Fluctuation Spectrum and Fluctuation-Induced Force

We begin with the question: How can we distinguish a suspension of pollen grains at thermal equilibrium from a suspension of active microswimmers? On the one hand, it has been shown that the breakdown of the fluctuation dissipation relation may be directly probed (22⇓–24), and, further, that novel fluctuation modes emerge out of equilibrium (25). On the other hand, an alternative way to characterize the system is via the wavenumber-dependent energy fluctuation spectrum. A natural means of monitoring the fluctuation spectrum (the spectrum of noise due to random forces in the particles’ dynamics) uses dynamic light scattering (26). A general feature of the macroscopic view of physical systems is that fluctuations are intrinsic due to statistical averaging over microscopic degrees of freedom. The magnitude of this intrinsic noise can, in general, be a function of the frequency and wavenumber—this fluctuation spectrum is one key signature of a particular physical system.

Although the fluctuation spectrum can be derived from microscopic kinetic processes, here we are interested in showing that the general properties of such spectra can provide a framework for understanding nonequilibrium behavior. Equilibrium thermal fluctuations, such as that for a Brownian suspension or Johnson–Nyquist noise (27), are usually associated with white noise corresponding to equipartition of energy between different modes. The key point here is that nonequilibrium processes have the potential to generate a nontrivial (for example, nonmonotonic) fluctuation spectrum by continuously injecting energy into particular modes of an otherwise homogenous medium. In the example of microswimmers, they create “active turbulence” by pumping energy preferentially into certain length scales of a homogeneous isotropic fluid (28).

The relation between fluctuation spectra and disjoining force may be examined by generalizing the classic calculation of Casimir (29). We consider an effectively one-dimensional system of two infinite, parallel plates separated by a distance L and immersed in a nonequilibrium medium. We assume that the fluctuations are manifested as waves and impart a radiative stress. We define the fluctuation spectrum asG(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 waves with wavenumber between k and k+δk (where k=|𝐤| is the magnitude of the wavevector), and with angle of incidence between θ and θ+δθ, isδF=G(k)δkcos2θδθ2π.[2]

One factor of cosine in Eq. 2 is due to projecting the momentum in the horizontal direction, the other factor of cosine is due to momentum being spread over an area larger than the cross-sectional length of the wave, and the factor of 2π accounts for the force per unit angle (see, e.g., ref. 30 for a more detailed derivation of Eq. 2). For isotropic fluctuations, we can consider δθ as an infinitesimal quantity, and, upon integrating from θ=−π/2 to π/2, we arrive atδF=14G(k)δk.[3]Outside of the plates, any wavenumber is permitted, and soFout=14∫0∞G(k)dk.[4]However, the waves traveling perpendicular to and between the plates are restricted to take only integer multiples of Δk=π/L, because the waves are reflected by each plate. The force imparted by the waves to the inner surface of the plates is thenFin=14∑m=1∞G(mΔk)Δk,[5]in one dimension. Thus, the net disjoining force for a one-dimensional system is given byFfluct=Fin−Fout=14∑m=1∞G(mΔk)Δk−14∫0∞G(k)dk.[6]Note that Ffluct≶0 for all plate separations L if the derivative G′(k)≶0 for all k: If a nonmonotonic force is observed, it necessarily implies a nonmonotonic spectrum. Furthermore, in higher dimensions, the continuous modes need to be integrated to compute the force between the plates.

Clearly, the fluctuation spectrum G(k) is the crucial quantity in our framework, and can, in principle, be calculated for different systems. We note that previous theoretical approaches have mostly focused on the stress tensor (31). For example, the effect of shaking protocols on force generation has been investigated theoretically for soft (32) and granular (33) media. More generally, nonequilibrium Casimir forces have been computed for reaction–diffusion models with a broken fluctuation–dissipation relation (34, 35), and spatial concentration (36) or thermal (37) gradients. Moving beyond specific models, however, we argue that there are important generic features of fluctuation-induced forces that can be fruitfully derived by considering the fluctuation spectrum and treating it as a phenomenological quantity.

Maritime Casimir Effect

We first illustrate the central result, Eq. 6, by applying it to the classical hydrodynamic example of ocean surface waves that are driven to a nonequilibrium steady state via wind–wave interactions. We treat the one-dimensional case in which the wind blows in a direction perpendicular to the plates (a simple model of ships on the sea), and hence waves traveling parallel to the plates are negligible. Observations (38) show that the spectrum G(k) is nonmonotonic (Fig. 1A). While various fits have been proposed (38, 39), these are untested at large and small wavenumber. Instead, we compute the force in Eq. 6 numerically, approximating the spectrum by a spline through the measured data points of Pierson and Moskowitz (38), and truncating for wavembers beyond their measured ranges. Fig. 1B shows that the resulting force is nonmonotonic and oscillatory as a function of L: The force can be repulsive (Ffluct>0) as well as attractive (Ffluct<0). Physically, the origin of the attractive force is akin to the Casimir force between metal plates—the presence of walls restricts the modes allowed in the interior, so that the energy density outside the walls is greater than that inside. This attractive “Maritime Casimir” force has been observed since antiquity (see, e.g., ref. 6, and references therein) and experimentally measured in a wavetank (40). However, the nonmonotonicity of the spectrum gives rise to an oscillatory force–displacement curve. In particular, the force is repulsive when one of the allowed discrete modes is close to the wavenumber at which the peak of the spectral density occurs (Fig. 1C): Here the sum overestimates the integral in Eq. 6, and the outward force is greater than the inward force. Thus, the local maxima in the repulsive force are approximately located atLn≈nπkmax,[7]where G′(kmax)=0; the separation between the force peaks is ΔL≈π/kmax. In a maritime context, our calculation implies that, if the separation between ships is L>π/kmax, the repulsive fluctuation force will keep the ships away from each other.

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Fig. 1.

(A) The energy spectrum of ocean waves is nonmonotonic. The data are taken from ref. 38 for a wind speed of 33.6 knots (≈17m/s). (B) The fluctuation-induced force per unit length. (Inset) The force (solid blue curve) is consistent with the asymptotic prediction Eq. 16 (dashed red line). (C) The disjoining force is the difference between the integral over the noise spectrum (area under the curve) and the Riemann sum (the shaded regions); crucially, the sum overestimates the integral (i.e., the force is repulsive) when one “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 underestimates the integral, leading to attraction (as in I and III). Note that the quantities on the axes are dimensionless.

To our knowledge, this prediction of a repulsive Maritime Casimir force has yet to be verified experimentally. Clearly quantitative measurement of this oscillatory hydrodynamic fluctuation force in an uncontrolled in situ ocean environment influenced by intermittency would be challenging, although the controlled laboratory framework used in pilot-wave hydrodynamics is ideally suited for direct experimental tests (e.g., ref. 41). We note that an oscillatory force has been observed in the acoustic analogue for which a nonmonotonic fluctuation spectrum was produced (42, 43). Moreover, one-dimensional filaments in a flowing 2D soap film are observed to oscillate in phase or out of phase depending on their relative separation (44), suggesting an oscillatory fluctuation-induced force; visualization of this instability reveals the presence of waves and coherent fluctuations as the mechanism for force generation, which is the basis of our approach.

We would expect that the fluctuation-induced force vanishes when the fluid is at thermal equilibrium. To test this, we note that a consequence of the equipartition theorem is that the energy spectrum for a 3D isotropic fluid at equilibrium is monotonic, and has the scaling (45)Geq(k)∝k2.[8]Noting that, in 3D, δk=δkxδkyδkz/(4πk2), Eq. 6 becomesFfluct=14π∫0∞dky∫0∞dkz(∑m=1∞Δkx−∫0∞dkx)=0,[9]where we have used the fact that the Riemann sum and integral agree exactly for a constant function. Checking this special case confirms that our approach can, in certain circumstances, distinguish between equilibrium and nonequilibrium: In the continuum hydrodynamic setting, a nonzero fluctuation-induced force implies nonequilibrium. We will comment on the UV divergence [divergence in G(k) as k→∞] in Eq. 8 and on the breakdown of continuum hydrodynamics in General Phenomenology of Narrow Unimodal Spectra below. We note that the result, Eqs. 8 and 9, applies only to isotropic one component fluids—thermal Casimir forces exist in systems such as liquid crystals (46) or liquid mixtures near criticality (47, 48).

General Phenomenology of Narrow Unimodal Spectra

Importantly, the phenomenology of nonmonotonic, and even oscillatory, forces is generic for sufficiently narrow, unimodal spectra. To see this, and to make some general quantitative predictions, we perform a Taylor expansion of a general unimodal spectrum, G(k), about its maximum at k=kmax, to find thatG(k)≈{G0[1−ν−2(k−kmax)2],|k−kmax|<ν0 otherwise,[10]where G0=G(kmax), G2=G″(kmax) and ν=−2G0/G2 is the peak width based on a parabolic approximation. In the narrow-peak limit (ν≪π/L, ν≪kmax), the force close to the nth peak is given byFn≈{G0π4L[1−ν−2(nπL−kmax)2]−G0ν3,|nπL−kmax|<ν,−G0ν3  otherwise.[11]From the simplified spectrum in Eq. 11 it may be shown that the n^th maximum is located at Lnmax=nπ/kmax+O((ν/kmax)2), and has magnitudeFn,max=G0π4L−G0ν3=G0kmax4n−G0ν3.[12]Thus, the maximum force is linear in inverse plate separation, and the force reaches its minimum whenkmax−nπL=ν.[13]Writing L=Lnmax+ln=nπ/kmax+ln, where ln is the half-width of the peak in force, we obtainln=nπ(1kmax−1ν+kmax)≈nπνkmax2.[14]Therefore, the width of the force maxima increases linearly with n, and the positions of the n^th mechanical equilibria (Ffluct=0) in the limit of narrowly peaked spectra (ν≪kmax) are given byLn,eq≈Ln±ln≈nπ(1kmax±νkmax2).[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 that only one term in the sum Eq. 5 is significant. As such, this approximation breaks down when the width of the rectangles (compare Fig. 1C) becomes comparable to the width of the peak in G(k) itself, i.e., when L∼1/ν. In consequence, the prediction of Eq. 12 that the force becomes monotonically negative for L>Lthres=3π/4ν will be incorrect. However, in the limit L≫π/ν, the Riemann sum in Eq. 6 is nonzero only for L(kmax−ν)/π⪅m⪅L(kmax+ν)/π; the force then continues to oscillate between attractive and repulsive and has the asymptotic decayFmin≈−π2G03ν1L2,[16]which is the minimum (or maximal attractive) force. The inverse square decay is shown in Fig. 1B, Inset.

These predictions are borne out by the numerical results for the Maritime Casimir effect discussed earlier (Fig. 1B), but, more importantly, form a phenomenological theory that can be applied to systems where the fluctuation spectrum is not known a priori: if force measurements are found to illustrate these scalings, then we suggest that the underlying spectrum is likely to be narrow and unimodal. (The scalings derived here are specialized to the case of interactions between plates, which is a reasonable approximation for the interaction between objects when their separation 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. This UV divergence is cured by noting that hydrodynamic fluctuations, as captured by the spectrum G(k), are suppressed at the molecular length scale k∼2π/σ where σ is the molecular diameter. Therefore, our analysis (Eq. 11) predicts an oscillatory fluctuation-induced force with a period that is comparable to the molecular diameter. This is indeed observed in confined equilibrium fluids (49), although, clearly, at the molecular length scale, our hydrodynamic description breaks down and other physical phenomena, such as proximity induded layering, become relevant. Importantly, while the oscillation wavelength of the disjoining force in equilibrium fluids can be nanoscopic, of the order of the molecular scale, the oscillation wavelength in active nonequilibrium systems can be much larger than the size of the active particle, because the mechanism of force generation lies in a nontrivial partition of energy.

Force Generation with Active Brownian Particles

Interestingly, our asymptotic results are in agreement with force generation in what one might consider to be the unrelated context of self-propelled active Brownian particles. Ni et al. (21) simulated self-propelled Brownian hard spheres confined between hard walls of length W and found an oscillatory decay in the disjoining force (Fig. 2A). Although this system is 2D, our analysis can be generalized: In two dimensions, δk=δkxδky/(2πk), and henceFin=14∑n=1∞Δk∫0∞G((nΔk)2+q2)2π(nΔk)2+q2dq.[17]However, we can redefineh(k)≡∫0∞G(q2+k2)2πq2+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 linear scaling shown in Fig. 2B also implies that the width of the peak scales linearly in L, as predicted by Eq. 14.) The ∼1/L2 decay expected for large L is not observed in these data, as the asymptotic approximations underlying Eq. 12 only break down for L≳Lthres≈12σ, with σν=0.2 estimated from the data. This agreement between the data and our asymptotic framework suggests that the underlying spectrum for active Brownian systems is narrow and nonmonotonic. (For smaller values of the active self-propulsion force f simulated in ref. 21, the peaks are less pronounced and are obscured by numerical noise.) The slight discrepancy with the linear fit at large n is likely due to the fact that our asymptotic scaling only holds in the regime L≪π/ν (note that π/ν≈15σ in Fig. 2B, and the linear fit deteriorates when L≳7σ, confirming that the value of ν estimated from fitting to the width and height of the force peak is at least of the correct order of magnitude). An additional source of the discrepancy may be that the signal-to-noise ratio decreases for increasing plate separation as the magnitude of the force becomes smaller.

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Fig. 2.

Comparison of our theory with the simulations of a 2D suspension of self-propelled Brownian spheres, confined between hard slabs, that interact via the Weeks–Chandler–Anderson potential (21). In A and B, the packing fraction in the bulk is ρσ2=0.4, where σ is the particle diameter, the wall length 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 as suggested by our asymptotic predictions [12] and [14], these data suggest that the underlying fluctuation spectrum is unimodal and has a narrow peak, with parameters 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 fits of Eq. 15 to the simulation data.) The positions of the stable (closed circles) and unstable (open circles) mechanical equilibria (when Ffluct=0) are given by Leq, and the dotted lines are theoretical predictions (Eq. 15). Inset shows the force maxima in A ∝1/L and agrees with Eq. 12. (C) For ideal noninteracting self-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.

Further analytical insights can be obtained by considering the limit of no excluded volume interaction between particles in which Ni et al. (21) observed that the disjoining pressure is attractive and decays monotonically with separation (similar results have been obtained by Ray et al. (15) for run-and-tumble active matter particles). This observation can be explained within our framework by noting that the self-propulsion of point particles induces a Gaussian colored noise ζ(t) satisfying (50)⟨ζ(t)⟩=0,⟨ζ(t)ζ(t′)⟩=f23e−2Dr|t−t′|,[19]where f is the active self-propulsion force and Dr is the rotational diffusion coefficient. In the frequency domain, the fluctuation spectrum S(ω) is the Fourier transform of the time correlation function and isS(ω)=4Drf2314Dr2+ω2.[20]The Lorentzian noise spectrum of Eq. 20 deviates from entropy-maximizing white noise. Assuming a linear dispersion relationship, ω∝k, we note that, because the spectrum of Eq. 20 is now monotonic, the difference between the integral and the Riemann sum, Eq. 6, is monotonic and ∼1/L. Now, the degree of freedom in the direction parallel to the plates can be integrated, yieldingFfluct∝−f2L,[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 simulations is consistent with this scaling: A decay ∝1/L is observed and, further, doubling the activity f increases the prefactor by a factor of 5.6, very nearly the predicted factor of 4. (We believe the slight discrepancy to be caused by the sampling noise, which, as seen in Fig. 2C, is especially significant at large plate separations and is sufficient to alter the estimate of the fitting parameter.) Since oscillatory force decay is only seen for finite, active particles, evidently the coupling between excluded volume interactions and active self-propulsion must be the cause of the oscillatory decay seen in Fig. 2A (and the nonmonotonicity of the inferred spectrum). In particular, the presence of excluded volume interactions gives rise to a length scale of energy injection—the particle diameter—and, indeed, the peak in the spectrum, kmax, is approximately the inverse particle diameter.

Nonmonotonic energy spectra are also found in the continuum hydrodynamic description of active particles (28, 51), as well as models of active swimmers in a fluid (52). For a wide class of such “active turbulent” systems, the fluctuation spectra take the analytical form (51)G(k)=E0kαe−βk2,[22]where E0, α, and β are constants that depend on the underlying microscopic model. This spectrum is narrowly peaked when α/β≫1/β, i.e., α≫1. Although Eq. 22 captures the fluctuations of the active species, but not the background fluid, numerical results show that the energy spectrum of the background fluid—the spectrum that enters into our framework—is also nonmonotonic (52). Therefore, our asymptotic framework, Eqs. 12–15, derived for a general unimodal spectrum, can also be applied to those systems. We note that the effective viscosity of an active fluid in a plane Couette geometry has been shown numerically (53) to be an oscillatory function of plate separation; this supports the oscillatory force framework reported here. Furthermore, oscillatory and long-range fluctuation-induced forces have been reported in other soft-matter systems, including inclusions in a shaken granular medium (33, 54) (where the density field of the granular medium is also directly shown to be inhomogeneous and oscillatory, qualitatively agreeing with our fluctuating modes framework) and rotating active particles on a monolayer (55). Experimental or numerical measurements of Casimir forces in active systems will serve as a test bed of our formalism.