Fluctuation spectra and force generation in non-equilibrium systems
Alpha A. Lee
Dominic Vella
John S. Wettlaufer
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 non-equilibrium 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 non-equilibrium 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 non-equilibrium interactions are encoded within the fluctuation spectrum. In this sense the noise becomes the signal.
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\begin{document}


\title{Fluctuation spectra and force generation in non-equilibrium systems}

\author{Alpha A. Lee}
%\email{alpha.lee@maths.ox.ac.uk}
\affiliation{John A. Paulson School of Engineering and Applied Sciences, Harvard University, Cambridge, MA 02138, USA} 
\affiliation{Mathematical Institute, Andrew Wiles Building, University of Oxford, Woodstock Road, Oxford OX2 6GG, UK}

\author{Dominic Vella}
%\email{dominic.vella@maths.ox.ac.uk}
\affiliation{Mathematical Institute, Andrew Wiles Building, University of Oxford, Woodstock Road, Oxford OX2 6GG, UK}

\author{John S. Wettlaufer}
%\email{john.wettlaufer@yale.edu}
\affiliation{Mathematical Institute, Andrew Wiles Building, University of Oxford, Woodstock Road, Oxford OX2 6GG, UK}
\affiliation{Yale University, New Haven, USA }
\affiliation{Nordita, Royal Institute of Technology and Stockholm University, SE-10691 Stockholm, Sweden} 


\begin{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 non-equilibrium 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 non-equilibrium 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 non-equilibrium interactions are encoded within the fluctuation spectrum. In this sense the noise becomes the signal.
\end{abstract} 


\makeatother
\maketitle

Force generation between passive inclusions in active, non-equilibrium systems underpins many phenomena in Nature. Bioinspired examples range from interactions between proteins on active membranes \cite{manneville1999activity,gov2004membrane} to swimmers confined by a soft boundary \cite{Spellings2015,paoluzzi2016,takatori2016acoustic}. On the large scale, such systems feature interactions between objects in a turbulent flow and ships on a stormy sea \cite{boersma1996maritime}. A fundamental physical question arising is whether there is a convenient physical framework that could describe force generation in the wide variety of out-of-equilibrum systems across different lengthscales? %Understanding those active forces is an important problem in biophysics as well as crucial for designing biomimetic and functional active materials. 

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 \cite{fily2012athermal,cates2012diffusive,redner2013structure,stenhammar2013continuum,buttinoni2013dynamical,cates2015motility}, and non-trivial behavior such as flocking and swarming is realizable in a non-equilibrium system \cite{toner1995}. Therefore, many studies focus on the microscopic physics of a particular active system to compute the force exerted on the embedded inclusions \cite[e.g.,][]{angelani2011effective,ray2014casimir,parra2014casimir,mallory2014anomalous,harder2014role,leite2016depletion,bechinger2016active}.  

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 non-monotonic, the force can oscillate between attraction and repulsion as a function of the separation between objects. By making simple approximations of the spectrum, we extract scaling properties of the fluctuation-induced force that compare favorably with recent simulations of the force between solid plates in a bath of self-propelling Brownian particles \cite{ni2015tunable}. 

%\dropcap{A}ctive, non-equilibrium systems are realized in many physical and biological processes. Examples range from external mechanical driving, as in the case of turbulence \cite{frisch1995turbulence}, to chemical gradients \cite{paxton2004catalytic,soto2014self} and high-energy chemical bonds \cite{golestanian2010synthetic,guix2014nano}, which many microswimmers, synthetic and natural alike, use as the means of propulsion \cite{ramaswamy2010mechanics,romanczuk2012active,cates2015motility}. Indeed, life itself is a particular case of such a non-equilibrium system. In such systems, non-equilibrium steady states are sustained by continuous energy input.
%
%The diverse physical mechanisms leading to non-equilibrium steady states have motivated many studies that focus on the microscopic physics of a particular system. Unlike the equilibrium counterpart, the continuous input of energy places convenient statistical concepts, such as 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 behaviour qualitatively different from the passive analogue \cite{fily2012athermal,cates2012diffusive,redner2013structure,stenhammar2013continuum,buttinoni2013dynamical,cates2015motility}. 
%
%A fundamental question to ask about those active systems is the force that it generates on passive inclusions. In biology, those active forces are ubiquitous and spans multiple lengthscales, from interactions between proteins on active membranes to swimmers in confined by a soft boundary. On an even larger scale, those forces are realised in settings such as objects in a turbulent flow or ships on a wavy sea. 

%\dropcap{A}ctive, non-equilibrium systems are realized in many physical and biological processes. Examples range from external mechanical driving, as in the case of turbulence \cite{frisch1995turbulence}, to chemical gradients \cite{paxton2004catalytic,soto2014self} and high-energy chemical bonds \cite{golestanian2010synthetic,guix2014nano}, which many microswimmers, synthetic and natural alike, use as the means of propulsion \cite{ramaswamy2010mechanics,romanczuk2012active,cates2015motility}. Indeed, life itself is a particular case of such a non-equilibrium system. In such systems, non-equilibrium steady states are sustained by continuous energy input.
%
%The diverse physical mechanisms leading to non-equilibrium steady states have motivated many studies that focus on the microscopic physics of a particular system. Unlike the equilibrium counterpart, the continuous input of energy places convenient statistical concepts, such as the partition function and the free energy, on more tenuous ground. In fact, theories and simulations of active Brownian particles show that self-propulsion induces complex phase behaviour qualitatively different from the passive analogue \cite{fily2012athermal,cates2012diffusive,redner2013structure,stenhammar2013continuum,buttinoni2013dynamical,cates2015motility}. However, there are very few general results that are broadly applicable to non-equilibrium systems; those that are known principally to pertain to a near equilibrium linear response \cite{mazur1963non,zwanzig2001nonequilibrium}, or to fluctuation relations for small systems \cite{jarzynski1997nonequilibrium,crooks1999entropy}.  
%
%We begin with the question: How can we distinguish a suspension of pollen at thermal equilibrium from a suspension of active microswimmers? A natural means of monitoring the fluctuation spectrum (the spectrum of noise due to random forces in the particles' dynamics) uses dynamic light scattering \cite{chu1974laser}. 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 --- 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 how the general properties of such spectra can provide a framework for understanding nonequilibrium behaviour. Equilibrium thermal fluctuations, such as that for a Brownian suspension or the Johnson--Nyquist noise \cite{nyquist1928thermal}, are usually associated with white noise corresponding to equipartition of energy between different modes. The key point here is that non-equilibrium processes have the potential to generate a nontrivial (even non-monotonic) 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 lengthscales of a homogeneous isotropic fluid \cite{wensink2012tubulence}.   

\section*{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? A natural means of monitoring the fluctuation spectrum (the spectrum of noise due to random forces in the particles' dynamics) uses dynamic light scattering \cite{chu1974laser}. 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 --- 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 the Johnson--Nyquist noise \cite{nyquist1928thermal}, are usually associated with white noise corresponding to equipartition of energy between different modes. The key point here is that non-equilibrium processes have the potential to generate a nontrivial (even non-monotonic) 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 lengthscales of a homogeneous isotropic fluid \cite{wensink2012meso}.   

%As energy can be difficult to define out of equilibrium, a natural macroscopic quantity is the disjoining force --- the force exerted by the medium on embedded bodies. 

The relation between fluctuation spectra and disjoining force may be examined by generalizing the classic calculation of Casimir \cite{casimir1948attraction}. We consider an effectively one dimensional system of two infinite, parallel plates separated by a distance $L$ and immersed in a non-equilibrium medium. We assume that the fluctuations are manifested by waves and neglect any damping and dispersion (in particular, we assume the absence of a mode-dependent dissipation mechanism).
%The fluctuation imparts a radiative stress, with the collective waves caused by the fluctuations being reflected by the plates. As the wavenumber can only take integer multiples of $\Delta k = \pi/L$ inside the plates because the waves are reflected by each plate. The radiation energy imparted by the waves to the inner surface of the plate then
%\begin{align}
%E_{\mathrm{in}} &=  \sum_{n=0}^{\infty} \Delta k \int_0^{\infty} \mathrm{d}k_y \; h\left(\sqrt{(n \Delta k)^2 + k_y^2}\right) \nonumber \\ 
%& = \sum_{n=0}^{\infty}  G(n \Delta k)  \Delta k, 
%\end{align}
%where $G(x) = \int_0^{\infty} \mathrm{d}k_y h\left(\sqrt{x^2 + k_y^2}\right) $ is the reduced energy spectrum. Note that $G(x)$ is non-monotonic if and only if the energy spectrum $h(k)$ is non-monotonic, thus the phenomenology of the system can be considered by studying $G(x)$. As the radiation energy outside the plate is a constant independent of plate separation, the fluctuation-induced force is given by
%\begin{equation}
%F_{\mathrm{fluct}} = \frac{\partial E_{\mathrm{in}}}{\partial L}. 
%\label{central_res}
%\end{equation}
The fluctuations impart a radiative stress. Defining the fluctuation spectrum 
\begin{equation}
G(k) \equiv \frac{\mathrm{d} E(k)}{ \mathrm{d}k}, 
\label{fluct_spec}
\end{equation} 
where $E(k)$ is the energy density of modes with wavenumber $k$, the radiation force per unit plate area due to waves with wavenumber between $k$ and $k+\delta k$ (where $k = |\mathbf{k}|$, the magnitude of the wavevector), with angle of incidence between $\theta$ and $\theta+\delta\theta$, is 
\begin{equation}
\delta F =  G(k) \delta k \cos^2 \theta \frac{\delta \theta}{2\pi}. 
\label{intensity_force}
\end{equation} 
One factor of $\mathrm{cosine}$ in equation (\ref{intensity_force}) is due to projecting the momentum in the horizontal direction, the other factor of $\mathrm{cosine}$ is due to momentum being spread over an area larger than the cross sectional length of the wave, and the factor of $2\pi$ accounts for the force per unit angle (see e.g.~\cite{landau1987fluid} for a derivation of Eq (\ref{intensity_force})). For isotropic fluctuations, we can consider $\delta \theta$ as an infinitesimal quantity and, upon integrating from $\theta = -\pi/2$ to $\pi/2$, we arrive at 
\begin{equation}
\delta F =  \frac{1}{4} G(k) \delta k. 
\end{equation}  

Outside the plates, any wavenumber is permitted and so
\begin{equation}
F_{\mathrm{out}} = \frac{1}{4}  \int_{0}^{\infty}G(k) \mathrm{d}k. 
\end{equation}
However,  the waves traveling perpendicular to and between the plates are restricted to take only integer multiples of $\Delta k = \pi/L$, because the waves are reflected by each plate. The force imparted by the waves to the inner surface of the plate is then
\begin{equation}
F_{\mathrm{in}} = \frac{1}{4} \sum_{n=1}^{\infty}  G (n \Delta k) \; \Delta k
\end{equation}
in one dimension. Thus, the \emph{net} disjoining force for a one dimensional system is given by 
\begin{equation}
\Ffluct= F_{\mathrm{in}} - F_{\mathrm{out}} = \frac{1}{4} \sum_{n=1}^{\infty}  G(n \Delta k) \; \Delta k - \frac{1}{4}\int_{0}^{\infty} G(k) \; \mathrm{d}k. 
\label{central_res}
\end{equation}
Note that $\Ffluct \lessgtr 0$ for all plate separations $L$ if the derivative $G'(k)\lessgtr 0$ for all $k$:  if a non-monotonic force is observed, it necessarily implies a non-monotonic 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 \cite{dean2010out}. For example, the effect of shaking protocols on force generation have been investigated theoretically for soft \cite{bartolo2003effective} and granular \cite{cattuto2006fluctuation} media. More generally, non-equilibrium Casimir forces have been computed for reaction-diffusion models with a broken fluctuation-dissipation relation \cite{brito2007generalized,rodriguez2011dynamical}, and spatial concentration \cite{spohn1983long} or thermal \cite{najafi2004forces} gradients. Moving beyond specific models, however, we argue that there are important \emph{generic} features of fluctuation-induced forces that can be fruitfully derived by considering the fluctuation spectrum, and treating it as a phenomenological quantity. 

\section*{Maritime Casimir Effect}

We first illustrate the central result, equation (\ref{central_res}), by applying it to the classic hydrodynamic example of ocean waves that are driven to a non-equilibrium steady state via wind-wave interactions. Empirically, $G(k)$ is measured to be non-monotonic and is well described by 
\begin{equation}
G(k) = \frac{\rho g \alpha }{2 k^3} \exp\left[ - \beta \left( \frac{k_0}{k} \right)^2 \right], 
\label{piersonspec}
\end{equation} 
where $\rho$ is the density of water, $g$ is gravitational acceleration, $k_0 = g/U^2$, $U$ is the wind speed, and $\alpha = 0.0081$ and $\beta = 0.74$ are fitted parameters \cite{pierson1964proposed}. We treat the one-dimensional case in which the wind blows in a direction perpendicular to the plates, hence waves traveling parallel to the plates are negligible.  Figure \ref{fluct_force}(a) shows that the resulting force is non-monotonic and oscillatory as a function of $L$: the force can be \emph{repulsive} ($\Ffluct >0$) as well as attractive ($\Ffluct < 0$). 
\begin{figure}
\centering 
\subfigure[]{\includegraphics[width=0.45\columnwidth]{fluct_force}}
\subfigure[]{\includegraphics[width=0.45\columnwidth]{new_scheme}}
\caption{(a) The fluctuation-induced force per unit length in the Maritime Casimir effect for different wind velocities, with the qualitative descriptors taken from the Beaufort scale. (b) 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, $\kmax\approx n\pi/L$ for some $n$; more often the sum underestimates the integral, leading to attraction. Note that the quantities on the axes are dimensionless.  }
\label{fluct_force}
\end{figure} 
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. In the limit $L \rightarrow 0$, fluctuations inside the plates are suppressed, and $\Ffluct = - F_{\mathrm{out}} = - \rho W \alpha U^4/( 16 \beta g)$. This attractive ``Maritime Casimir'' force has been observed since antiquity \citep[e.g.,][and refs therein]{boersma1996maritime} and experimentally measured in a wavetank \cite{denardo2009water}.  However, the \emph{non-monotonicity} of the spectrum gives rise to an \emph{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 (see Fig.~\ref{fluct_force}(b)): here the sum overestimates the integral in equation \eqref{central_res} and the outward force is greater than the inward force. Thus, the local maxima in the repulsive force are located at
\begin{equation}
L_{n} = n \frac{\pi}{k_{\mathrm{max}}}, 
\end{equation}
where $G'(k_{\mathrm{max}}) =0$; the separation between the force peaks is $\Delta L = \pi/k_{\mathrm{max}}$. In a maritime context, our calculation shows that if the separation between ships is $L>\pi/k_{\mathrm{max}} = \pi U^2 \sqrt{3/(2\beta)}/g $, the repulsive fluctuation force will keep the ships away from each other. 

Although quantitative measurement of this oscillatory hydrodynamic fluctuation force may be challenging, an oscillatory force has been observed in the acoustic analogue for which a non-monotonic fluctuation spectrum was produced \cite{larraza1998acoustic,larraza1998force}. Moreover, one-dimensional filaments in a flowing two-dimensional soap film with flow velocity above the flapping transition oscillate in phase or out of phase depending on their relative separation \cite{zhang2000flexible}, 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 note that the experimental framework used in pilot-wave hydrodynamics is ideally suited for direct experimental tests \cite[e.g.,][]{bush2015pilot}. 

We would expect that the fluctuation-induced force vanishes when the fluid is at thermal equilibrium. As a consequence of the equipartition theorem, the energy spectrum for a three-dimensional isotropic fluid at equilibrium is monotonic, and has the scaling \cite{forster1977large}
\begin{equation}
G_{\mathrm{eq}}(k) \propto k^2. 
\label{g_Eq}
\end{equation}
Noting that in 3D $\delta k = \delta k_x \delta k_y \delta k_z/(4 \pi k^2)$, equation (\ref{central_res}) becomes 
\begin{align}
\Ffluct&= \frac{1}{4 \pi} \int_0^{\infty} \mathrm{d}k_y \int_0^{\infty} \mathrm{d}k_z \left( \sum_{n=1}^{\infty}  \; \Delta k - \int_{0}^{\infty}  \; \mathrm{d}k \right) =0, \nonumber \\ 
\end{align}
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 non-equilibrium: in the continuum hydrodynamic setting, a non-zero fluctuation induced force implies non-equilibrium. We will comment on the ultraviolet divergence in Equation (\ref{g_Eq}) and the breakdown of continuum hydrodynamics in the section below.  

%We note that oscillatory forces exist in confined equilibrium fluids due to layering near the interface \cite{Horn1981}. However, this equilibrium layering force decays exponentially rather than the $\sim 1/L$ predicted by our theory and observed in simulations. 

\section*{General Phenomenology of Narrow Unimodal Spectra}

Importantly, the phenomenology of non-monotonic, and even oscillatory, forces is generic for sufficiently narrow, unimodal spectra. To see this, we perform a Taylor expansion of a general uni-modal spectrum, $G(k)$, about its maximum at $k = k_{\mathrm{max}}$; we find that
\begin{equation}
G(k) \approx \begin{cases}
G_0 \left[ 1 - \nu^{-2} (k-k_\mathrm{max})^2 \right], & |k-k_\mathrm{max}|<\nu \\ 
0 & \mathrm{otherwise}, 
\end{cases}
\end{equation}  
where $G_0 = G(k_{\mathrm{max}})$, $G_2 = G''(k_{\mathrm{max}})$ and $\nu = \sqrt{-2G_0/G_2}$ is the peak width based on a parabolic approximation. In the narrow-peak limit ($\nu \ll \pi/L$, $\nu \ll k_{\mathrm{max}}$), the force close to the $n^{th}$ peak is given by
%\begin{widetext}
\begin{equation}
F_{n} \approx  \begin{cases}
\frac{\pi G_0}{4 L} \left[ 1 - \nu^{-2} \left(\frac{n \pi}{L}-k_\mathrm{max} \right)^2 \right] - \frac{ G_0 \nu}{3}, & \left|\frac{n \pi}{L}-k_\mathrm{max} \right|<\nu, \\ 
 - \frac{ G_0 \nu}{3}  & \mathrm{otherwise}. 
\end{cases} 
\label{nthpeak}
\end{equation}  
%\end{widetext}
Equation (\ref{nthpeak}) shows that the $n^{th}$ maximum, located at $L = n \pi/k_\mathrm{max}$, has magnitude 
\begin{equation}
F_{n,\mathrm{max}} = \frac{G_0 \pi}{4 L} - \frac{ G_0 \nu }{3} = \frac{G_0 k_\mathrm{max}}{4 n} - \frac{ G_0 \nu}{3} , 
\label{max_asym_force}
\end{equation} 
and thus the maximum force is linear in inverse plate separation.  The force reaches a minimum when 
\begin{equation}
k_{\mathrm{max}} - \frac{n \pi}{ L} = \nu. 
\end{equation}
Writing  $L = L_{n} +l_n =  n \pi/k_\mathrm{max} + l_n$, where $ l_n$ is the half-width of the peak in force, we obtain
\begin{equation}
l_n  = n \pi \left(  \frac{1}{k_\mathrm{max}} - \frac{1}{\nu + k_\mathrm{max}} \right)\approx \frac{n \pi\nu}{\kmax^2}.
\label{peak_width}
\end{equation}
Therefore the width of the force maxima increases \emph{linearly} with $n$, and the positions of the $n^{th}$ mechanical equilibria ($\Ffluct =0$) in the limit of narrowly-peaked spectra ($\nu \ll k_{\mathrm{max}}$) are given by
%\begin{align}
%L_{n,eq} &=L_n \pm l_n \nonumber \\ 
%&=  n \pi \left( \frac{1}{k_{\mathrm{max}}}\pm \frac{1}{\nu+ k_\mathrm{max}} \mp \frac{1}{k_\mathrm{max}} \right),\nonumber \\
%&\approx n \pi \left( \frac{1}{k_{\mathrm{max}}}\mp \frac{\nu}{\kmax^2}\right).
%\label{sta_unsta_eq_eq}
%\end{align} 
\begin{equation}
L_{n,eq} =L_n \pm l_n\approx n \pi \left( \frac{1}{k_{\mathrm{max}}}\mp \frac{\nu}{\kmax^2}\right).
\label{sta_unsta_eq_eq}
\end{equation}
Here the positive (negative) branches correspond to unstable (stable) equilibria. Equations (\ref{max_asym_force}) and (\ref{peak_width}) predict that the force-displacement curve has peak repulsion $\propto 1/L$ and peak width $\propto n \propto L$. These predictions form a phenomenological theory that can be applied to systems where the fluctuation spectrum is not known \emph{a priori}: if force measurements are found to illustrate these scalings then we suggest that the underlying spectrum is likely to be narrow and uni-modal. 

We can now revisit the case of classic fluids at equilibrium. Obviously, the divergence in Equation (\ref{g_Eq}) as $k\rightarrow \infty$ is unphysical. This ultraviolet divergence is cured by noting that hydrodynamic fluctuations, as captured by the spectrum $G(k)$, are suppressed at the molecular lengthscale $k  \sim 2\pi/\sigma$ where $\sigma$ is the molecular diameter. Therefore, our analysis (Equation (\ref{nthpeak})) predicts an oscillatory fluctuation-induced force with a period that is comparable to the molecular diameter. This is indeed observed in confined equilibrium fluids  \cite{horn1981direct}, although clearly at the molecular lengthscale our hydrodynamic description breaks down and other physical phenomena, such as layering, become important. Importantly, while the oscillation wavelength of the disjoining force in equilibrium fluids is nanoscopic, of order the molecular scale, the oscillation wavelength in active non-equilibrium systems can be much larger than the size of the active particle, because the mechanism of force generation lies in a non trivial partition of energy. 

%Performing a Taylor expansion about $k = k_{\mathrm{max}}$, we find that (see Methods) the magnitude of the force of the $n^{th}$ maximum, located at $L_n = n \pi/k_\mathrm{max}$, is
%\begin{equation}
%F_{n,\mathrm{max}} = \frac{G_0 \pi}{4 L} - \frac{ \sqrt{2} G_0 }{3} \sqrt{\frac{G_0}{-G_2}} = \frac{G_0 k_\mathrm{max}}{4 n} - \frac{ \sqrt{2} G_0 }{3} \sqrt{\frac{G_0}{-G_2}}, 
%\label{max_asym_force}
%\end{equation} 
%where $G_0 = G(k_{\mathrm{max}})$ and $G_2 = G''(k_{\mathrm{max}})$. Similarly, the width of the $n^{th}$ maximum, defined as the distance between the two mechanical equilibria with $F=0$, is given by
%\begin{equation}
%l_n  = n \pi \left( \frac{1}{\sqrt{-2 G_0/G_2} + k_\mathrm{max}} - \frac{1}{k_\mathrm{max}} \right). 
%\label{peak_width}
%\end{equation}

%Interestingly, our results are in agreement with force generation in the rather disparate context of self-propelled active Brownian particles. Ni \emph{et al.} \cite{ni2015tunable} simulated self-propelled Brownian hard spheres confined between hard walls of length $W$ \footnote{In 2D, $\delta k = \delta k_x \delta k_y/(2\pi k)$, hence $F_{\mathrm{in}} = \frac{1}{4} \sum_{n=1}^{\infty} \Delta k \int_{0}^{\infty} \mathrm{d} q  \; \frac{G\left(\sqrt{(n \Delta k)^2 + q^2} \right)}{2 \pi \sqrt{(n \Delta k)^2 + q^2}}$. However, we can redefine $h(k) \equiv \int_{0}^{\infty}  \frac{G(\sqrt{q^2 + k^2})}{2 \pi \sqrt{q^2 + k^2}} \; \mathrm{d}q$ as an effective 1D spectrum, and substitute $h(k)$ in the place of $G(k)$ in Equation (\ref{central_res}), and proceed with the same asymptotic analysis for the narrow-peak limit.}. In those simulations, self-propulsion is described via a constant force $f$ in the direction $\hat{\mathbf{u}}_i(t)$ acting on the $i^{th}$ particle. The propulsion direction $\hat{\mathbf{u}}_i(t)$ undergoes rotational diffusion with diffusion coefficient $D_r = 3D_0/\sigma^2$, where $D_0$ is the translational diffusion coefficient and $\sigma$ the particle diameter.  Crucially, Ni \emph{et al.} found an oscillatory decay in the disjoining force (Fig.~\ref{abpa}a) that is quantitatively in agreement with our asymptotic scalings (\ref{max_asym_force}) and (\ref{peak_width}) (see Fig.~\ref{abpa}b). This agreement suggests that the underlying spectrum for active Brownian systems is narrow and non-monotonic \footnote{For smaller values of $f$ simulated in \cite{ni2015tunable}, the peaks are less pronounced and obscured by numerical noise.}. We note that oscillatory forces exist in confined equilibrium fluids due to layering near the interface \cite{Horn1981}. However, this equilibrium layering force decays exponentially rather then $\sim 1/L$ predicted by the theory and observed in simulations. 

\section*{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 \emph{et al.} \cite{ni2015tunable} simulated self-propelled Brownian hard spheres confined between hard walls of length $W$  and found an oscillatory decay in the disjoining force (Fig.~\ref{abpa}a). Although this system is two-dimensional, our analysis can be generalized: In 2D, $\delta k = \delta k_x \delta k_y/(2\pi k)$, and hence 
\begin{equation}
F_{\mathrm{in}} = \frac{1}{4} \sum_{n=1}^{\infty} \Delta k \int_{0}^{\infty}   \frac{G\left(\sqrt{(n \Delta k)^2 + q^2} \right)}{2 \pi \sqrt{(n \Delta k)^2 + q^2}} \mathrm{d} q.
\end{equation}
However, we can redefine 
\begin{equation} 
h(k) \equiv \int_{0}^{\infty}  \frac{G(\sqrt{q^2 + k^2})}{2 \pi \sqrt{q^2 + k^2}} \; \mathrm{d}q
\end{equation}
as an effective 1D spectrum and substitute $h(k)$ for $G(k)$ in Equation (\ref{central_res}). Performing the same asymptotic analysis as for the narrow-peak limit, the asymptotic scalings (\ref{max_asym_force}) and (\ref{peak_width}) are reproduced, in quantitative agreement with simulations (see Fig.~\ref{abpa}b). This agreement suggests that the underlying spectrum for active Brownian systems is narrow and non-monotonic \footnote{For smaller values of the active self-propulsion force $f$ simulated in \cite{ni2015tunable}, the peaks are less pronounced and obscured by numerical noise.}. 
\begin{figure*}
\centering
\subfigure[]{\includegraphics[width=0.4\columnwidth]{expt_abp}}
\subfigure[]{\includegraphics[width=0.4\columnwidth]{active_brownian}}
\subfigure[]{\includegraphics[width=0.4\columnwidth]{ideal_part}}
\caption{ 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 \cite{ni2015tunable}. In (a) and (b) the packing fraction in the bulk is $\rho \sigma^2 = 0.4$, where $\sigma$ is the particle diameter, the wall length is $W=10 \sigma$, and self-propulsion force  $f = 40 k_B T/\sigma$. (a) The raw force-displacement curve for $\rho \sigma^2 = 0.4$ from \cite{ni2015tunable}.  (b) When replotted as suggested by our asymptotic predictions (\ref{max_asym_force}) and (\ref{peak_width}) these data suggest that the underlying fluctuation spectrum is unimodal and has a narrow peak, with parameters $G_0 = 4.8 \times 10^{3}$ and $\nu = 0.2$. (As the peaks are spaced approximately $\sigma$ apart, we assume $k_{\mathrm{max}} = \pi/\sigma$.) The positions of the stable (closed circles) and unstable (open circles) mechanical equilibria (when $\Ffluct= 0$) are given by $x_{\mathrm{eq}}$, and the dotted lines are theoretical predictions (Eq. (\ref{sta_unsta_eq_eq})). The inset shows the force maxima in (a) $\propto 1/L$ and agrees with equation (\ref{max_asym_force}). (c) For ideal non-interacting self-propelled point particles, the function $A \sigma/L$ (black dotted line, \emph{c.f.,}  equation (\ref{f_fluct_ideal})) can be fitted (using $A$) to simulation data with $F \sigma^2/(W k_B T) = 40$ ($A = 182$) and $F \sigma^2/(W k_B T) = 20$ ($A = 31.6$). Here $W = 80 \sigma$.} 
\label{abpa}
\end{figure*} 

Further analytical insights can be obtained by considering the ideal particle limit in which Ni \emph{et al.}  \cite{ni2015tunable} observed that the disjoining pressure is attractive and decays monotonically with separation (similar results have been obtained by Ray \emph{et al.} \cite{ray2014casimir} for run-and-tumble active matter particles). 
%Although apparently at odds with our results, 
This observation can be explained within our framework by noting that the self-propulsion of point-particles induces a Gaussian colored noise  $\zeta(t)$ satisfying \cite{farage2015effective}
\begin{equation}
\left< \zeta(t)\right> = 0, \; \; \left< \zeta(t)\zeta(t')\right> = \frac{f^2}{3} e^{-2 D_r |t-t'|}, 
\end{equation} 
where $f$ is the active self-propulsion force and $D_r$ is the rotational diffusion coefficient. In the frequency domain, the fluctuation spectrum $S(\omega)$ is the Fourier transform of the time-correlation function and is
\begin{equation}
S(\omega) = \frac{4 D_r f^2}{3} \frac{1}{4 D_r^2 + \omega^2}.  
\label{S_colour}
\end{equation}
The Lorentzian noise spectrum of equation (\ref{S_colour}) deviates from the entropy-maximising white noise. Assuming a linear dispersion relation, we note first that the spectrum is now monotonic: we expect to see a monotonic force--displacement relation, as observed by Ni \emph{et al.} \cite{ni2015tunable}. Furthermore, the degree of freedom in the direction parallel to the plates can be integrated,  
%to give an effective 1D spectrum 
%\begin{equation}
%h(k) = \frac{f^2}{3 \pi  } \frac{\mathrm{arcsech}\left( \frac{v k}{ \sqrt{4 D_r^2 + v^2 k^2}}\right)}{\sqrt{4 D_r^2 + v^2 k^2}}
%\label{eff_1D}
%\end{equation} 
%and substituting equation (\ref{eff_1D}) into equation (\ref{central_res}) 
yielding  
\begin{equation}
\Ffluct \propto -\frac{f^2}{L}, 
\label{f_fluct_ideal}
\end{equation}
for large $L$. Fig.~\ref{abpa}(c) shows that the disjoining pressure obtained from simulations is consistent with this scaling: the decay $\propto 1/L$ and doubling the activity $f$ increases the prefactor by a factor of $5.6$, very nearly the predicted factor of $4$. Since oscillatory force decay is only seen for finite, active particles, it seems that the coupling between excluded volume interactions and active self-propulsion gives rise to a non-monotonic spectrum and the oscillatory decay seen in Fig.~\ref{abpa}(a). 

Non-monotonic energy spectra are also found in the continuum hydrodynamic description of active particles \cite{wensink2012meso,bratanov2015new}, as well as active swimmers in a fluid \cite{slomka2015generalized}. For a wide class of such ``active turbulent'' systems, the fluctuation spectra takes the analytical form \cite{bratanov2015new}
\begin{equation}
G(k) = E_0 k^{\alpha} e^{- \beta k^2 },
\label{bratanov}
\end{equation}
where $E_0$, $\alpha$ and $\beta $ are constants that depend on the underlying microscopic model. This spectrum is narrowly peaked when $\alpha/\beta\gg 1/\beta$, i.e. $\alpha\gg1$. Although Equation (\ref{bratanov}) 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 {\em also} non-monotonic \cite{slomka2015generalized}. Therefore, our asymptotic framework, (\ref{max_asym_force}) -- (\ref{sta_unsta_eq_eq}), 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 \cite{slomka2016geometry} to be an oscillatory function of plate separation; this supports the oscillatory force framework reported here. Nevertheless, experimental or numerical measurements of Casimir forces in active turbulent systems will serve as a test-bed of our formalism. 

\section*{Conclusion}

There are of course a plethora of ways to prepare non-equilibrium systems.  We suggest that an organizing principle for force generation is the fluctuation spectrum --- the active species drives a non-equipartition 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 non-equilibrium 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 non-equilibrium systems by providing the bridge between microscopic calculations \cite{takatori2014swim,solon2015pressure,yan2015swim}, measurements of the fluctuation spectra \cite{chu1974laser} and the varied measurements of Casimir interactions \cite{lamoreaux1997demonstration, munday2009measured, sushkov2011observation}.  

In particular, while the fluctuation spectrum of equilibrium fluids vanishes at the molecular scale, so that force oscillations are seen at the molecular lengthscale (e.g. \cite{horn1981direct}), it is the case that a hydrodynamic system with a force oscillation wavelength much larger than the molecular lengthscale {\em must be out of equilibrium} (because the thermal fluctuation spectrum, $G\sim k^2$, is monotonic). As a corollary, out-of-equilibrium systems can exhibit force oscillations with wavelengths significantly longer than the size of the active particles. More generally, because time reversal symmetry requires equilibrium \cite{pomeau1982symetrie}, it would appear prudent to examine the time correlations in the systems we have studied here. Additionally,  another form of an ``active fluid" can be constructed in a pure system using, for example, a thermally non-equilibrium steady state; temperature fluctuations in such a system have been observed to give rise to long-range Casimir-like behavior \cite{kirkpatrick2013giant,aminov2015fluctuation}. Hence, an intriguing possibility suggested by our analysis is that rather than tuning forces by controlling the nature (e.g., dielectric properties \cite{french2010long}) of the bounding walls, one can envisage actively controlling the fluctuation spectra of the intervening material.  Indeed, a natural speculation is that swimmers in biological (engineering) settings could (be designed to) actively control the forces they experience in confined geometries.


\acknowledgments
%Comments and suggestions from J.F. Brady, M. Cates, D. Frenkel and B.I. Halperin are gratefully acknowledged.  
This work was supported by an EPSRC Research Studentship and Fulbright Scholarship (AAL) and by the European Research Council (Starting Grant GADGET No.~637334 to DV). JSW acknowledges support from Swedish Research Council Grant No. 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).


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%\bibinfo{author}{\bibfnamefont{J.}~\bibnamefont{Slomka}},
% \bibnamefont{and} \bibinfo{author}{\bibfnamefont{J.}~\bibnamefont{Dunkel}},
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%  \bibinfo{author}{\bibfnamefont{Y.}~\bibnamefont{Fily}},
%  \bibinfo{author}{\bibfnamefont{A.}~\bibnamefont{Baskaran}},
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%  \bibinfo{author}{\bibfnamefont{W.}~\bibnamefont{Fily}}, \bibnamefont{and}
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%
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%  Parsegian}}]{munday2009measured}
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%  \bibinfo{author}{\bibfnamefont{V.~A.} \bibnamefont{Parsegian}},
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%
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%  \bibinfo{author}{\bibfnamefont{D.}~\bibnamefont{Dalvit}}, \bibnamefont{and}
%  \bibinfo{author}{\bibfnamefont{S.}~\bibnamefont{Lamoreaux}},
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%  \bibinfo{author}{\bibfnamefont{M.}~\bibnamefont{Kardar}},
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%  et~al.}}]{french2010long}
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%  \bibinfo{author}{\bibfnamefont{V.~A.} \bibnamefont{Parsegian}},
%  \bibinfo{author}{\bibfnamefont{R.}~\bibnamefont{Podgornik}},
%  \bibinfo{author}{\bibfnamefont{R.~F.} \bibnamefont{Rajter}},
%  \bibinfo{author}{\bibfnamefont{A.}~\bibnamefont{Jagota}},
%  \bibinfo{author}{\bibfnamefont{J.}~\bibnamefont{Luo}},
%  \bibinfo{author}{\bibfnamefont{D.}~\bibnamefont{Asthagiri}},
%  \bibinfo{author}{\bibfnamefont{M.~K.} \bibnamefont{Chaudhury}},
%  \bibinfo{author}{\bibfnamefont{Y.~M.} \bibnamefont{Chiang}},
%  \bibinfo{author}{\bibfnamefont{S.}~\bibnamefont{Granick}},
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%  \textbf{\bibinfo{volume}{82}}, \bibinfo{pages}{1887} (\bibinfo{year}{2010}).
%
%\end{thebibliography}

\end{document} 
casimir-pnas.bbl
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\bibitem [{\citenamefont {Parra-Rojas}\ and\ \citenamefont
  {Soto}(2014)}]{parra2014casimir}%
  \BibitemOpen
  \bibfield  {author} {\bibinfo {author} {\bibfnamefont {C.}~\bibnamefont
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\bibitem [{\citenamefont {Mallory}\ \emph {et~al.}(2014)\citenamefont
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\bibitem [{\citenamefont {Harder}\ \emph {et~al.}(2014)\citenamefont {Harder},
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  \BibitemOpen
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\bibitem [{\citenamefont {Leite}\ \emph {et~al.}(2016)\citenamefont {Leite},
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  \BibitemOpen
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\bibitem [{\citenamefont {Bechinger}\ \emph {et~al.}(2016)\citenamefont
  {Bechinger}, \citenamefont {Di~Leonardo}, \citenamefont {L{\"o}wen},
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  \BibitemOpen
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  }\href@noop {} {\bibfield  {journal} {\bibinfo  {journal} {Reviews of Modern
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  (\bibinfo {year} {2016})}\BibitemShut {NoStop}%
\bibitem [{\citenamefont {Ni}\ \emph {et~al.}(2015)\citenamefont {Ni},
  \citenamefont {Stuart},\ and\ \citenamefont {Bolhuis}}]{ni2015tunable}%
  \BibitemOpen
  \bibfield  {author} {\bibinfo {author} {\bibfnamefont {R.}~\bibnamefont
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\bibitem [{\citenamefont {Chu}(1974)}]{chu1974laser}%
  \BibitemOpen
  \bibfield  {author} {\bibinfo {author} {\bibfnamefont {B.}~\bibnamefont
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  (\bibinfo  {publisher} {Academic Press, New York},\ \bibinfo {year}
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\bibitem [{\citenamefont {Nyquist}(1928)}]{nyquist1928thermal}%
  \BibitemOpen
  \bibfield  {author} {\bibinfo {author} {\bibfnamefont {H.}~\bibnamefont
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  (\bibinfo {year} {1928})}\BibitemShut {NoStop}%
\bibitem [{\citenamefont {Wensink}\ \emph {et~al.}(2012)\citenamefont
  {Wensink}, \citenamefont {Dunkel}, \citenamefont {Heidenreich}, \citenamefont
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  \BibitemOpen
  \bibfield  {author} {\bibinfo {author} {\bibfnamefont {H.~H.}\ \bibnamefont
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  {author} {\bibfnamefont {K.}~\bibnamefont {Drescher}}, \bibinfo {author}
  {\bibfnamefont {R.~E.}\ \bibnamefont {Goldstein}}, \bibinfo {author}
  {\bibfnamefont {H.}~\bibnamefont {L{\"o}wen}}, \ and\ \bibinfo {author}
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\bibitem [{\citenamefont {Casimir}(1948)}]{casimir1948attraction}%
  \BibitemOpen
  \bibfield  {author} {\bibinfo {author} {\bibfnamefont {H.~B.}\ \bibnamefont
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\bibitem [{\citenamefont {Landau}\ and\ \citenamefont
  {Lifshitz}(1987)}]{landau1987fluid}%
  \BibitemOpen
  \bibfield  {author} {\bibinfo {author} {\bibfnamefont {L.~D.}\ \bibnamefont
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  (\bibinfo  {publisher} {Butterworth-Heinemann Ltd},\ \bibinfo {year}
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\bibitem [{\citenamefont {Dean}\ and\ \citenamefont
  {Gopinathan}(2010)}]{dean2010out}%
  \BibitemOpen
  \bibfield  {author} {\bibinfo {author} {\bibfnamefont {D.~S.}\ \bibnamefont
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  {Physical Review E}\ }\textbf {\bibinfo {volume} {81}},\ \bibinfo {pages}
  {041126} (\bibinfo {year} {2010})}\BibitemShut {NoStop}%
\bibitem [{\citenamefont {Bartolo}\ \emph {et~al.}(2003)\citenamefont
  {Bartolo}, \citenamefont {Ajdari},\ and\ \citenamefont
  {Fournier}}]{bartolo2003effective}%
  \BibitemOpen
  \bibfield  {author} {\bibinfo {author} {\bibfnamefont {D.}~\bibnamefont
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  }\href@noop {} {\bibfield  {journal} {\bibinfo  {journal} {Physical Review
  E}\ }\textbf {\bibinfo {volume} {67}},\ \bibinfo {pages} {061112} (\bibinfo
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\bibitem [{\citenamefont {Cattuto}\ \emph {et~al.}(2006)\citenamefont
  {Cattuto}, \citenamefont {Brito}, \citenamefont {Marconi}, \citenamefont
  {Nori},\ and\ \citenamefont {Soto}}]{cattuto2006fluctuation}%
  \BibitemOpen
  \bibfield  {author} {\bibinfo {author} {\bibfnamefont {C.}~\bibnamefont
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\bibitem [{\citenamefont {Brito}\ \emph {et~al.}(2007)\citenamefont {Brito},
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  \BibitemOpen
  \bibfield  {author} {\bibinfo {author} {\bibfnamefont {R.}~\bibnamefont
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  (\bibinfo {year} {2007})}\BibitemShut {NoStop}%
\bibitem [{\citenamefont {Rodriguez-Lopez}\ \emph {et~al.}(2011)\citenamefont
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  {Soto}}]{rodriguez2011dynamical}%
  \BibitemOpen
  \bibfield  {author} {\bibinfo {author} {\bibfnamefont {P.}~\bibnamefont
  {Rodriguez-Lopez}}, \bibinfo {author} {\bibfnamefont {R.}~\bibnamefont
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\bibitem [{\citenamefont {Spohn}(1983)}]{spohn1983long}%
  \BibitemOpen
  \bibfield  {author} {\bibinfo {author} {\bibfnamefont {H.}~\bibnamefont
  {Spohn}},\ }\href@noop {} {\bibfield  {journal} {\bibinfo  {journal} {Journal
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  \bibinfo {pages} {4275} (\bibinfo {year} {1983})}\BibitemShut {NoStop}%
\bibitem [{\citenamefont {Najafi}\ and\ \citenamefont
  {Golestanian}(2004)}]{najafi2004forces}%
  \BibitemOpen
  \bibfield  {author} {\bibinfo {author} {\bibfnamefont {A.}~\bibnamefont
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  {Europhysics Letters}\ }\textbf {\bibinfo {volume} {68}},\ \bibinfo {pages}
  {776} (\bibinfo {year} {2004})}\BibitemShut {NoStop}%
\bibitem [{\citenamefont {Pierson}\ and\ \citenamefont
  {Moskowitz}(1964)}]{pierson1964proposed}%
  \BibitemOpen
  \bibfield  {author} {\bibinfo {author} {\bibfnamefont {W.~J.}\ \bibnamefont
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  \bibinfo {pages} {5181} (\bibinfo {year} {1964})}\BibitemShut {NoStop}%
\bibitem [{\citenamefont {Denardo}\ \emph {et~al.}(2009)\citenamefont
  {Denardo}, \citenamefont {Puda},\ and\ \citenamefont
  {Larraza}}]{denardo2009water}%
  \BibitemOpen
  \bibfield  {author} {\bibinfo {author} {\bibfnamefont {B.~C.}\ \bibnamefont
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  }\href@noop {} {\bibfield  {journal} {\bibinfo  {journal} {American Journal
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\bibitem [{\citenamefont {Larraza}\ and\ \citenamefont
  {Denardo}(1998)}]{larraza1998acoustic}%
  \BibitemOpen
  \bibfield  {author} {\bibinfo {author} {\bibfnamefont {A.}~\bibnamefont
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\bibitem [{\citenamefont {Larraza}\ \emph {et~al.}(1998)\citenamefont
  {Larraza}, \citenamefont {Holmes}, \citenamefont {Susbilla},\ and\
  \citenamefont {Denardo}}]{larraza1998force}%
  \BibitemOpen
  \bibfield  {author} {\bibinfo {author} {\bibfnamefont {A.}~\bibnamefont
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\bibitem [{\citenamefont {Zhang}\ \emph {et~al.}(2000)\citenamefont {Zhang},
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  \BibitemOpen
  \bibfield  {author} {\bibinfo {author} {\bibfnamefont {J.}~\bibnamefont
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  {} {\bibfield  {journal} {\bibinfo  {journal} {Nature}\ }\textbf {\bibinfo
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  {2000})}\BibitemShut {NoStop}%
\bibitem [{\citenamefont {Bush}(2015)}]{bush2015pilot}%
  \BibitemOpen
  \bibfield  {author} {\bibinfo {author} {\bibfnamefont {J.~W.~M.}\
  \bibnamefont {Bush}},\ }\href@noop {} {\bibfield  {journal} {\bibinfo
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  {NoStop}%
\bibitem [{\citenamefont {Forster}\ \emph {et~al.}(1977)\citenamefont
  {Forster}, \citenamefont {Nelson},\ and\ \citenamefont
  {Stephen}}]{forster1977large}%
  \BibitemOpen
  \bibfield  {author} {\bibinfo {author} {\bibfnamefont {D.}~\bibnamefont
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\bibitem [{\citenamefont {Horn}\ and\ \citenamefont
  {Israelachvili}(1981)}]{horn1981direct}%
  \BibitemOpen
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\bibitem [{Note1()}]{Note1}%
  \BibitemOpen
  \bibinfo {note} {For smaller values of the active self-propulsion force $f$
  simulated in \cite {ni2015tunable}, the peaks are less pronounced and
  obscured by numerical noise.}\BibitemShut {Stop}%
\bibitem [{\citenamefont {Farage}\ \emph {et~al.}(2015)\citenamefont {Farage},
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  {Brader}}]{farage2015effective}%
  \BibitemOpen
  \bibfield  {author} {\bibinfo {author} {\bibfnamefont {T.}~\bibnamefont
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\bibitem [{\citenamefont {Bratanov}\ \emph {et~al.}(2015)\citenamefont
  {Bratanov}, \citenamefont {Jenko},\ and\ \citenamefont
  {Frey}}]{bratanov2015new}%
  \BibitemOpen
  \bibfield  {author} {\bibinfo {author} {\bibfnamefont {V.}~\bibnamefont
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  {} {\bibfield  {journal} {\bibinfo  {journal} {Proceedings of the National
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\bibitem [{\citenamefont {S{\l}omka}\ and\ \citenamefont
  {Dunkel}(2015)}]{slomka2015generalized}%
  \BibitemOpen
  \bibfield  {author} {\bibinfo {author} {\bibfnamefont {J.}~\bibnamefont
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\bibitem [{\citenamefont {S{\l}omka}\ and\ \citenamefont
  {Dunkel}(2016)}]{slomka2016geometry}%
  \BibitemOpen
  \bibfield  {author} {\bibinfo {author} {\bibfnamefont {J.}~\bibnamefont
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  {Dunkel}},\ }\href@noop {} {\bibfield  {journal} {\bibinfo  {journal} {arXiv
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\bibitem [{\citenamefont {Takatori}\ \emph {et~al.}(2014)\citenamefont
  {Takatori}, \citenamefont {Yan},\ and\ \citenamefont
  {Brady}}]{takatori2014swim}%
  \BibitemOpen
  \bibfield  {author} {\bibinfo {author} {\bibfnamefont {S.~C.}\ \bibnamefont
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  }\href@noop {} {\bibfield  {journal} {\bibinfo  {journal} {Physical Review
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  (\bibinfo {year} {2014})}\BibitemShut {NoStop}%
\bibitem [{\citenamefont {Solon}\ \emph {et~al.}(2015)\citenamefont {Solon},
  \citenamefont {Fily}, \citenamefont {Baskaran}, \citenamefont {Cates},
  \citenamefont {Kafri}, \citenamefont {Kardar},\ and\ \citenamefont
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  \BibitemOpen
  \bibfield  {author} {\bibinfo {author} {\bibfnamefont {A.~P.}\ \bibnamefont
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  {\bibfnamefont {Y.}~\bibnamefont {Kafri}}, \bibinfo {author} {\bibfnamefont
  {M.}~\bibnamefont {Kardar}}, \ and\ \bibinfo {author} {\bibfnamefont
  {J.}~\bibnamefont {Tailleur}},\ }\href@noop {} {\bibfield  {journal}
  {\bibinfo  {journal} {Nature Physics}\ }\textbf {\bibinfo {volume} {11}},\
  \bibinfo {pages} {673} (\bibinfo {year} {2015})}\BibitemShut {NoStop}%
\bibitem [{\citenamefont {Yan}\ and\ \citenamefont
  {Brady}(2015)}]{yan2015swim}%
  \BibitemOpen
  \bibfield  {author} {\bibinfo {author} {\bibfnamefont {W.}~\bibnamefont
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  (\bibinfo {year} {2015})}\BibitemShut {NoStop}%
\bibitem [{\citenamefont {Lamoreaux}(1997)}]{lamoreaux1997demonstration}%
  \BibitemOpen
  \bibfield  {author} {\bibinfo {author} {\bibfnamefont {S.~K.}\ \bibnamefont
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  {Physical Review Letters}\ }\textbf {\bibinfo {volume} {78}},\ \bibinfo
  {pages} {5} (\bibinfo {year} {1997})}\BibitemShut {NoStop}%
\bibitem [{\citenamefont {Munday}\ \emph {et~al.}(2009)\citenamefont {Munday},
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  {Parsegian}}]{munday2009measured}%
  \BibitemOpen
  \bibfield  {author} {\bibinfo {author} {\bibfnamefont {J.~N.}\ \bibnamefont
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  }\href@noop {} {\bibfield  {journal} {\bibinfo  {journal} {Nature}\ }\textbf
  {\bibinfo {volume} {457}},\ \bibinfo {pages} {170} (\bibinfo {year}
  {2009})}\BibitemShut {NoStop}%
\bibitem [{\citenamefont {Sushkov}\ \emph {et~al.}(2011)\citenamefont
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  \BibitemOpen
  \bibfield  {author} {\bibinfo {author} {\bibfnamefont {A.}~\bibnamefont
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  {\bibfield  {journal} {\bibinfo  {journal} {Nature Physics}\ }\textbf
  {\bibinfo {volume} {7}},\ \bibinfo {pages} {230} (\bibinfo {year}
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\bibitem [{\citenamefont {Pomeau}(1982)}]{pomeau1982symetrie}%
  \BibitemOpen
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  \citenamefont {Parsegian}, \citenamefont {Podgornik}, \citenamefont {Rajter},
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\end{thebibliography}%
