large-deviation techniques, we put forward a

particular set of biased ensembles which allows one toregulate the liquid structure in a controlled manner. Theexplicit form of the bias is motivated by the relationsbetween dissipation and structure that we derive for drivenliquids. At leading order, any bias in this class simply leadsto introducing additional interactions in the dynamics.Furthermore, a higher-order bias systematically constrainsthe trajectories to favor the formation of clusters. Based onminimal case studies, we sample the biased configurations,using state-of-the-art numerics [46,63–68], to illustrate theability to stabilize specific structures and collective effectsin a controlled manner.Since dynamical bias consists in favoring rare noise

fluctuations, the corresponding dynamics effectivelyprovides useful insights on how to promote atypicalconfigurations with an external drive. In practice, thedriving protocol should simply mimic the biased noiserealizations. This line of thought has already beenexploited for efficient sampling of the biased ensemble[45,65,113,114], where control forces make rare eventsbecome typical. Moreover, since our analytic frameworkencompasses the case of a specific bias for each pair ofparticle, it could potentially be regarded as a fruitful routeto promote the spontaneous self-assembly of complexstructures at the cost of energy dissipation. For instance,inspired by recent works [115,116], one might considerour approach to design energetic landscapes, in terms ofthe pair-specific bias parameters, which selectively stabi-lize some target molecules.Overall, these results illustrate how specifying the

amount of energy dissipated by nonequilibrium forcesallows one to constrain the dynamics and structure ofdriven liquids. This constraint paves the way towardscontrolling the emerging properties of such systems bytuning dissipation accurately. It remains to investigatewhether similar results can be obtained in more complexsystems which could, for instance, potentially includeanisotropic building blocks, such as driven chiral objectsor active liquid crystals [27,117,118].

ACKNOWLEDGMENTS

The authors acknowledge insightful discussions withMichael E. Cates, David Martin, Robert L. Jack, andVincenzo Vitelli. This work was granted access to theHPC resources of CINES/TGCC under the allocation 2018-A0042A10457 made by GENCI and of MesoPSL financedby the Region Ile de France and the project Equip@Meso(reference ANR-10-EQPX-29-01) of the programInvestissements d’Avenir supervised by the AgenceNationale pour la Recherche. S. V. and L. T. were supported


041026-14

by the University of Chicago Materials Research Scienceand Engineering Center, which is funded by NationalScience Foundation under Grant No. DMR-1420709.S. V. acknowledges support from the Sloan Foundationand startup funds from the University of Chicago. S. V. andL. T. acknowledge support from the National ScienceFoundation under Grant No. DMR-1848306. É. F. issupported by an Oppenheimer Research Fellowship fromthe University of Cambridge and a Junior ResearchFellowship from St. Catherine’s College.

APPENDIX A: NUMERICAL SIMULATIONS

In Sec. II A, a custom code of molecular dynamics, basedon the finite time difference, is used to perform thesimulations in a two-dimensional box 102σ × 102σ withperiodic boundary conditions. The time step is δt ¼ 10−4,and the initial condition is homogeneous. Parameter valuesare ρ0 ¼ 0.7, T ¼ 0, γ ¼ 1, f ¼ 3 × 10−2, τ ¼ 103,v0 ¼ 5, n ¼ 102, and σ ¼ 1.In Secs. II B and II C, numerical simulations of the

dynamics (1) are performed using the LAMMPS simula-tion package in a two-dimensional box 102σ × 102σ, whereσ is the particle diameter, with periodic boundary con-ditions at average density ρ0 ¼ 0.45. Our custom codeactually implements overdamped Langevin equations ofmotion with the finite time difference. It simply utilizes theefficient force computation routines that are built as a partof the molecular dynamics package. The system is firstrelaxed for 103 conjugate gradient descent steps and laterequilibrated during 50τ. We evaluate average values overten independent trajectories with duration 150τ. Thedensity pair correlations are constructed using ten inde-pendent trajectories, each with duration 50τ. We performerror analysis from the independent simulations and obtainnegligible errors for all the data in Figs. 2–4. The time stepis 5 × 10−4, and the bin size for computing the paircorrelations is 0.01σ. We perform simulations at othervalues of the time step f10−4; 10−5g and of the bin sizef5 × 10−3σ; 2 × 10−2σg to confirm that our calculations ofthe α coefficients are well converged. Parameter values areT ¼ 1, γ ¼ 102, v0 ¼ 1, and σ ¼ 1.In Sec. III C, a custom code of molecular dynamics,

based on the finite time difference, is used to perform thesimulations in a two-dimensional box 10σ × 10σ withperiodic boundary conditions. We bias the pair potentialbetween eight blue particles and 16 red particles. Tosample the biased ensemble, we use the cloning algorithmdescribed in Appendix A in Ref. [65]. The time intervalfor cloning is Δt ¼ 10δt, and the number of clones is1600. The time step is δt ¼ 10−4, the initial relaxationtime is 104Δt, and the total simulation time is 106Δt.Parameter values are T ¼ 1, γ ¼ 1, v0 ¼ 4 (Fig. 6),and σ ¼ 1.

In Sec. III D, a custom code of molecular dynamicsbased on the finite time difference is used to perform thesimulations. N ¼ 128 particles are simulated in a two-dimensional box of size 4σ × 4σ with periodic boundaryconditions. To sample the biased ensemble, we use thecloning algorithm described in Appendix A in Ref. [65].The time interval for cloning is Δt ¼ 10δt, and the numberof clones is 200. The time step is δt ¼ 10−3, and the totalsimulation time is 500Δt. Parameter values are V0 ¼ 2,μr ¼ 2, ρ0 ¼ 8, and σ ¼ 1.

APPENDIX B: DISSIPATION AND DIFFUSION

This Appendix is devoted to the derivation of thedissipation rate J and the diffusion coefficient D of adriven tracer, as defined in Sec. II. We employ a perturba-tive treatment at weak interactions, originally introducedfor a particle driven at constant force in Refs. [52,53]. Tothis aim, the tracer-bath interaction potential v is scaled by asmall dimensionless parameter h ≪ 1 in what follows.Besides, we focus on the regime of dilute tracers, so thatinteractions among them, either direct or mediated by thebath, can be safely neglected.The dynamic action associated with the tracer dynamics

(11) and (12) follows from standard path integral methods[99,100]. It can be separated into contributions from thefree tracer motion and from interactions, respectively,denoted by A0 and Aint:

A0 ¼Z

r̄0 · ½ið_r0 − Fd=γÞ þD0r̄0�dt;

Aint ¼h2

γ

Zdq

ð2πÞd jqj2jvðqÞj2

Z∞

−∞ds

Zs

−∞du

× e−DGjqj2KðqÞðs−uÞþiq·½r0ðsÞ−r0ðuÞ�

× r̄0ðsÞ ·�r̄0ðuÞγKðqÞ −

qγG

�; ðB1Þ

where D0 ¼ T=γ is the tracer diffusion coefficient in theabsence of interactions (v ¼ 0) and r̄0 is the processconjugated with the tracer position r0. For weak inter-actions h ≪ 1, any average value can be then expanded interms of h as h·i ¼ h·i0 − h2hAint·i0 þOðh4Þ, where h·i0 isthe average taken with respect to A0 only. As a result,determining the first correction from interactions in anyobservable amounts to computing the corresponding aver-age hAint·i0.Considering the dissipation rate per particle J =N ¼

h_r0i · Fd, the leading order is h_r0i0 · Fd ¼ jFdj2=γ ¼ f2=γ,and the first correction reads −h2hAint _r0i0 · Fd. Giventhe explicit form of Aint in Eq. (B1), the correlations ofinterest are


041026-15

h_r0ðtÞ½q · r̄0ðsÞ�eiq·½r0ðsÞ−r0ðuÞ�i0¼ iqδðt − sÞe−D0jqj2ðt−uÞþðiq=γÞ·

Rt

uFdðwÞdw;

h_r0ðtÞ½r̄0ðuÞ · r̄0ðsÞ�eiq·½r0ðsÞ−r0ðuÞ�i0¼ −iqδðt − sÞe−D0jqj2ðt−uÞþðiq=γÞ·

Rt

uFdðwÞdw; ðB2Þ

where we use that the tracer statistics is Gaussian in theabsence of interactions, following Refs. [52,53]. From thisresult, we get

J −f2=γ¼Nh2

dγ2

Zdq

ð2πÞd iq ·FdðtÞjqj2jvðqÞj2D0þDGKðqÞ

D0KðqÞ×Z

t

−∞due−jqj

2½D0þDGKðqÞ�ðt−uÞþðiq=γÞ·R

t

uFdðwÞdw

þOðh4Þ; ðB3Þ

where we use γG ¼ γ=ρ0 and DG ¼ ρ0D0. Expanding atsmall f, we deduce

J −f2=γ¼−Nh2

dγ3

Zdq

ð2πÞd jqj4jvðqÞj2D0þDGKðqÞ

D0KðqÞ×Z

t

−∞due−jqj2½D0þDGKðqÞ�ðt−uÞ

Zt

udwFdðtÞ ·FdðwÞ

þOðh4;f4Þ: ðB4Þ

Substituting the explicit expression of the deterministicdrive (2) in Eq. (B4) and then integrating over u and w, weobtain

J −f2=γ¼−NðhfÞ2

dγ3

Zdq

ð2πÞdjqj4jvðqÞj2

jqj4½D0þDGKðqÞ�2þω2

×D0þDGKðqÞ

D0KðqÞþOðh4;f4Þ: ðB5Þ

For the case of active drive with correlations (3), we exploitthe equivalence with a disordered drive detailed inSec. II A. Substituting the explicit drive (4) in Eq. (B4)and then averaging over disorder in the limit of manyoscillators (n ≫ 1), we get

J −f2=γ¼−NðhfÞ2

dγ3

Zdqdω0

ð2πÞdþ1jqj4jvðqÞj2ϕðω0Þ

jqj4½D0þDGKðqÞ�2þðω0Þ2

×D0þDGKðqÞ

D0KðqÞþOðh4;f4Þ; ðB6Þ

where ϕðω0Þ ¼ 2τ=½1þ ðω0τÞ2�, yielding

J −f2=γ¼−NτðhfÞ2

dγ3

Zdq

ð2πÞdjqj2jvðqÞj2D0KðqÞ

×1

τjqj2½D0þDGKðqÞ�þ1þOðh4;f4Þ: ðB7Þ

The asymptotic results for the rate of work _w ¼ f2=γ − J ,presented in Sec. II B for both deterministic and activedrives, follow directly.We now turn to deriving the diffusion coefficient D. It is

defined in terms of the mean-squared displacement (MSD)hΔr20ðtÞi¼h½hr0ðtÞi−r0ðtÞ�2i as D ¼ limt→∞hΔr20ðtÞi=2dt.At leading order, the MSD reads hΔr20ðtÞi0 ¼ 2dD0t. Toobtain the first order, we need to compute the followingcorrelations:

hΔr20ðtÞ½q · r̄0ðsÞ�eiq·½r0ðsÞ−r0ðuÞ�i0 ¼ −4ðD0=γÞjqj2ðs − uÞΘðt − sÞ e−D0jqj2ðt−uÞþðiq=γÞ·

Rt

uFdðwÞdw;

hΔr20ðtÞ½r̄0ðuÞ · r̄0ðsÞ�eiq·½r0ðsÞ−r0ðuÞ�i0 ¼ ð2=γ2ÞΘðt − sÞ½2D0jqj2ðs − uÞ − 1� e−D0jqj2ðt−uÞþðiq=γÞ·

Rt

uFdðwÞdw; ðB8Þ

where we use again that A0 is Gaussian in terms of r̄0, yielding

hΔr20ðtÞi − 2dD0t ¼2h2

γ2

Zdq


KðqÞZ

t

−∞ds

Zs

−∞duf2jqj2½D0 þDGKðqÞ�ðs − uÞ − 1g

× e−jqj2½D0þDGKðqÞ�ðs−uÞþðiq=γÞ·

Rs

uFdðwÞdw þOðh4Þ: ðB9Þ

Expanding at small f, we get

hΔr20ðtÞi − 2dDeqt ¼ −2h2

γ4

Zdq


KðqÞZ

t

−∞ds

Zs

−∞duf2jqj2½D0 þDGKðqÞ�ðs − uÞ − 1g

× e−jqj2½D0þDGKðqÞ�ðs−uÞZ

s

udw1dw2Fdðw1Þ · Fdðw2Þ þOðh4; f4Þ; ðB10Þ


041026-16

where Deq refers to the diffusion coefficient in the absenceof driving force (f ¼ 0). For the deterministic drive (2), theexplicit time integrations give

D −Deq ¼ðhfÞ2dγ4

Zdq


KðqÞ½D0 þDGKðqÞ�

×5jqj4½D0 þDGKðqÞ�2 þ ω2fjqj4½D0 þDGKðqÞ�2 þ ω2g2

þOðh4; f4Þ: ðB11ÞUsing the mapping in Sec. II A for the case of active drivewith correlations (3), we deduce

D −Deq ¼ðhfÞ2dγ4

Zdqdω0

ð2πÞdþ1jqj2jvðqÞj2ϕðω0Þ

KðqÞ½D0 þDGKðqÞ�

×5jqj4½D0 þDGKðqÞ�2 þ ðω0Þ2fjqj4½D0 þDGKðqÞ�2 þ ðω0Þ2g2

þOðh4; f4Þ; ðB12Þwhere again ϕðω0Þ ¼ 2τ=½1þ ðω0τÞ2�, yielding

D −Deq ¼τðhfÞ2dγ4

Zdq

ð2πÞdjvðqÞj2

KðqÞ½D0 þDGKðqÞ�2

×5τjqj2½D0 þDGKðqÞ� þ 3fτjqj2½D0 þDGKðqÞ� þ 1g2

þOðh4; f4Þ: ðB13Þ

Finally, we obtain the expressions in the asymptoticregimes, as reported in Sec. II B for both deterministicand active drives.

APPENDIX C: EQUIVALENCE OF BIASEDENSEMBLES

In this Appendix, we demonstrate the equivalencebetween specific dynamical biased ensembles. First, weconsider ensembles related to the equilibrium dynamics(28). Ensemble (a) corresponds to biasing with the factorexp½Pi;j κij R t0 EijðsÞds� in the path probability, where

Eij ¼1

γT

Xk

½T∇k −∇kV� ·∇kAðri − rjÞ: ðC1Þ

Ensemble (b) is associated with the first-order auxiliarydynamics, whose potential reads V þ 2Pi;j κijAðri − rjÞ,biased with exp½R t0 ε0ðsÞds�, where

ε0 ¼ 1γT

Xk

�Xi;j

κij∇kA½riðsÞ − rjðsÞ��2

: ðC2Þ

Obtaining the equivalence between (a) and (b) amounts toshowing that their path probabilities are similar. The

corresponding dynamic actions, denoted by AðσÞðtÞ ¼Pk

Rt0 A

ðσÞk ðsÞds for σ ¼ fa; bg, are given by

AðaÞk ¼1

4γT½γ_rk þ∇kV�2 − 1

2γ∇2kV

−1

γT

Xi;j

κij½T∇k −∇kV� ·∇kAðri − rjÞ ðC3Þ

and

AðbÞk ¼1

4γT

�γ_rk þ∇kV þ 2

Xi;j

κij∇kAðri − rjÞ�2

−1

2γ∇2k

�V þ 2

Xi;j

κijAðri − rjÞ�

−1

γT

�Xi;j

κij∇kAðri − rjÞ�2

: ðC4Þ

Expanding AðbÞk in Eq. (C4) and comparing with AðbÞk in

Eq. (C3), it appears that AðaÞ and AðbÞ are indeed equal upto a boundary term proportional to

Pi;j κijfAð½riðtÞ −

rjðtÞ� − A½rið0Þ − rjð0Þ�g which can be neglected at larget: This result establishes the equivalence between ensem-bles (a) and (b).We now turn to demonstrate the equivalence between

two ensembles related to the Vicsek-like dynamics (39).Ensemble (c) is biased with the factor exp½κ R t0 EθðsÞds�,where Eθ is defined in Eq. (40). Ensemble (d) correspondsto the first-order auxiliary dynamics, with the torque givenby ð1þ κÞT , biased with exp½R t0 εθðsÞds�, where

εθ¼ðκμrÞ24Dr

Xi;j;k

T ðθi−θk;ri−rkÞT ðθi−θj;ri−rjÞ: ðC5Þ

The dynamic actions for each ensemble, denoted by

BðσÞðtÞ ¼ Pi R t0 BðσÞi ðsÞds for σ ¼ fc; dg, are given by

BðcÞi ¼1

4Dr

�_θi − μr

Xj

T ðθi − θj; ri − rjÞ�2

þ μr2

∂∂θi

Xj

T ðθi − θj; ri − rjÞ

þ κμr2

Xj

� ∂∂θi þ

μrDr

Xk

T ðθi − θk; ri − rkÞ�T ðθi − θj; ri − rjÞ ðC6Þ


041026-17

and

BðdÞi ¼1

4Dr

�_θi − ð1þ κÞμr

Xj

T ðθi − θj; ri − rjÞ�2

þ ð1þ κÞμr2

∂∂θi

Xj

T ðθi − θj; ri − rjÞ

−ðκμrÞ24Dr

Xj;k

T ðθi − θk; ri − rkÞT ðθi − θj; ri − rjÞ: ðC7Þ

Expanding BðdÞi in Eq. (C6) and comparing with BðcÞi in

Eq. (C7), it appears that BðcÞ and BðdÞ differ only by a termproportional to

Pi;j

_θiT ðθi − θj; ri − rjÞ which can safelybe neglected at large t, thus proving the equivalencebetween ensembles (c) and (d).

[1] S. Toyabe, T. Okamoto, T. Watanabe-Nakayama, H.Taketani, S. Kudo, and E. Muneyuki, NonequilibriumEnergetics of a Single f1-Atpase Molecule, Phys. Rev.Lett. 104, 198103 (2010).

[2] É. Fodor, W.W. Ahmed, M. Almonacid, M. Bussonnier,N. S. Gov, M.-H. Verlhac, T. Betz, P. Visco, and F. vanWijland, Nonequilibrium Dissipation in Living Oocytes,Europhys. Lett. 116, 30008 (2016).

[3] C. Battle, C. Broedersz, M. Fakhri, V. Geyer, J. Howard, C.Schmidt, and F. MacKintosh, Broken Detailed Balance atMesoscopic Scales in Active Biological Systems, Science352, 604 (2016).

[4] F. S. Gnesotto, F. Mura, J. Gladrow, and C. P. Broedersz,Broken Detailed Balance and Non-Equilibrium Dynamicsin Living Systems: A Review, Rep. Prog. Phys. 81, 066601(2018).

[5] P. P. Lele, B. G. Hosu, and H. C. Berg, Dynamics ofMechanosensing in the Bacterial Flagellar Motor, Proc.Natl. Acad. Sci. U.S.A. 110, 11839 (2013).

[6] G. Lan, P. Sartori, S. Neumann, V. Sourjik, and Y. Tu, TheEnergy-Speed-Accuracy Trade-off in Sensory Adaptation,Nat. Phys. 8, 422 (2012).

[7] F. Wang, H. Shi, R. He, R. Wang, R. Zhang, and J. Yuan,Non-equilibrium Effect in the Allosteric Regulation of theBacterial Flagellar Switch, Nat. Phys. 13, 710 (2017).

[8] M. S. e Silva, M. Depken, B. Stuhrmann, M. Korsten, F. C.MacKintosh, and G. H. Koenderink, Active MultistageCoarsening of Actin Networks Driven by Myosin Motors,Proc. Natl. Acad. Sci. U.S.A. 108, 9408 (2011).

[9] T. Sanchez, D. T. N. Chen, S. J. DeCamp, M. Heymann,and Z. Dogic, Spontaneous Motion in HierarchicallyAssembled Active Matter, Nature (London) 491, 431(2012).

[10] L. Blanchoin, R. Boujemaa-Paterski, C. Sykes, and J.Plastino, Actin Dynamics, Architecture, and Mechanics inCell Motility, Physiol. Rev. 94, 235 (2014).

[11] M. Murrell, P. W. Oakes, M. Lenz, and M. L. Gardel,Forcing Cells into Shape: The Mechanics of ActomyosinContractility, Nat. Rev. Mol. Cell Biol. 16, 486 (2015).

[12] S. J. DeCamp, G. S. Redner, A. Baskaran, M. F. Hagan,and Z. Dogic, Orientational Order of Motile Defects inActive Nematics, Nat. Mater. 14, 1110 (2015).

[13] M. E. Cates and J. Tailleur, Motility-Induced Phase Sep-aration, Annu. Rev. Condens. Matter Phys. 6, 219 (2015).

[14] A. P. Solon, Y. Fily, A. Baskaran, M. E. Cates, Y. Kafri, M.Kardar, and J. Tailleur, Pressure Is Not a State Function forGeneric Active Fluids, Nat. Phys. 11, 673 (2015).

[15] M. Nguyen and S. Vaikuntanathan, Design Principles forNonequilibrium Self-Assembly, Proc. Natl. Acad. Sci.U.S.A. 113, 14231 (2016).

[16] É. Fodor, C. Nardini, M. E. Cates, J. Tailleur, P. Visco, andF. van Wijland, How Far from Equilibrium Is ActiveMatter?, Phys. Rev. Lett. 117, 038103 (2016).

[17] A. Murugan and S. Vaikuntanathan, Topologically Pro-tected Modes in Non-equilibrium Stochastic Systems, Nat.Commun. 8, 13881 (2017).

[18] C. Nardini, É. Fodor, E. Tjhung, F. van Wijland, J. Tailleur,and M. E. Cates, Entropy Production in Field Theorieswithout Time-Reversal Symmetry: Quantifying the Non-equilibrium Character of Active Matter, Phys. Rev. X 7,021007 (2017).

[19] M. Nguyen and S. Vaikuntanathan, Dissipation InducedTransitions in Elastic Strings, arXiv:1803.04368.

[20] M. C. Marchetti, J. F. Joanny, S. Ramaswamy, T. B. Liver-pool, J. Prost, M. Rao, and R. A. Simha, Hydrodynamics ofSoft Active Matter, Rev. Mod. Phys. 85, 1143 (2013).

[21] M. Han, J. Yan, S. Granick, and E. Luijten, EffectiveTemperature Concept Evaluated in an Active ColloidMixture, Proc. Natl. Acad. Sci. U.S.A. 114, 7513 (2017).

[22] C. Bechinger, R. Di Leonardo, H. Löwen, C. Reichhardt,G. Volpe, and G. Volpe, Active Particles in Complex andCrowded Environments, Rev. Mod. Phys. 88, 045006(2016).

[23] C. del Junco, L. Tociu, and S. Vaikuntanathan, EnergyDissipation and Fluctuations in a Driven Liquid, Proc.Natl. Acad. Sci. U.S.A. 115, 3569 (2018).

[24] É. Fodor and M. C. Marchetti, The Statistical Physics ofActive Matter: From Self-Catalytic Colloids to LivingCells, Physica (Amsterdam) 504A, 106 (2018).

[25] T. Vicsek, A. Czirók, E. Ben-Jacob, I. Cohen, and O.Shochet, Novel Type of Phase Transition in a System ofSelf-Driven Particles, Phys. Rev. Lett. 75, 1226 (1995).

[26] J. Tailleur and M. E. Cates, Statistical Mechanics ofInteracting Run-and-Tumble Bacteria, Phys. Rev. Lett.100, 218103 (2008).

[27] B. C. van Zuiden, J. Paulose, W. T. M. Irvine, D. Bartolo,and V. Vitelli, Spatiotemporal Order and Emergent EdgeCurrents in Active Spinner Materials, Proc. Natl. Acad.Sci. U.S.A. 113, 12919 (2016).

[28] S. C. Takatori and J. F. Brady, Towards a Thermodynamicsof Active Matter, Phys. Rev. E 91, 032117 (2015).

[29] A. P. Solon, J. Stenhammar, M. E. Cates, Y. Kafri, and J.Tailleur,Generalized Thermodynamics of Motility-Induced


041026-18

https://doi.org/10.1103/PhysRevLett.104.198103https://doi.org/10.1103/PhysRevLett.104.198103https://doi.org/10.1209/0295-5075/116/30008https://doi.org/10.1126/science.aac8167https://doi.org/10.1126/science.aac8167https://doi.org/10.1088/1361-6633/aab3edhttps://doi.org/10.1088/1361-6633/aab3edhttps://doi.org/10.1073/pnas.1305885110https://doi.org/10.1073/pnas.1305885110https://doi.org/10.1038/nphys2276https://doi.org/10.1038/nphys4081https://doi.org/10.1073/pnas.1016616108https://doi.org/10.1038/nature11591https://doi.org/10.1038/nature11591https://doi.org/10.1152/physrev.00018.2013https://doi.org/10.1038/nrm4012https://doi.org/10.1038/nmat4387https://doi.org/10.1146/annurev-conmatphys-031214-014710https://doi.org/10.1038/nphys3377https://doi.org/10.1073/pnas.1609983113https://doi.org/10.1073/pnas.1609983113https://doi.org/10.1103/PhysRevLett.117.038103https://doi.org/10.1038/ncomms13881https://doi.org/10.1038/ncomms13881https://doi.org/10.1103/PhysRevX.7.021007https://doi.org/10.1103/PhysRevX.7.021007https://arXiv.org/abs/1803.04368https://doi.org/10.1103/RevModPhys.85.1143https://doi.org/10.1073/pnas.1706702114https://doi.org/10.1103/RevModPhys.88.045006https://doi.org/10.1103/RevModPhys.88.045006https://doi.org/10.1073/pnas.1713573115https://doi.org/10.1073/pnas.1713573115https://doi.org/10.1016/j.physa.2017.12.137https://doi.org/10.1103/PhysRevLett.75.1226https://doi.org/10.1103/PhysRevLett.100.218103https://doi.org/10.1103/PhysRevLett.100.218103https://doi.org/10.1073/pnas.1609572113https://doi.org/10.1073/pnas.1609572113https://doi.org/10.1103/PhysRevE.91.032117

Phase Separation: Phase Equilibria, Laplace Pressure,and Change of Ensembles, New J. Phys. 20, 075001(2018).

[30] A. P. Solon, J. Stenhammar, M. E. Cates, Y. Kafri, and J.Tailleur, Generalized Thermodynamics of Phase Equilib-ria in Scalar Active Matter, Phys. Rev. E 97, 020602(R)(2018).

[31] T. F. F. Farage, P. Krinninger, and J. M. Brader, EffectiveInteractions in Active Brownian Suspensions, Phys. Rev. E91, 042310 (2015).

[32] R. Wittmann, U. M. B. Marconi, C. Maggi, and J. M.Brader, Effective Equilibrium States in the Colored-NoiseModel for Active Matter II. A Unified Framework forPhase Equilibria, Structure and Mechanical Properties,J. Stat. Mech. 2017, 113208 (2017).

[33] J. A. McLennan, Statistical Mechanics of the Steady State,Phys. Rev. 115, 1405 (1959).

[34] T. S. Komatsu and N. Nakagawa, Expression for theStationary Distribution in Nonequilibrium Steady States,Phys. Rev. Lett. 100, 030601 (2008).

[35] C. Maes, K. Netočný, and B. M. Shergelashvili, Non-equilibrium Relation between Potential and StationaryDistribution for Driven Diffusion, Phys. Rev. E 80, 011121(2009).

[36] C. Maes and K. Neton, Rigorous Meaning of MclennanEnsembles, J. Math. Phys. (N.Y.) 51, 015219 (2010).

[37] H. Touchette, The Large Deviation Approach to StatisticalMechanics, Phys. Rep. 478, 1 (2009).

[38] R. L. Jack and P. Sollich, Large Deviations and Ensemblesof Trajectories in Stochastic Models, Prog. Theor. Phys.Suppl. 184, 304 (2010).

[39] J. P. Garrahan, R. L. Jack, V. Lecomte, E. Pitard, K. vanDuijvendijk, and F. van Wijland, Dynamical First-OrderPhase Transition in Kinetically Constrained Models ofGlasses, Phys. Rev. Lett. 98, 195702 (2007).

[40] L. O. Hedges, R. L. Jack, J. P. Garrahan, and D. Chandler,Dynamic Order-Disorder in Atomistic Models of Struc-tural Glass Formers, Science 323, 1309 (2009).

[41] E. Pitard, V. Lecomte, and F. van Wijland, DynamicTransition in an Atomic Glass Former: A Molecular-Dynamics Evidence, Europhys. Lett. 96, 56002 (2011).

[42] T. Speck, A. Malins, and C. P. Royall, First-Order PhaseTransition in a Model Glass Former: Coupling of LocalStructure and Dynamics, Phys. Rev. Lett. 109, 195703(2012).

[43] T. Bodineau, V. Lecomte, and C. Toninelli, Finite SizeScaling of the Dynamical Free-Energy in a KineticallyConstrained Model, J. Stat. Phys. 147, 1 (2012).

[44] D. T. Limmer and D. Chandler, Theory of Amorphous Ices,Proc. Natl. Acad Sci. U.S.A. 111, 9413 (2014).

[45] T. Nemoto, R. L. Jack, and V. Lecomte, Finite-Size Scalingof a First-Order Dynamical Phase Transition: AdaptivePopulation Dynamics and an Effective Model, Phys. Rev.Lett. 118, 115702 (2017).

[46] J. Tailleur and J. Kurchan, Probing Rare Physical Tra-jectories with Lyapunov Weighted Dynamics, Nat. Phys. 3,203 (2007).

[47] T. Laffargue, K.-D. N. Thu-Lam, J. Kurchan, and J.Tailleur, Large Deviations of Lyapunov Exponents,J. Phys. A 46, 254002 (2013).

[48] F. Cagnetta, F. Corberi, G. Gonnella, and A. Suma, LargeFluctuations and Dynamic Phase Transition in a System ofSelf-Propelled Particles, Phys. Rev. Lett. 119, 158002(2017).

[49] T. Nemoto, É. Fodor, M. E. Cates, R. L. Jack, and J.Tailleur, Optimizing Active Work: Dynamical Phase Tran-sitions, Collective Motion, and Jamming, Phys. Rev. E 99,022605 (2019).

[50] R. Chetrite and H. Touchette, Nonequilibrium Micro-canonical and Canonical Ensembles and Their Equiva-lence, Phys. Rev. Lett. 111, 120601 (2013).

[51] D. S. Dean, Langevin Equation for the Density of a Systemof Interacting Langevin Processes, J. Phys. A. 29, L613(1996).

[52] V. Démery and D. S. Dean, Perturbative Path-IntegralStudy of Active- and Passive-Tracer Diffusion in Fluctu-ating Fields, Phys. Rev. E 84, 011148 (2011).

[53] V. Démery, O. Bénichou, and H. Jacquin, GeneralizedLangevin Equations for a Driven Tracer in Dense SoftColloids: Construction and Applications, New J. Phys. 16,053032 (2014).

[54] T. Harada and S.-i. Sasa, Equality Connecting EnergyDissipation with a Violation of the Fluctuation-ResponseRelation, Phys. Rev. Lett. 95, 130602 (2005).

[55] D. Mizuno, C. Tardin, C. F. Schmidt, and F. C.MacKintosh, Nonequilibrium Mechanics of Active Cytos-keletal Networks, Science 315, 370 (2007).

[56] É. Fodor, M. Guo, N. S. Gov, P. Visco, D. A. Weitz, and F.van Wijland, Activity-Driven Fluctuations in Living Cells,Europhys. Lett. 110, 48005 (2015).

[57] H. Turlier, D. A. Fedosov, B. Audoly, T. Auth, N. S. Gov,C. Sykes, J.-F. Joanny, G. Gompper, and T. Betz, Equi-librium Physics Breakdown Reveals the Active Nature ofRed Blood Cell Flickering, Nat. Phys. 12, 513 (2016).

[58] W.W. Ahmed, É. Fodor, M. Almonacid, M. Bussonnier,M.-H. Verlhac, N. S. Gov, P. Visco, F. van Wijland, and T.Betz, Active Mechanics Reveal Molecular-Scale ForceKinetics in Living Oocytes, Biophys. J. 114, 1667 (2018).

[59] T. R. Gingrich, G. M. Rotskoff, and J. M. Horowitz, In-ferring Dissipation from Current Fluctuations, J. Phys. A50, 184004 (2017).

[60] É. Roldán, J. Barral, P. Martin, J. M. R. Parrondo, and F.Jülicher, Arrow of Time in Active Fluctuations,arXiv:1803.04743.

[61] I. A. Martínez, G. Bisker, J. M. Horowitz, and J. M. R.Parrondo, Inferring Broken Detailed Balance in theAbsence of Observable Currents, Nat. Commun. 10,3542 (2019).

[62] J. Li, J. M. Horowitz, T. R. Gingrich, and N. Fakhri,Quantifying Dissipation Using Fluctuating Currents,Nat. Commun. 10, 1666 (2019).

[63] C. Giardinà, J. Kurchan, and L. Peliti, Direct Evaluation ofLarge-Deviation Functions, Phys. Rev. Lett. 96, 120603(2006).

[64] P. I. Hurtado and P. L. Garrido, Test of the AdditivityPrinciple for Current Fluctuations in a Model of HeatConduction, Phys. Rev. Lett. 102, 250601 (2009).

[65] T. Nemoto, F. Bouchet, R. L. Jack, and V. Lecomte,Population-Dynamics Method with a MulticanonicalFeedback Control, Phys. Rev. E 93, 062123 (2016).


041026-19

https://doi.org/10.1088/1367-2630/aaccddhttps://doi.org/10.1088/1367-2630/aaccddhttps://doi.org/10.1103/PhysRevE.97.020602https://doi.org/10.1103/PhysRevE.97.020602https://doi.org/10.1103/PhysRevE.91.042310https://doi.org/10.1103/PhysRevE.91.042310https://doi.org/10.1088/1742-5468/aa8c37https://doi.org/10.1103/PhysRev.115.1405https://doi.org/10.1103/PhysRevLett.100.030601https://doi.org/10.1103/PhysRevE.80.011121https://doi.org/10.1103/PhysRevE.80.011121https://doi.org/10.1063/1.3274819https://doi.org/10.1016/j.physrep.2009.05.002https://doi.org/10.1143/PTPS.184.304https://doi.org/10.1143/PTPS.184.304https://doi.org/10.1103/PhysRevLett.98.195702https://doi.org/10.1126/science.1166665https://doi.org/10.1209/0295-5075/96/56002https://doi.org/10.1103/PhysRevLett.109.195703https://doi.org/10.1103/PhysRevLett.109.195703https://doi.org/10.1007/s10955-012-0458-1https://doi.org/10.1073/pnas.1407277111https://doi.org/10.1103/PhysRevLett.118.115702https://doi.org/10.1103/PhysRevLett.118.115702https://doi.org/10.1038/nphys515https://doi.org/10.1038/nphys515https://doi.org/10.1088/1751-8113/46/25/254002https://doi.org/10.1103/PhysRevLett.119.158002https://doi.org/10.1103/PhysRevLett.119.158002https://doi.org/10.1103/PhysRevE.99.022605https://doi.org/10.1103/PhysRevE.99.022605https://doi.org/10.1103/PhysRevLett.111.120601https://doi.org/10.1088/0305-4470/29/24/001https://doi.org/10.1088/0305-4470/29/24/001https://doi.org/10.1103/PhysRevE.84.011148https://doi.org/10.1088/1367-2630/16/5/053032https://doi.org/10.1088/1367-2630/16/5/053032https://doi.org/10.1103/PhysRevLett.95.130602https://doi.org/10.1126/science.1134404https://doi.org/10.1209/0295-5075/110/48005https://doi.org/10.1038/nphys3621https://doi.org/10.1016/j.bpj.2018.02.009https://doi.org/10.1088/1751-8121/aa672fhttps://doi.org/10.1088/1751-8121/aa672fhttps://arXiv.org/abs/1803.04743https://doi.org/10.1038/s41467-019-11051-whttps://doi.org/10.1038/s41467-019-11051-whttps://doi.org/10.1038/s41467-019-09631-xhttps://doi.org/10.1103/PhysRevLett.96.120603https://doi.org/10.1103/PhysRevLett.96.120603https://doi.org/10.1103/PhysRevLett.102.250601https://doi.org/10.1103/PhysRevE.93.062123

[66] U. Ray, G. Kin-Lic Chan, and D. T. Limmer, ExactFluctuations of Nonequilibrium Steady States fromApproximate Auxiliary Dynamics, Phys. Rev. Lett. 120,210602 (2018).

[67] K. Klymko, P. L. Geissler, J. P. Garrahan, and S. Whitelam,Rare Behavior of Grow

Documents

PHYSICAL REVIEW X 9, 041026 (2019)...dynamical bias. The explicit form of the bias is inspired by the results of Sec. II connecting dissipation to many-body interactions. Using analytical