Fluctuation spectra and force generation in nonequilibrium systems Alpha 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, New Haven, CT 06520; eDepartment of Physics, Yale University, New Haven, CT 06520; and fNordic Institute for Theoretical Physics, Royal Institute of Technology 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 passive Fluctuation Spectrum and Fluctuation-Induced Force inclusions immersed in an active bath: from proteins on active We begin with the question: How can we distinguish a suspen- membranes to microscopic swimmers confined by boundaries. The sion of pollen grains at thermal equilibrium from a suspension nonequilibrium forces exerted by the active bath on the inclu- of active microswimmers? On the one hand, it has been shown sions or boundaries often regulate function, and such forces may that the breakdown of the fluctuation dissipation relation may also be exploited in artificial active materials. Nonetheless, the be directly probed (22–24), and, further, that novel fluctuation general phenomenology of these active forces remains elusive. modes emerge out of equilibrium (25). On the other hand, an We show that the fluctuation spectrum of the active medium, alternative way to characterize the system is via the wavenumber- the partitioning of energy as a function of wavenumber, controls dependent energy fluctuation spectrum. A natural means of the phenomenology of force generation. We find that, for a nar- monitoring the fluctuation spectrum (the spectrum of noise due row, unimodal spectrum, the force exerted by a nonequilibrium to random forces in the particles’ dynamics) uses dynamic light system on two embedded walls depends on the width and the scattering (26). A general feature of the macroscopic view of position of the peak in the fluctuation spectrum, and oscillates physical systems is that fluctuations are intrinsic due to statistical between repulsion and attraction as a function of wall separation. averaging over microscopic degrees of freedom. The magnitude SCIENCES We examine two apparently disparate examples: the Maritime of this intrinsic noise can, in general, be a function of the fre- APPLIED PHISICAL Casimir effect and recent simulations of active Brownian particles. quency and wavenumber—this fluctuation spectrum is one key A key implication of our work is that important nonequilibrium signature of a particular physical system. interactions are encoded within the fluctuation spectrum. In this Although the fluctuation spectrum can be derived from micro- sense, the noise becomes the signal. scopic kinetic processes, here we are interested in showing that the general properties of such spectra can provide a frame- Casimir effect j nonequilibrium physics j fluctuations j active matter work for understanding nonequilibrium behavior. Equilibrium thermal fluctuations, such as that for a Brownian suspension or Johnson–Nyquist noise (27), are usually associated with white orce generation between passive inclusions in active, non- noise corresponding to equipartition of energy between different Fequilibrium systems underpins many phenomena in nature. modes. The key point here is that nonequilibrium processes have Bioinspired examples in which such interactions might arise the potential to generate a nontrivial (for example, nonmono- range from proteins on active membranes (1, 2) to swimmers tonic) fluctuation spectrum by continuously injecting energy into confined by a soft boundary (3–5). On the large scale, such sys- particular modes of an otherwise homogenous medium. In the tems feature interactions between objects in a turbulent flow and example of microswimmers, they create “active turbulence” by ships on a stormy sea (6). A fundamental physical question that pumping energy preferentially into certain length scales of a arises is whether there is a convenient physical framework that homogeneous isotropic fluid (28). could describe force generation in the wide variety of out-of- equilibrum systems across different length scales. Significance The salient challenge is that, unlike an equilibrium system, the continuous input of energy places convenient and general sta- Understanding force generation in nonequilibrium systems tistical concepts, underlying the partition function and the free is a significant challenge in statistical and biological physics. energy, on more tenuous ground. For example, theories and sim- We show that force generation in nonequilibrium systems is ulations of active Brownian particles show that self-propulsion encoded in their energy fluctuation spectra. In particular, a induces complex phase behavior qualitatively different from the nonequipartition of energy, which is only possible in active passive analogue (7–12), and nontrivial behavior such as flock- systems, can lead to a nonmonotonic fluctuation spectrum. ing and swarming is realizable in a nonequilibrium system (13). For a narrow, unimodal spectrum, we find that the force Therefore, many studies focus on the microscopic physics of a exerted by a nonequilibrium system on two embedded walls particular active system to compute the force exerted on the depends on the width and the position of the peak in the embedded inclusions (e.g., refs. 14–20). fluctuation spectrum, and oscillates between repulsion and In this paper, we show that the force generated by an active sys- attraction as a function of wall separation. Our results agree tem on passive objects is determined by the partition of energy with recent molecular dynamics simulations of active Brown- in the active system, given mathematically by the wavenumber ian particles, and shed light on the old riddle of the Maritime dependence of energy fluctuations within it. A key prediction Casimir effect. is that, if the energy fluctuation spectrum is nonmonotonic, the force can oscillate between attraction and repulsion as a function Author contributions: A.A.L., D.V., and J.S.W. designed research, performed research, of the separation between objects. By making simple approxima- analyzed data, and wrote the paper. tions of a narrow, unimodal spectrum, we extract scaling prop- The authors declare no conflict of interest. erties of the fluctuation-induced force that compare well with This article is a PNAS Direct Submission. recent simulations of the force between solid plates in a bath of 1To whom correspondence may be addressed. Email: [email protected], self-propelling Brownian particles (21). [email protected], or [email protected]. www.pnas.org/cgi/doi/10.1073/pnas.1701739114 PNAS Early Edition j 1 of 6 Downloaded by guest on October 1, 2021 The relation between fluctuation spectra and disjoining force driven to a nonequilibrium steady state via wind–wave inter- may be examined by generalizing the classic calculation of actions. We treat the one-dimensional case in which the wind Casimir (29). We consider an effectively one-dimensional sys- blows in a direction perpendicular to the plates (a simple model tem of two infinite, parallel plates separated by a distance L and of ships on the sea), and hence waves traveling parallel to the immersed in a nonequilibrium medium. We assume that the fluc- plates are negligible. Observations (38) show that the spec- tuations are manifested as waves and impart a radiative stress. trum G(k) is nonmonotonic (Fig. 1A). While various fits have We define the fluctuation spectrum as been proposed (38, 39), these are untested at large and small dE(k) wavenumber. Instead, we compute the force in Eq. 6 numeri- G(k) ≡ ; [1] cally, approximating the spectrum by a spline through the mea- dk sured data points of Pierson and Moskowitz (38), and truncating where E(k) is the energy density of modes with wavenumber k. for wavembers beyond their measured ranges. Fig. 1B shows that Hence the radiation force per unit plate area, δF , due to waves the resulting force is nonmonotonic and oscillatory as a function k k + δk k = jkj with wavenumber between and (where is of L: The force can be repulsive (Ffluct > 0) as well as attractive the magnitude of the wavevector), and with angle of incidence (Ffluct < 0). Physically, the origin of the attractive force is akin to between θ and θ + δθ, is the Casimir force between metal plates—the presence of walls δθ restricts the modes allowed in the interior, so that the energy δF = G(k)δk cos2θ : [2] 2π density outside the walls is greater than that inside. This attrac- One factor of cosine in Eq. 2 is due to projecting the momen- tive “Maritime Casimir” force has been observed since antiquity tum in the horizontal direction, the other factor of cosine is due (see, e.g., ref. 6, and references therein) and experimentally mea- to momentum being spread over an area larger than the cross- sured in a wavetank (40). However, the nonmonotonicity of the sectional length of the wave, and the factor of 2π accounts for the spectrum gives rise to an oscillatory force–displacement curve. force per unit angle (see, e.g., ref. 30 for a more detailed deriva- In particular, the force is repulsive when one of the allowed dis- tion of Eq. 2). For isotropic fluctuations, we can consider δθ as crete modes is close to the wavenumber at which the peak of the an infinitesimal quantity, and, upon integrating from θ = −π=2 spectral density occurs (Fig. 1C): Here the sum overestimates to π=2, we arrive at the integral in Eq. 6, and the outward force is greater than the inward force. Thus, the local maxima in the repulsive force are 1 δF = G(k)δk: [3] approximately located at 4 π Outside of the plates, any wavenumber is permitted, and so Ln ≈ n ; [7] kmax Z 1 1 0 Fout = G(k)dk: [4] where G (kmax) = 0; the separation between the force peaks 4 0 is ∆L ≈ π=kmax.