# Fluctuation spectra and force generation in nonequilibrium systems

^{a}School of Engineering and Applied Sciences, Harvard University, Cambridge, MA 02138;^{b}Mathematical Institute, University of Oxford, Oxford OX2 6GG, United Kingdom;^{c}Department of Geology and Geophysics, Yale University, New Haven, CT 06520;^{d}Department of Mathematics, Yale University, New Haven, CT 06520;^{e}Department of Physics, Yale University, New Haven, CT 06520;^{f}Nordic Institute for Theoretical Physics, Royal Institute of Technology and Stockholm University, SE-10691 Stockholm, Sweden

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

## Significance

Understanding force generation in nonequilibrium systems is a significant challenge in statistical and biological physics. We show that force generation in nonequilibrium systems is encoded in their energy fluctuation spectra. In particular, a nonequipartition of energy, which is only possible in active systems, can lead to a nonmonotonic fluctuation spectrum. For a narrow, unimodal spectrum, we find that the force exerted by a nonequilibrium system on two embedded walls depends on the width and the position of the peak in the fluctuation spectrum, and oscillates between repulsion and attraction as a function of wall separation. Our results agree with recent molecular dynamics simulations of active Brownian particles, and shed light on the old riddle of the Maritime Casimir effect.

## Abstract

Many biological systems are appropriately viewed as passive inclusions immersed in an active bath: from proteins on active membranes to microscopic swimmers confined by boundaries. The nonequilibrium forces exerted by the active bath on the inclusions or boundaries often regulate function, and such forces may also be exploited in artificial active materials. Nonetheless, the general 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, controls the phenomenology of force generation. We find that, for a narrow, unimodal spectrum, the force exerted by a nonequilibrium system on two embedded walls depends on the width and the position of the peak in the fluctuation spectrum, and oscillates between repulsion and attraction as a function of wall separation. We examine two apparently disparate examples: the Maritime Casimir effect and recent simulations of active Brownian particles. A key implication of our work is that important nonequilibrium interactions are encoded within the fluctuation spectrum. In this sense, the noise becomes the signal.

Force generation between passive inclusions in active, nonequilibrium systems underpins many phenomena in nature. Bioinspired examples in which such interactions might arise range from proteins on active membranes (1, 2) to swimmers confined by a soft boundary (3⇓–5). On the large scale, such systems feature interactions between objects in a turbulent flow and ships on a stormy sea (6). A fundamental physical question that arises is whether there is a convenient physical framework that could 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, the continuous input of energy places convenient and general statistical concepts, underlying the partition function and the free energy, on more tenuous ground. For example, theories and simulations of active Brownian particles show that self-propulsion induces complex phase behavior qualitatively different from the passive analogue (7⇓⇓⇓⇓–12), and nontrivial behavior such as flocking and swarming is realizable in a nonequilibrium system (13). Therefore, many studies focus on the microscopic physics of a particular active system to compute the force exerted on the embedded inclusions (e.g., refs. 14⇓⇓⇓⇓⇓–20).

In this paper, we show that the force generated by an active system on passive objects is determined by the partition of energy in the active system, given mathematically by the wavenumber dependence of energy fluctuations within it. A key prediction is that, if the energy fluctuation spectrum is nonmonotonic, the force can oscillate between attraction and repulsion as a function of the separation between objects. By making simple approximations of a narrow, unimodal spectrum, we extract scaling properties of the fluctuation-induced force that compare well with recent simulations of the force between solid plates in a bath of self-propelling Brownian particles (21).

## 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

One factor of **2** is due to projecting the momentum in the horizontal direction, the other factor of **2**). For isotropic fluctuations, we can consider

Clearly, the fluctuation spectrum

## 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 *A*). 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. 1*B* shows that the resulting force is nonmonotonic and oscillatory as a function of *C*): 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 at

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)**6** becomes**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, **11** it may be shown that the ^{th} maximum is located at ^{th} mechanical equilibria (**12** and **14** predict that the force–displacement curve has peak repulsion

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. 1*C*) becomes comparable to the width of the peak in **12** that the force becomes monotonically negative for **6** is nonzero only for *B*, *Inset*.

These predictions are borne out by the numerical results for the Maritime Casimir effect discussed earlier (Fig. 1*B*), 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 **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 *A*). Although this system is 2D, our analysis can be generalized: In two dimensions, **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. 2*B* also implies that the width of the peak scales linearly in **14**.) The **12** only break down for *B*, and the linear fit deteriorates when

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 **20** deviates from entropy-maximizing white noise. Assuming a linear dispersion relationship, **20** is now monotonic, the difference between the integral and the Riemann sum, Eq. **6**, is monotonic and *C* shows that the disjoining pressure obtained from simulations is consistent with this scaling: A decay *C*, 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. 2*A* (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,

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)**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.

## Conclusion

There are, of course, a plethora of ways to prepare nonequilibrium systems. We suggest that an organizing principle for force generation is the fluctuation spectrum—the active species drives a nonequipartition of energy. By adopting this top-down view, we computed the relationship between the disjoining pressure and the fluctuation spectrum, and verified our approach by considering two seemingly disparate nonequilibrium physical systems: the Maritime Casimir effect, which is driven by wind–water interactions, and the forces generated by confined active Brownian particles. Our framework affords crucial insight into the phenomenology of both driven and active nonequilibrium systems by providing the bridge between microscopic calculations (56⇓–58), measurements of the fluctuation spectra (26), and the varied measurements of Casimir interactions (59⇓–61). Although this article is motivated by biological and biomimetic settings, measurements of the nonequilibrium electromagnetic Casimir effect, such as the force that an (active) oscillating charge exerts on a neighboring charge, may also test our theory.

In particular, while the fluctuation spectrum of equilibrium fluids vanishes at the molecular scale, so that force oscillations are seen at the molecular length scale (e.g., ref. 49), it is the case that a hydrodynamic system with a force oscillation wavelength much larger than the molecular length scale must be out of equilibrium (because the thermal fluctuation spectrum,

## Acknowledgments

This work was supported by an Engineering and Physical Sciences Research Council Research Studentship, Fulbright Scholarship, and George F. Carrier Fellowship (to A.A.L.) and by the European Research Council Starting Grant 637334 (to D.V.). J.S.W. acknowledges support from Swedish Research Council Grant 638-2013-9243, a Royal Society Wolfson Research Merit Award, and the 2015 Geophysical Fluid Dynamics Summer Study Program at the Woods Hole Oceanographic Institution (National Science Foundation and the Office of Naval Research under OCE-1332750).

## Footnotes

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^{1}To whom correspondence may be addressed. Email: john.wettlaufer{at}yale.edu, alphalee{at}g.harvard.edu, or dominic.vella{at}maths.ox.ac.uk.

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.

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