Fluctuation spectra and force generation in nonequilibrium ...

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Fluctuation spectra and force generation innonequilibrium systemsAlpha A. Leea,1, Dominic Vellab,1, and John S. Wettlauferb,c,d,e,f,1

aSchool of Engineering and Applied Sciences, Harvard University, Cambridge, MA 02138; bMathematical Institute, University of Oxford, Oxford OX2 6GG,United Kingdom; cDepartment of Geology and Geophysics, Yale University, New Haven, CT 06520; dDepartment of Mathematics, Yale University, NewHaven, CT 06520; eDepartment of Physics, Yale University, New Haven, CT 06520; and fNordic Institute for Theoretical Physics, Royal Institute ofTechnology and Stockholm University, SE-10691 Stockholm, Sweden

Edited by David A. Weitz, Harvard University, Cambridge, MA, and approved July 10, 2017 (received for review January 31, 2017)

Many biological systems are appropriately viewed as passiveinclusions immersed in an active bath: from proteins on activemembranes to microscopic swimmers confined by boundaries. Thenonequilibrium forces exerted by the active bath on the inclu-sions or boundaries often regulate function, and such forces mayalso be exploited in artificial active materials. Nonetheless, thegeneral phenomenology of these active forces remains elusive.We show that the fluctuation spectrum of the active medium,the partitioning of energy as a function of wavenumber, controlsthe phenomenology of force generation. We find that, for a nar-row, unimodal spectrum, the force exerted by a nonequilibriumsystem on two embedded walls depends on the width and theposition of the peak in the fluctuation spectrum, and oscillatesbetween repulsion and attraction as a function of wall separation.We examine two apparently disparate examples: the MaritimeCasimir effect and recent simulations of active Brownian particles.A key implication of our work is that important nonequilibriuminteractions are encoded within the fluctuation spectrum. In thissense, the noise becomes the signal.

Casimir effect | nonequilibrium physics | fluctuations | active matter

Force generation between passive inclusions in active, non-equilibrium systems underpins many phenomena in nature.

Bioinspired examples in which such interactions might ariserange from proteins on active membranes (1, 2) to swimmersconfined by a soft boundary (3–5). On the large scale, such sys-tems feature interactions between objects in a turbulent flow andships on a stormy sea (6). A fundamental physical question thatarises is whether there is a convenient physical framework thatcould describe force generation in the wide variety of out-of-equilibrum systems across different length scales.

The salient challenge is that, unlike an equilibrium system, thecontinuous input of energy places convenient and general sta-tistical concepts, underlying the partition function and the freeenergy, on more tenuous ground. For example, theories and sim-ulations of active Brownian particles show that self-propulsioninduces complex phase behavior qualitatively different from thepassive analogue (7–12), and nontrivial behavior such as flock-ing and swarming is realizable in a nonequilibrium system (13).Therefore, many studies focus on the microscopic physics of aparticular active system to compute the force exerted on theembedded inclusions (e.g., refs. 14–20).

In this paper, we show that the force generated by an active sys-tem on passive objects is determined by the partition of energyin the active system, given mathematically by the wavenumberdependence of energy fluctuations within it. A key predictionis that, if the energy fluctuation spectrum is nonmonotonic, theforce can oscillate between attraction and repulsion as a functionof the separation between objects. By making simple approxima-tions of a narrow, unimodal spectrum, we extract scaling prop-erties of the fluctuation-induced force that compare well withrecent simulations of the force between solid plates in a bath ofself-propelling Brownian particles (21).

Fluctuation Spectrum and Fluctuation-Induced ForceWe begin with the question: How can we distinguish a suspen-sion of pollen grains at thermal equilibrium from a suspensionof active microswimmers? On the one hand, it has been shownthat the breakdown of the fluctuation dissipation relation maybe directly probed (22–24), and, further, that novel fluctuationmodes emerge out of equilibrium (25). On the other hand, analternative way to characterize the system is via the wavenumber-dependent energy fluctuation spectrum. A natural means ofmonitoring the fluctuation spectrum (the spectrum of noise dueto random forces in the particles’ dynamics) uses dynamic lightscattering (26). A general feature of the macroscopic view ofphysical systems is that fluctuations are intrinsic due to statisticalaveraging over microscopic degrees of freedom. The magnitudeof this intrinsic noise can, in general, be a function of the fre-quency and wavenumber—this fluctuation spectrum is one keysignature of a particular physical system.

Although the fluctuation spectrum can be derived from micro-scopic kinetic processes, here we are interested in showing thatthe general properties of such spectra can provide a frame-work for understanding nonequilibrium behavior. Equilibriumthermal fluctuations, such as that for a Brownian suspension orJohnson–Nyquist noise (27), are usually associated with whitenoise corresponding to equipartition of energy between differentmodes. The key point here is that nonequilibrium processes havethe potential to generate a nontrivial (for example, nonmono-tonic) fluctuation spectrum by continuously injecting energy intoparticular modes of an otherwise homogenous medium. In theexample of microswimmers, they create “active turbulence” bypumping energy preferentially into certain length scales of ahomogeneous isotropic fluid (28).

Significance

Understanding force generation in nonequilibrium systemsis a significant challenge in statistical and biological physics.We show that force generation in nonequilibrium systems isencoded in their energy fluctuation spectra. In particular, anonequipartition of energy, which is only possible in activesystems, can lead to a nonmonotonic fluctuation spectrum.For a narrow, unimodal spectrum, we find that the forceexerted by a nonequilibrium system on two embedded wallsdepends on the width and the position of the peak in thefluctuation spectrum, and oscillates between repulsion andattraction as a function of wall separation. Our results agreewith recent molecular dynamics simulations of active Brown-ian particles, and shed light on the old riddle of the MaritimeCasimir effect.

Author contributions: A.A.L., D.V., and J.S.W. designed research, performed research,analyzed data, and wrote the paper.

The authors declare no conflict of interest.

This article is a PNAS Direct Submission.1To whom correspondence may be addressed. Email: [email protected],[email protected], or [email protected].

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The relation between fluctuation spectra and disjoining forcemay be examined by generalizing the classic calculation ofCasimir (29). We consider an effectively one-dimensional sys-tem of two infinite, parallel plates separated by a distance L andimmersed in a nonequilibrium medium. We assume that the fluc-tuations are manifested as waves and impart a radiative stress.We define the fluctuation spectrum as

G(k) ≡ dE(k)

dk, [1]

where E(k) is the energy density of modes with wavenumber k .Hence the radiation force per unit plate area, δF , due to waveswith wavenumber between k and k + δk (where k = |k| isthe magnitude of the wavevector), and with angle of incidencebetween θ and θ + δθ, is

δF = G(k)δk cos2θδθ

2π. [2]

One factor of cosine in Eq. 2 is due to projecting the momen-tum in the horizontal direction, the other factor of cosine is dueto momentum being spread over an area larger than the cross-sectional length of the wave, and the factor of 2π accounts for theforce per unit angle (see, e.g., ref. 30 for a more detailed deriva-tion of Eq. 2). For isotropic fluctuations, we can consider δθ asan infinitesimal quantity, and, upon integrating from θ = −π/2to π/2, we arrive at

δF =1

4G(k)δk . [3]

Outside of the plates, any wavenumber is permitted, and so

Fout =1

4

∫ ∞0

G(k)dk . [4]

However, the waves traveling perpendicular to and between theplates are restricted to take only integer multiples of ∆k =π/L, because the waves are reflected by each plate. The forceimparted by the waves to the inner surface of the plates is then

Fin =1

4

∞∑m=1

G(m∆k) ∆k , [5]

in one dimension. Thus, the net disjoining force for a one-dimensional system is given by

Ffluct = Fin − Fout =1

4

∞∑m=1

G(m∆k) ∆k − 1

4

∫ ∞0

G(k) dk .

[6]

Note that Ffluct ≶ 0 for all plate separations L if the deriva-tive G ′(k) ≶ 0 for all k : If a nonmonotonic force is observed, itnecessarily implies a nonmonotonic spectrum. Furthermore, inhigher dimensions, the continuous modes need to be integratedto compute the force between the plates.

Clearly, the fluctuation spectrum G(k) is the crucial quantityin our framework, and can, in principle, be calculated for differ-ent systems. We note that previous theoretical approaches havemostly focused on the stress tensor (31). For example, the effectof shaking protocols on force generation has been investigatedtheoretically for soft (32) and granular (33) media. More gen-erally, nonequilibrium Casimir forces have been computed forreaction–diffusion models with a broken fluctuation–dissipationrelation (34, 35), and spatial concentration (36) or thermal (37)gradients. Moving beyond specific models, however, we arguethat there are important generic features of fluctuation-inducedforces that can be fruitfully derived by considering the fluctuationspectrum and treating it as a phenomenological quantity.

Maritime Casimir EffectWe first illustrate the central result, Eq. 6, by applying it to theclassical hydrodynamic example of ocean surface waves that are

driven to a nonequilibrium steady state via wind–wave inter-actions. We treat the one-dimensional case in which the windblows in a direction perpendicular to the plates (a simple modelof ships on the sea), and hence waves traveling parallel to theplates are negligible. Observations (38) show that the spec-trum G(k) is nonmonotonic (Fig. 1A). While various fits havebeen proposed (38, 39), these are untested at large and smallwavenumber. Instead, we compute the force in Eq. 6 numeri-cally, approximating the spectrum by a spline through the mea-sured data points of Pierson and Moskowitz (38), and truncatingfor wavembers beyond their measured ranges. Fig. 1B shows thatthe resulting force is nonmonotonic and oscillatory as a functionof L: The force can be repulsive (Ffluct > 0) as well as attractive(Ffluct < 0). Physically, the origin of the attractive force is akin tothe Casimir force between metal plates—the presence of wallsrestricts the modes allowed in the interior, so that the energydensity outside the walls is greater than that inside. This attrac-tive “Maritime Casimir” force has been observed since antiquity(see, e.g., ref. 6, and references therein) and experimentally mea-sured in a wavetank (40). However, the nonmonotonicity of thespectrum gives rise to an oscillatory force–displacement curve.In particular, the force is repulsive when one of the allowed dis-crete modes is close to the wavenumber at which the peak of thespectral density occurs (Fig. 1C): Here the sum overestimatesthe integral in Eq. 6, and the outward force is greater than theinward force. Thus, the local maxima in the repulsive force areapproximately located at

Ln ≈ nπ

kmax, [7]

where G ′(kmax) = 0; the separation between the force peaksis ∆L ≈ π/kmax. In a maritime context, our calculation impliesthat, if the separation between ships is L > π/kmax, the repulsivefluctuation force will keep the ships away from each other.

To our knowledge, this prediction of a repulsive MaritimeCasimir force has yet to be verified experimentally. Clearly quan-titative measurement of this oscillatory hydrodynamic fluctuationforce in an uncontrolled in situ ocean environment influenced byintermittency would be challenging, although the controlled lab-oratory framework used in pilot-wave hydrodynamics is ideallysuited for direct experimental tests (e.g., ref. 41). We note that anoscillatory force has been observed in the acoustic analogue forwhich a nonmonotonic fluctuation spectrum was produced (42,43). Moreover, one-dimensional filaments in a flowing 2D soapfilm are observed to oscillate in phase or out of phase depend-ing on their relative separation (44), suggesting an oscillatoryfluctuation-induced force; visualization of this instability revealsthe presence of waves and coherent fluctuations as the mecha-nism for force generation, which is the basis of our approach.

We would expect that the fluctuation-induced force vanisheswhen the fluid is at thermal equilibrium. To test this, we note thata consequence of the equipartition theorem is that the energyspectrum for a 3D isotropic fluid at equilibrium is monotonic,and has the scaling (45)

Geq(k) ∝ k2. [8]

Noting that, in 3D, δk = δkx δkyδkz/(4πk2), Eq. 6 becomes

Ffluct =1

4π

∫ ∞0

dky

∫ ∞0

dkz

(∞∑

m=1

∆kx −∫ ∞

0

dkx

)= 0, [9]

where we have used the fact that the Riemann sum and inte-gral agree exactly for a constant function. Checking this specialcase confirms that our approach can, in certain circumstances,distinguish between equilibrium and nonequilibrium: In thecontinuum hydrodynamic setting, a nonzero fluctuation-inducedforce implies nonequilibrium. We will comment on the UVdivergence [divergence in G(k) as k → ∞] in Eq. 8 and on

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10-30 1 2 3 4 5

107

0

0.5

1

1.5

2

2.5

3A

B

C

103 104 105

10-2

100

102

104

1040 0.5 1 1.5 2 2.5 3 3.5 4

-2000

-1500

-1000

-500

0

500

1000

Fig. 1. (A) The energy spectrum of ocean waves is nonmonotonic. The dataare taken from ref. 38 for a wind speed of 33.6 knots (≈ 17m/s). (B) Thefluctuation-induced force per unit length. (Inset) The force (solid blue curve)is consistent with the asymptotic prediction Eq. 16 (dashed red line). (C) The

the breakdown of continuum hydrodynamics in General Phe-nomenology of Narrow Unimodal Spectra below. We note thatthe result, Eqs. 8 and 9, applies only to isotropic one compo-nent fluids—thermal Casimir forces exist in systems such as liq-uid crystals (46) or liquid mixtures near criticality (47, 48).

General Phenomenology of Narrow Unimodal SpectraImportantly, the phenomenology of nonmonotonic, and evenoscillatory, forces is generic for sufficiently narrow, unimodalspectra. To see this, and to make some general quantitative pre-dictions, we perform a Taylor expansion of a general unimodalspectrum, G(k), about its maximum at k = kmax, to find that

G(k) ≈{G0

[1− ν−2(k − kmax)2], |k − kmax| < ν

0 otherwise,[10]

where G0 = G(kmax), G2 = G ′′(kmax) and ν =√−2G0/G2

is the peak width based on a parabolic approximation. In thenarrow-peak limit (ν � π/L, ν � kmax), the force close to thenth peak is given by

Fn ≈

G0π

4L

[1− ν−2

(nπL

− kmax

)2]−

G0ν

3,∣∣∣nπL

− kmax

∣∣∣ < ν,

−G0ν

3otherwise.

[11]

From the simplified spectrum in Eq. 11 it may be shown that then th maximum is located at Lmax

n = nπ/kmax + O((ν/kmax)2),and has magnitude

Fn,max =G0π

4L− G0ν

3=

G0kmax

4n− G0ν

3. [12]

Thus, the maximum force is linear in inverse plate separation,and the force reaches its minimum when

kmax −nπ

L= ν. [13]

Writing L = Lmaxn +ln = nπ/kmax+ln , where ln is the half-width

of the peak in force, we obtain

ln = nπ

(1

kmax− 1

ν + kmax

)≈ nπν

k2max

. [14]

Therefore, the width of the force maxima increases linearly withn , and the positions of the n th mechanical equilibria (Ffluct = 0)in the limit of narrowly peaked spectra (ν � kmax) are given by

Ln,eq ≈ Ln ± ln ≈ nπ

(1

kmax± ν

k2max

). [15]

Here the positive (negative) branches correspond to stable(unstable) equilibria. Eqs. 12 and 14 predict that the force–displacement curve has peak repulsion ∝ 1/L and peak width∝ n ∝ L for L� π/ν.

The asymptotic prediction Eq. 12 arises from assuming thatonly one term in the sum Eq. 5 is significant. As such, this approx-imation breaks down when the width of the rectangles (com-pare Fig. 1C) becomes comparable to the width of the peak inG(k) itself, i.e., when L ∼ 1/ν. In consequence, the predic-tion of Eq. 12 that the force becomes monotonically negativefor L > Lthres = 3π/4ν will be incorrect. However, in thelimit L � π/ν, the Riemann sum in Eq. 6 is nonzero only forL(kmax − ν)/π / m / L(kmax + ν)/π; the force then continues

disjoining force is the difference between the integral over the noise spec-trum (area under the curve) and the Riemann sum (the shaded regions); cru-cially, the sum overestimates the integral (i.e., the force is repulsive) whenone “grid point” is sufficiently close to the maximum in the distribution,kmax ≈ nπ/L for some integer n (as in II and IV); more often, the sum under-estimates the integral, leading to attraction (as in I and III). Note that thequantities on the axes are dimensionless.

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to oscillate between attractive and repulsive and has the asymp-totic decay

Fmin ≈ −π2G0

3ν

1

L2, [16]

which is the minimum (or maximal attractive) force. The inversesquare decay is shown in Fig. 1B, Inset.

These predictions are borne out by the numerical results forthe Maritime Casimir effect discussed earlier (Fig. 1B), but, moreimportantly, form a phenomenological theory that can be appliedto systems where the fluctuation spectrum is not known a priori:if force measurements are found to illustrate these scalings, thenwe suggest that the underlying spectrum is likely to be narrowand unimodal. (The scalings derived here are specialized to thecase of interactions between plates, which is a reasonable approx-imation for the interaction between objects when their separa-tion is much less than their radii of curvature.)

We can now revisit the case of classical fluids at equilibrium.Obviously, the divergence in Eq. 8 as k→∞ is unphysical. ThisUV divergence is cured by noting that hydrodynamic fluctu-ations, as captured by the spectrum G(k), are suppressed atthe molecular length scale k ∼ 2π/σ where σ is the moleculardiameter. Therefore, our analysis (Eq. 11) predicts an oscillatoryfluctuation-induced force with a period that is comparable to themolecular diameter. This is indeed observed in confined equilib-rium fluids (49), although, clearly, at the molecular length scale,our hydrodynamic description breaks down and other physicalphenomena, such as proximity induded layering, become rele-vant. Importantly, while the oscillation wavelength of the disjoin-ing force in equilibrium fluids can be nanoscopic, of the order ofthe molecular scale, the oscillation wavelength in active nonequi-librium systems can be much larger than the size of the activeparticle, because the mechanism of force generation lies in a non-trivial partition of energy.

Force Generation with Active Brownian ParticlesInterestingly, our asymptotic results are in agreement with forcegeneration in what one might consider to be the unrelated con-text of self-propelled active Brownian particles. Ni et al. (21) sim-ulated self-propelled Brownian hard spheres confined betweenhard walls of length W and found an oscillatory decay in thedisjoining force (Fig. 2A). Although this system is 2D, our anal-ysis can be generalized: In two dimensions, δk = δkx δky/(2πk),and hence

A B C

Fig. 2. Comparison of our theory with the simulations of a 2D suspension of self-propelled Brownian spheres, confined between hard slabs, that interactvia the Weeks–Chandler–Anderson potential (21). In A and B, the packing fraction in the bulk is ρσ2 = 0.4, where σ is the particle diameter, the walllength is W = 10σ, and self-propulsion force f = 40kBT/σ. (A) The raw force–displacement curve for ρσ2 = 0.4 from ref. 21. (B) When replotted assuggested by our asymptotic predictions [12] and [14], these data suggest that the underlying fluctuation spectrum is unimodal and has a narrow peak, withparameters G0≈ 4.8 × 103 and ν≈ 0.2/σ. (As the peaks are spaced approximately σ apart, we assume kmax =π/σ, and G0 and ν are obtained from fitsof Eq. 15 to the simulation data.) The positions of the stable (closed circles) and unstable (open circles) mechanical equilibria (when Ffluct = 0) are given byLeq, and the dotted lines are theoretical predictions (Eq. 15). Inset shows the force maxima in A ∝ 1/L and agrees with Eq. 12. (C) For ideal noninteractingself-propelled point particles, the function Aσ/L (black dotted line; see Eq. 21) can be fitted (using A) to simulation data with Fσ2/(WkBT) = 40 (A = 182)and Fσ2/(WkBT) = 20 (A = 31.6). Here W = 80σ.

Fin =1

4

∞∑n=1

∆k

∫ ∞0

G

(√(n∆k)2 + q2

)2π√

(n∆k)2 + q2

dq . [17]

However, we can redefine

h(k) ≡∫ ∞

0

G(√

q2 + k2)

2π√

q2 + k2dq [18]

as an effective 1D spectrum and substitute h(k) for G(k) in Eq.6. Performing the same asymptotic analysis as for the narrow-peak limit, the asymptotic scalings [12] and [14] are reproduced,in quantitative agreement with simulations. (We note that the lin-ear scaling shown in Fig. 2B also implies that the width of thepeak scales linearly in L, as predicted by Eq. 14.) The ∼1/L2

decay expected for large L is not observed in these data, as theasymptotic approximations underlying Eq. 12 only break down forL & Lthres ≈ 12σ, with σν = 0.2 estimated from the data. Thisagreement between the data and our asymptotic framework sug-gests that the underlying spectrum for active Brownian systems isnarrow and nonmonotonic. (For smaller values of the active self-propulsion force f simulated in ref. 21, the peaks are less pro-nounced and are obscured by numerical noise.) The slight dis-crepancy with the linear fit at large n is likely due to the fact thatour asymptotic scaling only holds in the regime L � π/ν (notethat π/ν ≈ 15σ in Fig. 2B, and the linear fit deteriorates whenL & 7σ, confirming that the value of ν estimated from fittingto the width and height of the force peak is at least of the cor-rect order of magnitude). An additional source of the discrep-ancy may be that the signal-to-noise ratio decreases for increasingplate separation as the magnitude of the force becomes smaller.

Further analytical insights can be obtained by considering thelimit of no excluded volume interaction between particles inwhich Ni et al. (21) observed that the disjoining pressure is attrac-tive and decays monotonically with separation (similar resultshave been obtained by Ray et al. (15) for run-and-tumble activematter particles). This observation can be explained within ourframework by noting that the self-propulsion of point particlesinduces a Gaussian colored noise ζ(t) satisfying (50)

〈ζ(t)〉 = 0,⟨ζ(t)ζ(t ′)

⟩=

f 2

3e−2Dr |t−t′|, [19]

where f is the active self-propulsion force and Dr is therotational diffusion coefficient. In the frequency domain, the

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fluctuation spectrum S(ω) is the Fourier transform of the timecorrelation function and is

S(ω) =4Dr f

2

3

1

4D2r + ω2

. [20]

The Lorentzian noise spectrum of Eq. 20 deviates from entropy-maximizing white noise. Assuming a linear dispersion relation-ship, ω ∝ k , we note that, because the spectrum of Eq. 20is now monotonic, the difference between the integral and theRiemann sum, Eq. 6, is monotonic and∼1/L. Now, the degree offreedom in the direction parallel to the plates can be integrated,yielding

Ffluct ∝ −f 2

L, [21]

for large L. Hence, we expect to see a monotonic force–displacement relation, as observed by Ni et al. (21). Indeed, Fig.2C shows that the disjoining pressure obtained from simulationsis consistent with this scaling: A decay ∝ 1/L is observed and,further, doubling the activity f increases the prefactor by a fac-tor of 5.6, very nearly the predicted factor of 4. (We believe theslight discrepancy to be caused by the sampling noise, which,as seen in Fig. 2C, is especially significant at large plate sep-arations and is sufficient to alter the estimate of the fittingparameter.) Since oscillatory force decay is only seen for finite,active particles, evidently the coupling between excluded vol-ume interactions and active self-propulsion must be the causeof the oscillatory decay seen in Fig. 2A (and the nonmono-tonicity of the inferred spectrum). In particular, the presenceof excluded volume interactions gives rise to a length scale ofenergy injection—the particle diameter—and, indeed, the peakin the spectrum, kmax, is approximately the inverse particlediameter.

Nonmonotonic energy spectra are also found in the continuumhydrodynamic description of active particles (28, 51), as well asmodels of active swimmers in a fluid (52). For a wide class ofsuch “active turbulent” systems, the fluctuation spectra take theanalytical form (51)

G(k) = E0kαe−βk

2

, [22]

where E0, α, and β are constants that depend on the underly-ing microscopic model. This spectrum is narrowly peaked whenα/β� 1/β, i.e., α� 1. Although Eq. 22 captures the fluctua-tions of the active species, but not the background fluid, numer-ical results show that the energy spectrum of the backgroundfluid—the spectrum that enters into our framework—is alsononmonotonic (52). Therefore, our asymptotic framework, Eqs.12–15, derived for a general unimodal spectrum, can also beapplied to those systems. We note that the effective viscosityof an active fluid in a plane Couette geometry has been shownnumerically (53) to be an oscillatory function of plate separation;this supports the oscillatory force framework reported here. Fur-thermore, oscillatory and long-range fluctuation-induced forceshave been reported in other soft-matter systems, including inclu-sions in a shaken granular medium (33, 54) (where the den-sity field of the granular medium is also directly shown to be

inhomogeneous and oscillatory, qualitatively agreeing with ourfluctuating modes framework) and rotating active particles ona monolayer (55). Experimental or numerical measurements ofCasimir forces in active systems will serve as a test bed of ourformalism.

ConclusionThere are, of course, a plethora of ways to prepare nonequilib-rium systems. We suggest that an organizing principle for forcegeneration is the fluctuation spectrum—the active species drivesa nonequipartition of energy. By adopting this top-down view, wecomputed the relationship between the disjoining pressure andthe fluctuation spectrum, and verified our approach by consid-ering two seemingly disparate nonequilibrium physical systems:the Maritime Casimir effect, which is driven by wind–water inter-actions, and the forces generated by confined active Brownianparticles. Our framework affords crucial insight into the phe-nomenology of both driven and active nonequilibrium systemsby providing the bridge between microscopic calculations (56–58), measurements of the fluctuation spectra (26), and the var-ied measurements of Casimir interactions (59–61). Although thisarticle is motivated by biological and biomimetic settings, mea-surements of the nonequilibrium electromagnetic Casimir effect,such as the force that an (active) oscillating charge exerts on aneighboring charge, may also test our theory.

In particular, while the fluctuation spectrum of equilibriumfluids vanishes at the molecular scale, so that force oscillationsare seen at the molecular length scale (e.g., ref. 49), it is the casethat a hydrodynamic system with a force oscillation wavelengthmuch larger than the molecular length scale must be out ofequilibrium (because the thermal fluctuation spectrum, G ∼ k2,is monotonic). As a corollary, out-of-equilibrium systems canexhibit force oscillations with wavelengths significantly longerthan the size of the active particles. More generally, because timereversal symmetry requires equilibrium (62), it would appearprudent to examine the time correlations in the systems we havestudied here. Additionally, another form of an “active fluid” canbe constructed in a pure system using, for example, a thermallynonequilibrium steady state; temperature fluctuations in such asystem have been observed to give rise to long-range Casimir-like behavior (63, 64). Hence, an intriguing possibility suggestedby our analysis is that, rather than tuning forces by controllingthe nature [e.g., dielectric properties (65)] of the bounding walls,one can envisage actively controlling the fluctuation spectra ofthe intervening material. Indeed, a natural speculation is thatswimmers in biological (engineering) settings could (be designedto) actively control the forces they experience in confinedgeometries.

ACKNOWLEDGMENTS. This work was supported by an Engineering andPhysical Sciences Research Council Research Studentship, Fulbright Schol-arship, and George F. Carrier Fellowship (to A.A.L.) and by the EuropeanResearch Council Starting Grant 637334 (to D.V.). J.S.W. acknowledges sup-port from Swedish Research Council Grant 638-2013-9243, a Royal SocietyWolfson Research Merit Award, and the 2015 Geophysical Fluid Dynam-ics Summer Study Program at the Woods Hole Oceanographic Institution(National Science Foundation and the Office of Naval Research under OCE-1332750).

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