Structural Nonequilibrium Forces in Driven Colloidal Systems

Nico C. X. Stuhlmüller, Tobias Eckert, Daniel de las Heras,* and Matthias Schmidt†

Theoretische Physik II, Physikalisches Institut, Universität Bayreuth, D-95440 Bayreuth, Germany

(Received 22 December 2017; revised manuscript received 13 July 2018; published 31 August 2018)

We identify a structural one-body force field that sustains spatial inhomogeneities in nonequilibriumoverdamped Brownian many-body systems. The structural force is perpendicular to the local flowdirection, it is free of viscous dissipation, it is microscopically resolved in both space and time, and it canstabilize density gradients. From the time evolution in the exact (Smoluchowski) low-density limit,Brownian dynamics simulations, and a novel power functional approximation, we obtain a quantitativeunderstanding of viscous and structural forces, including memory and shear migration.

DOI: 10.1103/PhysRevLett.121.098002

It is a very significant challenge of statistical physics torationalize and predict nonequilibrium structure formationfrom a microscopic starting point. Primary examplesinclude shear banding [1–3], where spatial regions ofdifferent shear rates coexist, and laning transitions inoppositely driven colloids [4,5], where regions of differentflow direction occur, as well as migration effects ininhomogeneous shear flow [6–11]. In computer simula-tions, discriminating true steady states from slow initialtransients can be difficult [12,13]. Often the nonequili-brium structuring effects have associated long timescalesand strong correlations [14,15]. The underlying equilib-rium phase diagram and bulk structure might already becomplex and interfere with the genuine nonequilibriumeffects [16,17].To identify commonalities of all of the above situations,

we investigate here a representative model situation. We putforward a systematic classification of the occurring non-equilibrium forces and identify a structural force compo-nent, which is able to sustain density gradients withoutcreating dissipation. The structural force is solely due to theinteraction between the particles, and it is hence of a naturedifferent than that of the lift forces in hydrodynamics. Werationalize our findings by constructing an explicit powerfunctional approximation (15) below.We restrict ourselves to overdamped Brownian dynamics

and consider the microscopically resolved position- andtime-dependent one-body density ρðr; tÞ and one-bodycurrent Jðr; tÞ. Both fields are related via the (exact)continuity equation ∂ρ=∂t ¼ −∇ · J; here ∇ indicatesthe derivative with respect to position r. The velocityprofile is simply the ratio vðr; tÞ ¼ Jðr; tÞ=ρðr; tÞ.The exact time evolution, with no hydrodynamic inter-

actions present, can then be expressed by the one-bodyforce balance equation

γv ¼ fint þ fext − kBT∇ ln ρ; ð1Þ

where γ is the single-particle friction constant againstthe (implicit) solvent, fintðr; tÞ is the internal forcefield, fextðr; tÞ is a one-body external force field that, ingeneral, drives the system out of equilibrium, and−kBT∇ ln ρðr; tÞ≡ faidðr; tÞ is the adiabatic ideal gas con-tribution due to the free thermal diffusion; here kB is theBoltzmann constant and T is absolute temperature. Theinternal force field fintðr; tÞ arises from the interparticleinteraction potential uðrNÞ, where rN ≡ r1;…; rN denotesthe set of all position coordinates, and it can be expressedas an average over the interparticle one-body forcedensity “operator”

Fint ¼ −X

i

δðr − riÞ∇iuðrNÞ; ð2Þ

where the sum is over all particles, δð·Þ indicates theDirac distribution, and ∇i denotes the derivative withrespect to ri. Using the probability distribution ΨðrN; tÞof finding microstate rN at time t, the average is builtaccording to

fintðr; tÞ ¼ ρðr; tÞ−1Z

drNΨðrN; tÞFint; ð3Þ

where the normalization is performed using the time-dependent one-body density, defined as the average

ρðr; tÞ ¼Z

drNΨðrN; tÞρ; ð4Þ

where ρ ¼P

iδðr − riÞ is the density operator.The internal force field (3) can be further systematically

decomposed [18,19] into adiabatic excess (faxc) and super-adiabatic one-body contributions (fsup), according to

fint ¼ faxc þ fsup: ð5Þ

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Here the excess (over ideal gas) adiabatic force field faxc isthe internal force field in a hypothetical equilibrium systemthat has the same density profile as the real nonequilibriumsystem at time t. Hence,

faxcðr; tÞ ¼ ρðr; tÞ−1Z

drNΨad;tðrNÞFint; ð6Þ

where the average is over a canonical equilibrium distri-bution Ψad;tðrNÞ for the (unchanged) interparticle interac-tion potential uðrNÞ, but under the influence of ahypothetical external (“adiabatic”) one-body potentialVad;tðrÞ, which is constructed in order to yield in equilib-rium the same one-body density as occurs in the dynamicalsystem at time t [18,19], i.e.,

ρðr; tÞ ¼ ρad;tðrÞ≡Z

drNΨad;tðrNÞρ: ð7Þ

The excess adiabatic force field is hence uniquely specifiedby (6) and (7); computer simulations permit direct access[19]. The force splitting (5) then defines the superadiabaticforce field.Here we demonstrate that fsupðr; tÞ further splits natu-

rally and systematically into different contributions, whichcorrespond to different physical effects. We have shownbefore that fsupðr; tÞ contains viscous force contributions[20]. These are of dissipative nature in that they workagainst the colloidal motion (i.e., antiparallel to the flowdirection). Here we focus on the component of the super-adiabatic force, which is perpendicular to the local flowdirection evðr; tÞ, where v ¼ jvjev; note that also Jkev. Wehence define the (normal) structural force field f⊥supðr; tÞ asthe component perpendicular to the local flow direction,

f⊥sup ¼ fsup − fsup · evev: ð8Þ

In contrast to the viscous force, the structural force isnondissipative, since the associated power density vanishesidentically everywhere, J · f⊥sup ¼ jJjev · f⊥sup ¼ 0.The structural force plays a vital role in nonequilibrium,

as it can stabilize density gradients. In order to demonstratethis effect, we consider a two-dimensional toy system ofGaussian core particles [21] in an inhomogeneous externalshear field. The pair interaction potential is ϵ expð−r2=σ2Þ,where r is the distance between both particles, and ϵ > 0 isthe energy cost at zero separation. We use ϵ and σ as theenergy and the length scale, respectively. N particles arelocated in a square box of length L with periodic boundaryconditions and (unit vector) directions ex and ey along thesquare box. The driving occurs along ex according to aninhomogeneous external shear field,

fextðy; tÞ ¼ f0 sinð2πy=LÞθðtÞex; ð9Þ

where f0 is a constant that controls the magnitude of thedriving force, and θð·Þ indicates the Heaviside (step)function, such that the force is instantaneously switchedon at time t ¼ 0. Ultimately, the system reaches a steadystate with a density gradient along ey, i.e., ρðr;∞Þ ¼ ρðyÞ,as we will see below. The density gradient is then solelysustained by the structural force f⊥sup.In order to study the system on the Fokker-Planck level,

we solve numerically the exact many-body Smoluchowskiequation (SE) for overdamped Brownian motion. The timeevolution of the probability distribution Ψ is given exactlyby the many-body continuity equation

∂ΨðrN; tÞ∂t ¼ −

X

i

∇i · viΨðrN; tÞ: ð10Þ

where vi is the velocity of particle i, given by the forcebalance

γvi ¼ −∇i½uðrNÞ þ kBT lnΨðrN; tÞ& þ fextðri; tÞ: ð11Þ

We solve (10) and (11) numerically using a (standard)operator splitting approach [22]. Each spatial coordinate isdiscretized in increments Δx ¼ σ=5; we use a time stepΔt=τ ¼ 5 × 10−3 with timescale τ ¼ σ2γ=ϵ. This methodprovides exact results of the nonequilibrium dynamics upto numerical inaccuracies. For d space dimensions, thedimension of configuration space is Nd, which limits theapplicability of the method to systems with small numbersof particles. Here, we consider N ¼ 2 in d ¼ 2, whichrenders the numerical field Nd ¼ 4 dimensional; hence,including time, we solve a 4þ 1-dimensional numericalproblem. As we show, despite the limited number ofparticles, all relevant forces are present.In order to analyze larger systems, we use Brownian

dynamics (BD), i.e., integrating in time the Langevinequation of motion, which corresponds to (10) and (11),

γdriðtÞdt

¼ −∇iuðrNÞ þ fextðri; tÞ þ χ iðtÞ; ð12Þ

where χ i is a delta-correlated Gaussian random force. Weuse a time step dt=τ ¼ 10−4 and histogram bins of sizeΔx ¼ σ=20. Density and force density profiles are obtainedby averaging over a total time of ∼109τ in steady state.In both SE and BD we use the iterative scheme of

Refs. [19,23] to construct the adiabatic external potentialVad;tðrÞ. The superadiabatic force then follows immediatelyfrom (5) since both fint and faxc can be directly calculated(sampled) in SE (BD). We have used the recently developedforce sampling method [24] to improve the sampling of thedensity profile in BD. We expect, on principal grounds, thatthe SE and BD results agree (for identical values of systemsize and particle number, here N ¼ 2).

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A schematic showing all forces in steady state is shownin Fig. 1(a). The stationary density and force profilesobtained by solving the Smoluchowski equation (N ¼ 2)and using BD simulations (N ¼ 25) are shown in Figs. 1(b)and 1(c), respectively. A net flow exists along þex fory < L=2, since the external force is only partially balancedby a superadiabatic force of viscous nature fksup along −ex(see middle panels). Hence, in this situation, ev ¼ ex. Theviscous force has roughly the same shape as the externalforce but with reversed direction. The inhomogeneousexternal force (9) fextðyÞ creates a density gradient in ey(see top panels), since the particles migrate to the low shearrate regions. The adiabatic ideal (i.e., diffusive) andadiabatic excess (i.e., due to internal interaction) forcesact along ey and both try to relax the density gradient (seebottom panels). Both adiabatic forces are, however, exactlybalanced by a structural superadiabatic force f⊥sup. Thepresence of the structural force hence renders the inhomo-geneities of the density profile stationary in time.We next analyze the system more systematically by

comparing the behavior of the structural force to that of theviscous force. We show in Fig. 2(a) the amplitudes of theviscous force fksup and the structural force f⊥sup (measuredfrom maximum to baseline) as a function of the amplitudeof the external driving f0. For small driving, the viscousforce scales linearly with f0. This behavior is expected,since for weak driving the velocity is proportional to the

strength of the driving and the viscous force is proportionalto the velocity (Newtonian rheology). The structuralforce, on the other hand, scales quadratically with f0and hence also with the velocity in the small drivingregime. Both forces saturate for high values of f0 [seethe inset in Fig. 2(a)]. Figure 2(b) shows fksup and f⊥sup as afunction of the average density ρ0, revealing again profounddifferences between viscous and structural forces. Theviscous force increases linearly with increasing ρ0 at lowdensities and it saturates at high densities. In contrast, thestructural force is nonmonotonic and exhibits a maximumat an intermediate density.We rationalize these findings by developing a theory

within the power functional approach [18]. Here theadiabatic and the superadiabatic contributions to the inter-nal force field (5) are obtained via functional differentiationof two generating functionals,

faxcðr; tÞ ¼ −∇ δFexc½ρ&δρðr; tÞ

; ð13Þ

fsupðr; tÞ ¼ −δPexc

t ½ρ; J&δJðr; tÞ

: ð14Þ

Here Fexc½ρ& is the intrinsic excess (over ideal) Helmholtzequilibrium free energy functional of density functionaltheory, and Pexc

t ½ρ; J& is the excess (again over ideal)

(a) (b) (c)

FIG. 1. (a) Illustration of the setup of a sinusoidal inhomogeneous shear flow induced by the external force field (9). Shown are thesuperadiabatic viscous force, which opposes the externally induced flow, the adiabatic excess, and ideal forces, which tend to relax thedensity gradient, as well as the superadiabatic structural nonequilibrium force, which restores the force balance in the y direction.(b) Steady-state density and force profiles obtained by numerically solving the Smoluchowski equation in a system with N ¼ 2,L=σ ¼ 10, and kBT=ϵ ¼ 0.4. (Top) Density profile for average density ρ0 ¼ N=L2 ¼ 0.02σ−2 as a function of y. (Middle) Forces actingalong ex as a function of y: external force fext of imposed amplitude f0 ≈ 0.25ϵ=σ (green dashed line) and superadiabatic viscous force(solid blue line) fksup. (Bottom) Forces acting along ey as a function of y: adiabatic ideal (diffusion) faid (orange dashed line), adiabaticexcess faxc (violet dotted line), and superadiabatic structural force f⊥sup (solid red line). (c) Same as (b) but for a system with N ¼ 25,L=σ ¼ 10, ρ0 ¼ 0.25σ−2, kBT=ϵ ¼ 1, and f0 ≈ 3.14ϵ=σ. Data obtained using BD simulations. Forces marked with an asterisk in (b) and(c) have been multiplied by 103 for clarity. The black circles in (b) and (c) indicate the theoretical prediction (18) of the superadiabaticviscous and structural forces.

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superadiabatic functional of power functional theory [18].Dynamical density functional theory (DDFT) [26–28] isobtained by setting Pexc

t ¼ 0; hence, no superadiabaticforces (neither viscous nor structural) occur in DDFT. InRef. [20], using a change of variables in power functionaltheory from the current J to the velocity gradient ∇v, it isshown that the excess superadiabatic functional can berepresented as a functional of ∇v. Following the sameline, we consider a temporally non-Markovian but spa-tially local form,

Pexct ¼

Zdr!Z

t

0dt0ntt0ð∇ × vÞ · ð∇ × v0Þ

−Z

t

0dt0

Zt

0dt00mtt0t00ð∇ · vÞð∇ × v0Þ · ð∇ × v00Þ

";

ð15Þ

where we have used the shorthand notation v0 ¼ vðr; t0Þ andv00 ¼ vðr; t00Þ. The factors ntt0 and mtt0t00 are density-depen-dent temporal convolution kernels; the subscripts indicatethe time arguments. Here the dependence is on thedifferences t − t0 and t − t00; the specific form of the kernelswill depend on the form of the interparticle interactionpotential uðrNÞ. In (15), we have left away bilinear andhigher contributions in ∇ · v [20]; these are important forcompressional flow, but not for the present shear setup.Furthermore, as ρ is practically constant in the casesconsidered, contributions in ∇ρ have also been omitted.The superadiabatic force field is then obtained by

inserting (15) into (14), where the derivative is carriedout at fixed density, and hence δ=δJ ¼ ρ−1δ=δv.Furthermore, the spatial derivatives in (15) can be suitablyrearranged by (spatial) integration by parts. The resultingforce density is

ρfsupðr; tÞ ¼Z

t

0dt0∇ · ntt0∇v0

−Z

t

0dt0

Zt

0dt00∇mtt0t00ð∇ × v0Þ · ð∇ × v00Þ:

ð16Þ

In steady state and for the case of constant density profile,(16) reduces to

ρfsupðrÞ ¼ η∇2v − χ∇ð∇ × vÞ2; ð17Þ

where the prefactors η ¼ limt→∞Rt0 dt

0ntt0 and χ ¼limt→∞

Rt0 dt

0 R t0 dt

00mtt0t00 are moments of the convolutionkernels, which depend on the overall density. We canidentify η with the shear viscosity [20], such that the firstterm in (17) represents the viscous force. From carrying outthe normal projection (8), the second contribution in (17)yields the structural force f⊥supðr; tÞ. Note that, while theforms (16) and (17) could possibly be postulated based onsymmetry considerations alone, in the current framework,the generator (15) constitutes the more fundamental object,as the form of the force field follows via the functionalderivative (14). Note that for the ideal gas (u ¼ 0) Pexc

t ¼ 0and fsup ¼ 0 by construction [18].In accordance with our numerical results, we assume that

the flow field is dominated by the external force; hence, weapproximate (1) by vðy; tÞ ≈ fextðy; tÞ=γ. Insertion of (9)into (17) then yields

fsupðyÞ ¼ −f0ηk2

γρsinðkyÞex þ

f20χk3

γρsinð2kyÞey; ð18Þ

where k ¼ 2π=L. In Figs. 1(b) and 1(c) we show thecomparison between the predicted (black circles) and thecomputed superadiabatic forces using SE and BD (solid

(a) (b) (c)

FIG. 2. Scaled amplitude of the superadiabatic forces fα'sup ¼ 103fαsupσ=ϵ with α ∈ fk;⊥g, as a function of (a) the amplitude of theexternal driving force f0, (b) the average density ρ0 ¼ N=L2, and (c) the time t. (a) Results for L=σ ¼ 10, N ¼ 2, and kBT=ϵ ¼ 0.4. (b)Results in a system with L=σ ¼ 10, f0 ≈ 3.14ϵ=σ, kBT=ϵ ¼ 1, and varying N. (c) Note the logarithmic scale in the main plot. Theexternal driving force is switched on at t ¼ 0. Results for L=σ ¼ 10, N ¼ 2, kBT=ϵ ¼ 0.4, and f0 ≈ 0.25ϵ=σ. In all panels, the viscous(structural) superadiabatic force is represented with blue circles (red squares), as indicated in the upper legend. Full (empty) symbolscorrespond to SE (BD) calculations. The blue dashed (red dotted) line indicates a linear (quadratic) fit to the data, as detailed in theSupplemental Material [25]. Solid lines are guides to the eye. The inset in each figure shows a linear plot of the same quantities as thosein the main plot, but over an extended region.

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lines). The values of the response coefficients η and χ[cf. (18)] have been adjusted to fit each amplitude; see theSupplemental Material [25] for details.The theory then predicts the shape of both viscous and

structural forces without further adjustable parameters, andit is in excellent agreement with the results from SE andBD. In addition, the linear (quadratic) scaling of the viscous(structural) force with the velocity [Fig. 2(a)] is alsoaccounted by the theory [cf. (17)]. Because of the saturationof both superadiabatic forces [inset of Fig. 2(a)], it isnecessary to analyze very small driving to obtain the correctscaling. Also, at low average density ρ0 the viscous force isproportional to ρ0 [see Fig. 2(b)], which according to (18)implies η ∝ ρ20, as expected [8]. See the SupplementalMaterial [25] for results for strong driving conditions andfor a theoretical prediction of the density profile.Memory plays a vital role in nonequilibrium systems, as

we show in Fig. 2(c), by investigating the transient timeevolution after switching on the driving at t ¼ 0 [cf. (9)].Both superadiabatic force contributions vanish in equilib-rium (t ≤ 0) and saturate in steady state (t → ∞). At shorttimes after switching on the driving force, the viscous(structural) force is linear (quadratic) in t, in full agreementwith the non-Markovian form of (15). See theSupplemental Material [25] for an analysis of the scalingof the amplitude of the forces with wave number k. Here wehave still taken ev ¼ ex as a (very good) approximation.We conclude that the structural force is a primary

candidate for a universal mechanism that leads to non-equilibrium structure formation. Examples of systems wherea force acts perpendicular to the flow include shear banding[1–3], colloidal lane formation [4,5], and effective inter-actions in active spinning particles [29,30]. The theory thatwe present here operates in a self-contained way on the one-body level of correlation functions and hence is differentfrom the approach of Refs. [7,11], where a dynamic closureon the two-body level via modeling of Brownian “scatter-ing” is postulated. Note that the treatment of Refs. [7,11]relies on the density distribution as the fundamental variable;in contrast, our theory predicts the behavior of the systemdirectly from the velocity field [cf. (16) and (17)].We have focused here on the viscous and structural

contributions to the superadiabatic force in shear-type flow.In compressional flow, where ∇ · v ≠ 0, further force con-tributions can occur. Calculating the values for the transportcoefficients η and χ within the current theory is desirable,based, e.g., on the two-body level of correlation functions[31]. Furthermore, interesting research tasks for the futureinclude investigating the effects of inhomogeneous temper-ature fields [32] and the possible emergence of time-periodicstates (“time crystals” [33]) in inertial dynamics [34].

This work is supported by the German ResearchFoundation (DFG) via SCHM 2632/1-1.

N. C. X. S. and T. E. contributed equally to this work.

*delasheras.daniel@gmail.com; www.danieldelasheras.com†Matthias.Schmidt@uni-bayreuth.de

[1] J. K. G. Dhont, Phys. Rev. E 60, 4534 (1999).[2] J. K. G. Dhont, M. P. Lettinga, Z. Dogic, T. A. J. Lenstra, H.

Wang, S. Rathgeber, P. Carletto, L. Willner, H. Frielinghaus,and P. Lindner, Faraday Discuss. 123, 157 (2003).

[3] H. Jin, K. Kang, K. H. Ahn, and J. K. G. Dhont, Soft Matter10, 9470 (2014).

[4] J. Chakrabarti, J. Dzubiella, and H. Löwen, Phys. Rev. E 70,012401 (2004).

[5] C. W. Wächtler, F. Kogler, and S. H. L. Klapp, Phys. Rev. E94, 052603 (2016).

[6] M.Krüger and J.M.Brader, Europhys.Lett.96, 68006 (2011).[7] J. M. Brader and M. Krüger, Mol. Phys. 109, 1029 (2011).[8] J. Reinhardt, F. Weysser, and J. M. Brader, Europhys. Lett.

102, 28011 (2013).[9] A. A. Aerov and M. Krüger, J. Chem. Phys. 140, 094701

(2014).[10] A. A. Aerov and M. Krüger, Phys. Rev. E 92, 042301 (2015).[11] A. Scacchi, M. Krüger, and J. M. Brader, J. Phys. Condens.

Matter 28, 244023 (2016).[12] J. Zausch, J. Horbach, M. Laurati, S. U. Egelhaaf, J. M.

Brader, Th. Voigtmann, and M. Fuchs, J. Phys. Condens.Matter 20, 404210 (2008).

[13] Ch. J. Harrer, D. Winter, J. Horbach, M. Fuchs, and Th.Voigtmann, J. Phys. Condens. Matter 24, 464105 (2012).

[14] R. Ni, M. A. Cohen Stuart, and P. G. Bolhuis, Phys. Rev.Lett. 114, 018302 (2015).

[15] U. Zimmermann, F. Smallenburg, and H. Löwen, J. Phys.Condens. Matter 28, 244019 (2016).

[16] D. de las Heras, N. Doshi, T. Cosgrove, J. Phipps, D. I.Gittins, J. S. van Duijneveldt, and M. Schmidt, Sci. Rep. 2,789 (2012).

[17] B. Ruzicka, E. Zaccarelli, L. Zulian, R. Angelini, M.Sztucki, A. Moussad, T. Narayanan, and F. Sciortino,Nat. Mater. 10, 56 (2011).

[18] M. Schmidt and J. M. Brader, J. Chem. Phys. 138, 214101(2013).

[19] A. Fortini, D. de las Heras, J. M. Brader, and M. Schmidt,Phys. Rev. Lett. 113, 167801 (2014).

[20] D. de las Heras and M. Schmidt, Phys. Rev. Lett. 120,028001 (2018).

[21] See, e.g., A. J. Archer and R. Evans, Phys. Rev. E 64,041501 (2001); A. J. Archer, J. Phys. Condens. Matter 17,1405 (2005).

[22] W. H. Press, S. A. Teukolsky, W. T. Vetterling, and B. P.Flannery,NumericalRecipes: TheArt of ScientificComputing,3rd ed. (Cambridge University Press, Cambridge, England,2007).

[23] E. Bernreuther and M. Schmidt, Phys. Rev. E 94, 022105(2016).

[24] D. de las Heras and M. Schmidt, Phys. Rev. Lett. 120,218001 (2018).

[25] See Supplemental Material at http://link.aps.org/supplemental/10.1103/PhysRevLett.121.098002 for de-scriptions of superadiabatic force profiles under strongdriving conditions, scaling of the superadiabatic forces withthe inverse box length, fitting details, and theoreticalprediction of the density profile.

[26] R. Evans, Adv. Phys. 28, 143 (1979).

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[27] U. M. B. Marconi and P. Tarazona, J. Chem. Phys. 110,8032 (1999).

[28] A. J. Archer and R. Evans, J. Chem. Phys. 121, 4246 (2004).[29] N. H. P. Nguyen, D. Klotsa, M. Engel, and S. C. Glotzer,

Phys. Rev. Lett. 112, 075701 (2014).[30] J. L. Aragones, J. P. Steimel, and A. Alexander-Katz, Nat.

Commun. 7, 11325 (2016).[31] J. M. Brader and M. Schmidt, J. Chem. Phys. 139, 104108

(2013); 140, 034104 (2014); J. Phys. Condens. Matter 27,194106 (2015).

[32] For recent work, see, e.g., G. Falasco and K. Kroy, Phys.Rev. E 93, 032150 (2016).

[33] J. Zhang et al., Nature (London) 543, 217 (2017);R. Moessner and S. L. Sondhi, Nat. Phys. 13, 424(2017).

[34] Power functional theory for quantum and for classicalNewtonian dynamics is formulated, respectively, in M.Schmidt, J. Chem. Phys. 143, 174108 (2015); 148,044502 (2018).

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Supplemental Material

Structural nonequilibrium forces in driven colloidal systems

Nico C. X. Stuhlmuller,1 Tobias Eckert,1 Daniel de las Heras,1 and Matthias Schmidt1

1Theoretische Physik II, Physikalisches Institut, Universitat Bayreuth, D-95440 Bayreuth, Germany

Superadiabatic force profiles under strong driving

conditions

The examples presented in the main text address thebehaviour of the system in response to relatively weak ex-ternal perturbations. Here we show that under strong ex-ternal driving conditions, both superadiabatic force pro-files change significantly. We show in Supplemental Fig. 1the viscous and the structural superadiabatic forces forboth weak and for strong external driving conditions.We find that the direction of both force fields remainsunchanged upon increasing the driving. Hence, for thecases considered here, the viscous force opposes alwaysthe external force and the structural force sustains (coun-teracts) the density gradient. However, although the onlydi↵erence between both driving conditions is the magni-tude of the external driving, the superadiabatic (position-dependent) force profiles change significantly their shapedepending on the strength of the driving. The powerfunctional approximation presented in Eq. (15) of themain text is intended to deal with low/moderate drivingand does not fully reproduce the characteristics of thesuperadiabatic forces at high driving conditions. Notethat the magnitude of the superadiabatic force is rela-tively small as compared to the magnitude of the exter-nal force, even for the case of weak driving (Fig. 1 of themain text). Therefore, the velocity field basically followsthe external driving, and the superadiabatic forces arethen given by Eq. (18) of the main text. That is, thetheory does not reproduce the shape change of the su-peradiabatic forces with the magnitude of the externaldriving (Supplemental Fig. 1). This was expected sincein Eq. (15) we have only incorporated the first two termsof the expansion of P exc

t in powers of rv. Therefore, oneexpects Eq. (15) to be valid under weak driving condi-tions. The correct treatment of strong driving conditionsrequires the addition of higher order terms to Eq. (15). Itis straightforward to incorporate such terms by rewritingthe excess superadiabatic functional as

P exc =

Zdrh⌘(r⇥v)2 � �(r·v)(r⇥v)2

ih(rv), (1)

where h(rv) is a scalar-valued function, which we chooseto have the following form:

h(rv) =1

1 + c0

(r⇥ v)2, (2)

where c0

is a positive constant which controls thestrength of the influence of the flow; setting c

0

= 0 re-covers the form of Eq. (15) of the main text. In Eq. (1)we have assumed steady state conditions and thereforeeliminated the time integrals. The first two terms of a se-ries expansion of Eq. (1) in rv have the same functionalstructure as that presented in Eq. (15) of the main text.In addition, Eq. (1) incorporates higher order contribu-tions in rv. We find that the superadiabatic force profilethat result upon functional di↵erentiation of Eq. (1) is inexcellent agreement with the simulation results both forweak and for strong driving conditions (see black circlesin Supplemental Fig. 1). For both cases shown in thefigure, we have set c

0

= 3.5 ·10�3 ⌧2 and fit the values of⌘ and �. The values of ⌘ and of � determine the overallamplitude of the viscous and of the structural superadia-batic force, respectively. We find that the values of ⌘ and� seem to depend on the magnitude of the external driv-ing, which is a clear indication that Eq. (1) constitutesonly a first approximation.

Scaling of the superadiabatic forces with the inverse

box length

We present a scaling analysis of the magnitude of theviscous and of the structural superadiabatic force fieldswith the amplitude of the external driving and the overalldensity in the main text. Here, we focus on the scalingof these force fields with the wavenumber k = 2⇡/L thatcharacterizes the inhomogeneous shear profile, cf. Eq. (9)of the main text. In practice we change k by changingthe box size L, ensuring that the other relevant variablesare unchanged, e.g. keeping the mean density constantby accordingly modifying the number of particles, N .Our theory, cf. Eq. (18) of the main text, predicts dif-

ferent types of scaling with k for the viscous (quadraticin k) and for the structural (cubic in k) superadiabaticforces. We fix the amplitude of the external driving(f

0

⇡ 0.30✏/�) and the average density of the system(⇢

0

�2 = 0.02). Therefore, according to Eq. (18) of themain text, the only remaining dependence of the ampli-tude of the superadiabatic forces is on k. In SupplementalFig. 2 we show the amplitude of the superadiabatic forcesas a function of the wavenumber. The smallest systemcorresponds to N = 2 and H/� = 10, and the largest oneto N = 100 and H/� = 70.7. Each curve shows three dif-ferent regimes. For small boxes (large wavenumber) weobserve a saturation of the superadiabatic forces, sim-

2

Supplemental Fig. 1. Steady-state density and force profiles according to BD simulations in a system with N = 25, L/� = 10,

and kB

T/✏ = 1.0. Results are shown for two strengths of the external driving force: (a) f0 ⇡ 1.5✏/�, weak external driving, and

(b) f0 ⇡ 15✏/�, strong external driving. (Top panels) Density profile, measured from the average density ⇢0 = N/L2= 0.25��2

,

as a function of y. (Middle panel) Superadiabatic viscous force acting along e

x

as a function of y obtained by BD simulations

(solid-blue line) and theoretically (black circles). (Bottom panel) Superadiabatic structural force acting along e

y

as a function

of y obtained by BD simulations (solid-red line) and theoretically (black circles).

ilar to the saturation e↵ect observed by increasing theexternal driving or the density (see Fig. 2 of the maintext). At intermediate wavenumbers the forces scale lin-early with k. Finally, for long boxes (small wavenumbers)the curves depart significantly from a linear behaviour.Note that both curves must pass through the origin ofcoordinates. The low driving conditions that are implicitin going to small values of k render the analysis of thedata di�cult. Nevertheless, the scaling of the viscousforce is fully consistent with the quadratic prediction ofthe theory, Eq. (18) of the main text. Also, it is evidentfrom the data that the structural force is not linear ink for su�ciently large boxes (small wavenumbers). Theprecision of our data, however, does not allow us to con-firm the cubic scaling predicted by the theory, althoughthe data is also clearly not in contradiction to the ana-lytical result. We hence cannot rule out the contributionof further terms to P exc

t . In particular, a term involvingthe combination (r ·v)(v ·v) produces a structural forcewith the same shape, but a linear scaling with k. Weconclude that revealing the true scaling of the superadi-abatic forces is a major challenge. Due to the saturatione↵ect, the forces increase linearly in a broad range of sys-tem sizes. Only for very small perturbations (very largesystem sizes in this case) the true scaling behaviour isrevealed.

Supplemental Fig. 2. Scaled amplitude of the superadia-

batic forces

ˆf⇤sup = 10

3ˆfsup�/✏ as a function of the (scaled)

wavenumber k� = 2⇡�/L. The blue circles (red squares)

correspond to the viscous (structural) superadiabatic force,

as indicated. The lines are guides to the eye. Data ob-

tained with BD simulations for systems with kB

T/✏ = 0.4and f0 ⇡ 0.30✏/�. The number of particles and the length

of the box vary across the data points, such that the average

density ⇢0�2= 0.02 is kept constant. Both curves must pass

through the origin.

3

Representation of numerical results via fit functions

The fit parameters in Fig. 2a of the main text are:A

1

= 3.7 · 10�4 (for the viscous force, linear in velocity),A

2

= 3.0 · 10�5�/✏ (for the structural force, quadratic invelocity). The fit has been done in the region of weakdriving f

0

�/✏ . 0.3.Regarding the scaling of the superadiabatic forces with

the total density, Fig. 2b of the main text (f0

⇡ 3.14✏/�),a good numerical representation in the range 0 < ⇢

0

�2 <0.1 is given by ⌘(⇢

0

) ⇡ 1.7 · 10�2⇢20

✏⌧�2 and �(⇢0

) ⇡10�2⇢2

0

✏⌧2�2 � 0.4 · 10�2⇢30

✏⌧2�4.We reemphasize that the true scaling of viscous and

structural forces is only revealed at weak driving condi-tions and low densities. At intermediate driving and/ortotal density both forces are linear in velocity and to-

tal density due to a, yet to be investigated, saturationmechanism of the superadiabatic forces.

Predicting the density profile

We can obtain a prediction of the density profile fromthe theory, by projecting the force balance equation(Eq. (1) of the main text) onto ey and observing thatv · ey = 0 = f

ext

· ey. We obtain kBT@⇢/@y = fsup,y

upon neglecting the adiabatic excess contribution fadx

,as this is small against the ideal term, |f

adx

| ⌧ |kBTr⇢|in the low density case N = 2. Integrating in y givesto lowest order in the di↵erence ⇢� ⇢

0

the result ⇢(y) =⇢0

� f2

0

�k2 cos(2ky)/(2�kBT ), which describes the low-density case (Fig. 1b of the main text) quite well.