Energy dissipation and fluctuations in a driven liquid

Significance

Characterizing how energy consumption changes the properties of model nonequilibrium systems, particularly those with time-dependent steady states, remains an open problem—one that is key to advancing our understanding of a wide range of experimental and biological systems. Here, we derive a relation between the work done on a system of particles driven by time-periodic forces and the force fluctuations in the system. This relation gives us information about how pushing the system out of equilibrium modifies its structure and transport properties and leads to phase separation. Our results suggest ways to explore how energy dissipation can be used to tune the properties of nonequilibrium materials.

Abstract

Minimal models of active and driven particles have recently been used to elucidate many properties of nonequilibrium systems. However, the relation between energy consumption and changes in the structure and transport properties of these nonequilibrium materials remains to be explored. We explore this relation in a minimal model of a driven liquid that settles into a time periodic steady state. Using concepts from stochastic thermodynamics and liquid state theories, we show how the work performed on the system by various nonconservative, time-dependent forces—this quantifies a violation of time reversal symmetry—modifies the structural, transport, and phase transition properties of the driven liquid.

Minimal models of active matter have provided an analytically and computationally tractable test bed to study nonequilibrium systems. Phase transitions in some classes of model systems composed of self-propelled particles are beginning to be characterized (1–3). Recent work has also studied nucleation phenomena (4, 5) and obtained expressions for pressure and other mechanical properties of active media (6, 7). Despite these advances, understanding the connections between energy consumption and the structural properties of these systems remains a challenging problem (8).

In our work, we explore these connections in a model nonequilibrium liquid driven by time-periodic forces. The rotational dynamics that result from the driving have similarities with a range of systems, including colloids in a periodically changing magnetic or electric field (9–12), shaken plastic particles and chiral wires (13, 14), and chemical and biological microswimmers with active rotational degrees of freedom (15, 16). Despite their relevance to this wide range of experimental and biological systems, model systems with rotational dynamics have only recently begun to be studied (17–19).

The central points of this paper are as follows: First, we describe the class of driven liquids considered in this paper and their associated phase diagrams. Second, we show that the density fluctuations in our nonequilibrium system are surprisingly well-described by Gaussian statistics (20). Within this effective description, we are able to write down simple scaling relations for the amount of work performed on the system due to the nonequilibrium driving forces. These analytical predictions are validated by simulation data from many-particle systems. Third, we derive a relation that shows how the rate of work done on the system changes the fluctuations in the conservative forces experienced by the particles. In other words, our relation describes how energy dissipation changes the structural properties of the nonequilibrium material. This relation between work and force fluctuations can be viewed as an instantiation of the Harada–Sasa relation that connects work performed in a nonequilibrium system to a breakdown of the fluctuation dissipation relation (21, 22). We demonstrate this relation numerically for a variety of driven systems to firmly establish its general nature. Fourth, we use a minimal model to demonstrate how a breakdown of the fluctuation dissipation relation can change the diffusion constant of a system. While our driven liquid is substantially more complex than the minimal model, the model provides intuition for the interplay between dissipation and transport. In particular, it shows how a violation of the fluctuation dissipation relation can lead to an increase in the diffusion constant. Finally, we show that such an enhancement of the diffusion constant can drive phase transitions in the liquid. Taken together, our results elucidate how a violation of time reversal symmetry can be used to alter the structural, transport, and phase transition properties of a liquid.

Phase Behavior of a Driven System

Our model system is composed of purely repulsive 2D disks whose positions evolve in time according to driven Brownian dynamics:

$${\dot{\text{𝐫}}}_i(t)=\frac{1}{\gamma }\left({\text{𝐅}}_{\mathrm{c,i}}(t)+{\text{𝐅}}_{\mathrm{d}}(t)\right)+{𝜼}_i(t),$$ [1]

where $\gamma$ is the friction constant and ${\text{𝐅}}_{\text{c,i}}$ is the conservative force on particle $i$ due to the Weeks–Chandler–Andersen interaction potential (23), and ${𝜼}_i(t)=({\eta }_{i,x}(t),{\eta }_{i,y}(t))$ are Gaussian-distributed random variables with $⟨{𝜼}_i(t)⟩=0$ and $⟨{\eta }_{i,\mu }(t){\eta }_{j,\nu }(t\prime )⟩=2D_0{\delta }_{i,j}{\delta }_{\mu ,\nu }\delta (t-t\prime )$, where $D_0$ is the bare diffusion constant. In all of our simulations and calculations, we set $\beta =1/k_{B}T=1$. All results reported here are for a number density of $\rho =N/L^2=0.45$. In addition to the conservative forces, half of the particles are driven by an external force acting on the center of mass of the particle whose direction changes periodically in time according to

$${\text{𝐅}}_{\text{d}}=A(\sin \theta {\widehat{e}}_x+\cos \theta {\widehat{e}}_y)$$ [2]

$$\theta =2\pi t/\tau ,$$ [3]

where A is an amplitude with units of force. For the other half of the particles, ${\text{𝐅}}_{\text{d}}=0$. Thus, in the zero-temperature limit, a single driven particle will trace a circle in the plane. ${\text{𝐅}}_{\text{d}}$ is the same for all driven particles, so that a pure system of driven particles, if we move to a frame of reference that is rotating with ${\text{𝐅}}_{\text{d}}$, will look the same as the equilibrium system. This model was motivated in part by a recent experimental active matter system developed by Han, Yan, Granick, and Luijten (10).

The driving force in Eq. 2 is characterized by a period $\tau$ and an amplitude $A$. The latter is quantified by the Peclet number, a dimensionless measure of the ratio of advective to diffusive velocity in the system. Here we define it as $Pe=\frac{A/\gamma }{D_0/r_0}$. For the results reported here, we choose units so that the bare, single-particle diffusion coefficient ($D_0$), particle size ($r_0$), and friction coefficient ($\gamma$) are all set to 1. The period of the driving force is measured in units of time set by $t_0=r_0^2/D_0$. We found three distinct regimes in the part of $Pe,\tau$ space that we studied here (Fig. 1), which we characterized using an order parameter $\mathrm{\Omega }$ (defined in Materials and Methods) that measures the degree of mixing. At low $Pe$, driven and passive particles remain mixed and the system is homogeneous. As $Pe$ is increased with $\tau$ fixed at one of the values in Fig. 1, Top, they segregate into regions of purely driven or passive particles; the change is characterized by a very steep increase in $\mathrm{\Omega }$ (Fig. S1). Similar to other strongly damped active systems with rotating dynamics, the interfaces in the phase-separated system have no particular orientation, and there is a particle current along the interface that decays rapidly into the bulk (24). The steady state is time-periodic with a period $\tau$. As $Pe$ is increased further, the system undergoes a transition to a mixed phase characterized by large variations in the local particle number density. As $\tau$ is increased, the segregated state persists over a smaller range of $Pe$. The nature of the transitions in this system and its relation to the one presented in ref. 10 are discussed in Supporting Information, section 1.

Fig. 1.

Phase separation of driven disks. (Top) Phase diagram of the system described by Eqs. 1–3 at number density $\rho =0.45$. At low $Pe$, the system is mixed (plus signs). As $Pe$ is increased, the system undergoes a transition to a phase-separated state (points). As $Pe$ is increased further, there is a transition to a mixed state characterized by large variations in local density (crosses). The locations of the transitions are defined as the maxima of $\partial \mathrm{\Omega }/\partial Pe$ (black lines). The green line is proportional to $P{e}^{2}b(\tau )$, where $b(\tau )$ is defined in Eq. 14. The purple line is proportional to the work done per cycle given by Eq. 7 using the fitting parameter ${\tau }_G$ extracted from the fits in Fig. 3. Contours were drawn using interpolated values. (Bottom) Snapshots of the final frame of a simulation in different regions of the phase diagram. Driven particles are colored red, and undriven particles are colored blue. (Bottom Left) At $Pe=15,\tau =0.1$. (Bottom Center) At $Pe=70,\tau =0.15$. (Bottom Right) At $Pe=150,\tau =0.2$.

In this work, we investigate how the nonequilibrium forces modify the structural and transport properties of the liquid leading up to the lower transition from region $a$ to $b$. Specifically, we are interested in the connection between the amount of energy input to the system and the resultant steady state. As a first step, in the next section we show that the density fluctuations in the driven liquid are surprisingly well-defined by Gaussian statistics. This result helps us obtain simple expressions for the amount of energy that is put into the system by the nonequilibrium forces as a function of $Pe$ and $\tau$. Subsequently, we show how this energy input renormalizes structural and transport properties of the system, eventually resulting in phase separation. These results provide a framework to understand how violation of time-reversal symmetry due to nonequilibrium forces can be used to modulate material properties.

Gaussian Density Fluctuations in the Driven Liquid

In many equilibrium liquids ranging from simple hard spheres to water, the statistics of fluctuations in the number of particles inside a small probe volume are surprisingly Gaussian (20, 25). This property has enabled the development of quantitatively accurate theories for certain thermodynamic properties such as solvation energies (25–27). A Gaussian description for density fluctuations also implies that changes in density due to external forces can be effectively captured within a linear response theory. In our context, linear response could help obtain relations between the driving forces and the energy supplied to the system.

To determine whether a Gaussian theory can describe the fluctuations out of equilibrium, we measured the statistics of number density fluctuations in the driven liquid in a small probe volume (20, 25) (Supporting Information, section 1). We find that they are indeed Gaussian to a good approximation at many points on the phase diagram (below the line where the system first phase separates). One example histogram is shown in Fig. 2. In Fig. S2, we show additional data to demonstrate that the statistics remain Gaussian even for points that come close to the phase transition line and for points above the second phase transition.

Fig. 2.

Number density probability distribution for a system with $Pe=20,\tau =0.1$. Points are the measured probability of finding N particles in a randomly selected probe volume of radius $1.75r_0$. The line is a Gaussian distribution with mean $⟨N⟩=4.32$ and variance ${\sigma }^2=1.25$. Error bars are smaller than the symbols. The density statistics are Gaussian to $\sim \pm 3\sigma$ from the mean. In Fig. S2, we show additional data that demonstrate that the statistics are Gaussian even for the values of $Pe$ and $\tau$ represented in Figs. 3–5 that come closest to the phase transition.

This finding is particularly unexpected since intuition and experience give us little reason to expect that the statistics would remain Gaussian out of equilibrium—for example, “giant” (i.e., non-Gaussian) number fluctuations have been observed in some active matter systems (28–30). However, we find Gaussian fluctuations for systems where on the order of $4kT$ of work is done per particle per cycle.

The Gaussian nature of density fluctuations allows us to predict how the work performed on the nonequilibrium liquid scales with $Pe$ and $\tau$. Here, and in the rest of the paper, we use the term work to denote the change in the energy of the system due to the action of nonconservative forces. This definition of work differs from commonly used conventions (21). Specifically, we define the rate at which work is done on the system by the nonconservative forces as

$$⟨\dot{w}⟩=-\frac{1}{N}\sum _{i=1}^N\frac{1}{\tau }{\int }_0^{\tau }\frac{⟨{\text{𝐅}}_{\mathrm{c},i}(t)⟩\cdot {\text{𝐅}}_{\mathrm{d},i}(t)}{\gamma }dt.$$ [4]

As we will show in the subsequent sections, this definition of work proves convenient to quantify the influence of the driving forces on the structural and transport properties of the system.

Given the Gaussian nature of density fluctuations, it is reasonable to speculate that the average restoring force, $⟨{\text{𝐅}}_{\mathrm{c},\mathrm{i}}(t)⟩$ in Eq. 4, exerted by the system in response to the driving forces, is a linear function of $Pe$. In the context of equilibrium liquids, Gaussian density fluctuations imply, within certain approximations (20, 31–33), that the coarse-grained field $\delta \rho (\text{𝐫},t)=\rho (\text{𝐫},t)-\overline{\rho }$, where $\overline{\rho }$ is the bulk density of the liquid, satisfies the following Gaussian equation of motion:

$$\dot{\delta \rho }(\text{𝐫},t)=-\frac{K_G}{{\gamma }_G}\delta \rho (\text{𝐫},t)+{\eta }_G(\text{𝐫},t).$$ [5]

Here $K_G$ is an effective spring constant, ${\gamma }_G$ is an effective friction, and ${\eta }_G$ is an effective noise with statistics $⟨{\eta }_G(t)⟩=0$ and $⟨{\eta }_G(t){\eta }_G(t\prime )⟩=2\delta (t-t\prime )\delta (\text{𝐫}-\text{𝐫}\prime )D_G$, with $D_G$ an effective diffusion. We provide an expression for $K_G$ in Supporting Information, section 3. We note that Eq. 5 does not conserve the density as it strictly should. The equivalent density conserving EOMs are available elsewhere (33); as we will demonstrate, Eq. 5 is more than adequate for our purposes. Eq. 5 can be driven out of equilibrium by adding an extra driving term $Pe{F}_{G,\mathrm{d}}(t)$ to it:

$$\dot{\delta \rho }(\text{𝐫},t)=-\frac{K_G}{{\gamma }_G}\delta \rho (\text{𝐫},t)+\frac{Pe{F}_{G,\mathrm{d}}(t)}{{\gamma }_G}+{\eta }_G(\text{𝐫},t).$$ [6]

Such a driving force has the effect of changing the local density and can mimic events where an active particle is driven into surrounding passive particles. The response of the system to this driving, $⟨\delta \rho ⟩$, scales linearly with $Pe$ (34). In this way, due to the Gaussian density fluctuations, we anticipate that the work performed on the system will be a quadratic function of $Pe$.

The scaling of $⟨\dot{w}⟩$ with $\tau$ can be similarly inferred using a simple scaling argument that assumes events are exponentially decorrelated with a characteristic time scale. Specifically, for values of $\tau$ much larger than the decorrelation time for fluctuations in the system, we simply anticipate that the work performed per cycle grows linearly with $\tau$. For values of $\tau$ much smaller than a correlation time, the work done per cycle can be a nonlinear function of $\tau$.

These scaling trends can be combined into an expression for $⟨\dot{w}⟩$ in terms of $Pe$ and $\tau$.

$$⟨\dot{w}⟩\propto P{e}^2\left[1-\frac{{\tau }_G}{\tau }[1-e^{-\tau /{\tau }_G}]\right].$$ [7]

The scaling of the work performed in Eq. 7 with $Pe$ and $\tau$ follows the expected trends outlined above, with ${\gamma }_G/K_G\equiv {\tau }_G$ acting as a correlation time. In Fig. 3, we show that $⟨\dot{w}⟩$ in the atomistic simulations does indeed scale as predicted in Eq. 7, with ${\tau }_G$ as a fitting parameter. We have checked that the scaling also holds for different driving protocols, including cases where the particles are not phase locked or have random phases and in systems with unequal number fraction of driven and undriven particles (Fig. S8).

Fig. 3.

Rate of work done per particle as a function of (Top) $Pe$ and (Bottom) $\tau$ by the driving forces in the full many-particle simulation, for $Pe$ ranging from 2.5 to 15 and $\tau$ ranging from 0.025 to 0.3. Lines are fits to Eq. 7. The parameter ${\tau }_G$ was fit separately to each line; there is a 5% variation in its value between fits. Error bars are smaller than the points.

Dissipation Modifies the Statistics of Force Fluctuations

We now demonstrate how the work done on the system by the nonequilibrium forces affects its microscopic statistics. We begin by noting that according to Floquet theory (35), the nonequilibrium steady state induced by time-periodic driving forces is also time-periodic. For the system to achieve such a time-periodic steady state, the increase in the internal energy of the system due to the total work, $w$, over each cycle has to be dissipated as heat. The average rate of heat emitted by the driven system (per particle) can be conveniently expressed in terms of the following stochastic integral, interpreted in the Stratonovich sense (36):

$$⟨\dot{q}⟩=-\frac{1}{N}\sum _{i=1}^N\frac{1}{\tau }{\int }_0^{\tau }⟨{\text{𝐅}}_{\mathrm{c},\mathrm{i}}\cdot {\dot{\text{𝐫}}}_{\mathrm{c},\mathrm{i}}⟩dt,$$ [8]

where ${\text{𝐅}}_{\mathrm{c},\mathrm{i}}$ denotes the conservative force vector for particle $i$, ${\dot{\text{𝐫}}}_{\mathrm{c},\mathrm{i}}$ is the rate of change of the position vector of particle $i$ due to the conservative and random forces only, the sum is over all of the particles in the system, and $\tau$ is the period of the driving. The average rate of heat emission $⟨\dot{q}⟩$ should equal the negative of the average rate at which work is performed on the system, $-⟨\dot{w}⟩$, given by Eq. 4.

Indeed, by adding together Eqs. 4 and 8, we obtain an energy balance expression,

$$\frac{1}{\tau }{\int }_0^{\tau }⟨\frac{dU}{dt}⟩dt=-\frac{1}{N}\sum _{i=1}^N\frac{1}{\tau }{\int }_0^{\tau }⟨{\text{𝐅}}_{\mathrm{c},\mathrm{i}}\cdot {\dot{\text{𝐫}}}_i⟩dt=⟨\dot{q}⟩+⟨\dot{w}⟩=0,$$ [9]

where ${\dot{\text{𝐫}}}_{\mathrm{i}}$ is the rate of change of the position vector of particle $i$ defined in Eq. 1. Expanding this energy balance equation, we find that $⟨\dot{w}⟩$ can be expressed in terms of the statistics of force fluctuations,

$$⟨\dot{w}⟩\propto ⟨{\text{𝐅}}^2⟩-{⟨{\text{𝐅}}^2⟩}_0,$$ [10]

where

$$⟨{\text{𝐅}}^2⟩=\frac{1}{N}\sum _{i=1}^N⟨{\text{𝐅}}_{\mathrm{c},\mathrm{i}}^2⟩$$ [11]

and

$${⟨{\text{𝐅}}^2⟩}_0=-\frac{\gamma }{2N}\sum _{i=1}^N\underset{t\to 0}{lim}⟨{\text{𝐅}}_{\mathrm{c},\mathrm{i}}(t)\cdot {\eta }_i(0)⟩.$$ [12]

In Supporting Information, section 5, we show that Eq. 10 does not depend on the convention (Itô or Stratonovich) used to interpret the stochastic integral. In the above equations, ${⟨{\text{𝐅}}^2⟩}_0$ captures the response of the nonequilibrium liquid following a random perturbation, while $⟨{\text{𝐅}}^2⟩$ describes the force fluctuations on a tagged particle in its nonequilibrium steady state. In Supporting Information, section 4, we show that in equilibrium the two terms in Eq. 10 are equal as a consequence of fluctuation dissipation theorem. Out of equilibrium, the work performed is positive, and the relation between fluctuations and response breaks down. In this case, the difference between fluctuations and response is predicted exactly by flux of heat, or alternately the rate at which work is performed, as illustrated in Eq. 10. We note that this expression is completely insensitive to the choice of nonconservative driving forces in Eq. 1. We also note that Eq. 10 is an instantiation of the Harada–Sasa relation (21, 22). Here we have obtained it for our nonequilibrium system following simple thermodynamic arguments.

We find numerically that the work performed in the many-particle system is indeed related to the breakdown of the equilibrium relation between fluctuations and response (Fig. 4). We verified Eq. 10 for a wide range of amplitudes and time periods of the driving force in the homogeneous part of the phase diagram. We have also verified that this result holds for driving forces in which the particles are not phase locked, have random phases, and in systems with unequal number fraction of driven and undriven particles (Fig. S8). The relation also holds when applied separately to driven and undriven particles in all of these cases—in other words, the work performed on average due to the driven particles predicts the change in the force fluctuations of the undriven particles. We now study the implications of this result for the diffusion constant of the nonequilibrium system.

Fig. 4.

Deviation from fluctuation–dissipation theorem as a function of the rate of work done per particle. Colors indicate values of $\tau$. The data collapse on to the line predicted by Eq. 10. Error bars are smaller than the points.

Enhanced Diffusion Due to Nonconservative Forces

In Brownian dynamics, the diffusion can be written as follows in terms of force autocorrelation functions:

$$D-D_0=\frac{1}{d}{\int }_0^{\mathrm{\infty }}\left[⟨\frac{\text{𝐅}(0)\cdot \text{𝐅}(t)}{{\gamma }^2}⟩+⟨\frac{𝜼(0)\cdot \text{𝐅}(t)}{\gamma }⟩\right]dt,$$ [13]

where $d$ denotes the dimensionality of the system. It is reasonable to ask whether the change in the force correlations that accompanies a breakdown of the fluctuation dissipation relation, Eq. 10, affects the diffusion coefficient of the driven liquid.

From simulations, we indeed find that the diffusion constant gets renormalized due to the driving forces and increases with the amplitude of the driving force after a transient region. The renormalization of the diffusion constant is very well-described by the functional form

$$D-D_{eq}=-a(\tau )Pe+b(\tau )P{e}^2$$ [14]

(Fig. 5). Understanding the basis for the renormalization of $D$ due to the nonconservative forces is important (37, 38)—as we will show, in the present context, it can help explain how the energy input due to the nonconservative forces can drive phase separation.

Fig. 5.

Scaling of the diffusion coefficient of a tagged particle in the liquid, $D$, with the amplitude of the driving force, $Pe$. The fit is of the form $D=D_{eq}-aPe+bP{e}^2$, where $a$ and $b$ depend on $\tau$. $D_{eq}$ is the equilibrium diffusion constant (Fig. S3). The intersection of the fits with the dotted line indicating $D_{eq}$ shift to the left, indicating that as $\tau$ increases, $D$ is a quadratic function of $Pe$ over a larger range. Colors indicate values of $\tau$. Error bars are smaller than the points.

To qualitatively understand the observed dependence of $D$ on the nonconservative forces, we consider a minimal model of a tracer particle diffusing in a fluid according to the equation of motion

$$\dot{\text{𝐫}}=\frac{h}{\gamma }\stackrel{\sim }{𝐅}(𝐫)+\stackrel{\sim }{𝜼}(t),$$ [15]

where $\stackrel{\sim }{\text{𝐅}}(\text{𝐫})$ is a spatially dependent force that can be designed to model the forces acting on a tagged particle in the liquid, $h$ is a parameter that tunes the coupling between the fluid and the tracer particle, and $\stackrel{\mathbf{\sim }}{𝜼}(t)$ is a Gaussian $\delta$-correlated white noise. To ensure no net drift, we constrain $\int \stackrel{\sim }{\text{𝐅}}(\text{𝐫})d\text{𝐫}=0$. We imagine sampling over many realizations of the force $\stackrel{\sim }{\text{𝐅}}(\text{𝐫})$ from a distribution to model the forces exerted by the fluid on a tracer particle. In the liquid considered in the previous sections, the statistics of force fluctuations on a tagged particle satisfy $⟨{\text{𝐅}}^2⟩={⟨{\text{𝐅}}^2⟩}_0$ when the system is in equilibrium. Equilibrium dynamics in Eq. 15 are achieved whenever $\stackrel{\sim }{\text{𝐅}}(\text{𝐫})$ can be expressed as a gradient of a potential, $\stackrel{\sim }{\text{𝐅}}(\text{𝐫})=-\nabla \stackrel{\sim }{U}(\text{𝐫})$. In such cases, it can be demonstrated that the equivalent relation $⟨{\stackrel{\sim }{\text{𝐅}}}^2⟩={⟨{\stackrel{\sim }{\text{𝐅}}}^2⟩}_0$ holds, where the averages $⟨\mathrm{\dots }⟩$ are taken both over the statistics of the random noise $\stackrel{\mathbf{\sim }}{𝜼}(t)$ and over many realizations of the force. The system can be driven out of equilibrium by ensuring that the force in Eq. 15 has a nonconservative component, $\stackrel{\sim }{\text{𝐅}}(\text{𝐫})={\stackrel{\sim }{\text{𝐅}}}_{\mathrm{c}}(\text{𝐫})+Pe{\stackrel{\sim }{\text{𝐅}}}_{\mathrm{d}}(\text{𝐫})$, where ${\stackrel{\sim }{\text{𝐅}}}_{\mathrm{c}}(\text{𝐫})=-\nabla \stackrel{\sim }{U}(\text{𝐫})$, ${\stackrel{\sim }{\text{𝐅}}}_{\mathrm{d}}(\text{𝐫})=\nabla \times \stackrel{\sim }{\text{𝐀}}(\text{𝐫})$. Like in the previous sections, $Pe$ tunes the magnitude of the nonconservative force.

A rough qualitative mapping between the forces in the minimal model and the forces in the atomistic liquid can be obtained using the following reasoning. The nonequilibrium driving in the atomistic liquid alters the pair correlation function between a driven and an undriven particle in two ways. First, the scaling of $⟨w⟩$ with $P{e}^2$ reveals that due to the driving forces the pair correlation function develops an anisotropic, time-dependent component that is also proportional to $Pe$. The nonconservative force introduced in the minimal model, $Pe{\stackrel{\sim }{\text{𝐅}}}_{\mathrm{d}}$, is meant to qualitatively simulate the effect of the anisotropic component of the atomistic pair correlation function. Second, the isotropic component of the pair correlation function can also be affected by the driving forces (Fig. S5). The potential $\stackrel{\sim }{U}$ in the minimal model is meant to simulate the effects of the isotropic component of the steady state pair correlation function.

In the presence of the nonconservative forces in the minimal model, as in the many-particle driven liquid, a breakdown of the fluctuation dissipation relation is predicted by the total amount of entropy dissipated by the system (22). Specifically, we use a perturbation theory (39, 40) (detailed in Supporting Information, sections 6–9) to show that to $O(h^2)$,

$$⟨\dot{\sigma }⟩=⟨{\stackrel{\sim }{\text{𝐅}}}^2⟩-{⟨{\stackrel{\sim }{\text{𝐅}}}^2⟩}_0=\frac{P{e}^2{h}^2}{V}\int \frac{d\text{𝐪}}{{(2\pi )}^d}⟨{\stackrel{\sim }{\text{𝐅}}}_{\mathrm{d}}(\text{𝐪})\cdot {\stackrel{\sim }{\text{𝐅}}}_{\mathrm{d}}(-\text{𝐪})⟩,$$ [16]

where $⟨\dot{\sigma }⟩$ denotes the average rate of entropy dissipation, analogous to $⟨\dot{w}⟩$ in the many-body liquid, d denotes the dimensionality of the minimal model, and V denotes the volume sampled by the particle in the minimal model.

For this minimal model, we obtained expressions for the diffusion constant in Eq. 15 to quadratic order in the parameters $h$ and $hPe$:

$$D-D_0=\alpha \frac{h^2}{V}\left(P{e}^2\int \frac{d\text{𝐪}}{{(2\pi )}^d}\frac{⟨{\stackrel{\sim }{\text{𝐅}}}_d(\text{𝐪})\cdot {\stackrel{\sim }{\text{𝐅}}}_d(-\text{𝐪})⟩}{{|\text{𝐪}|}^2}-\int ⟨{\stackrel{\sim }{U}}^2(\text{𝐫})⟩d\text{𝐫}\right),$$ [17]

where $\alpha =D_0/(d{(k_{B}T)}^2)$, and we set $⟨\stackrel{\sim }{U}⟩=0$ without loss of generality so that $⟨{\stackrel{\sim }{U}}^2⟩$ is simply the variance of energy fluctuations. In instances where the spectrum of force fluctuations is strongly peaked at a particular wave vector ${\text{𝐪}}^{\ast }$, the diffusion constant can be simply related to the average entropy dissipation rate:

$$D-D_0=\alpha \frac{⟨\dot{\sigma }⟩}{{|{\text{𝐪}}^{\ast }|}^2}-\alpha h^2⟨{\stackrel{\sim }{U}}^2⟩.$$ [18]

The dynamics of our driven liquid, specified by Eq. 1, are substantially more complicated than the minimal model considered. Nonetheless, the expressions in Eqs. 17 and 18 provide useful insight. By using the variance $⟨{\stackrel{\sim }{U}}^2⟩$ as a measure of the microscopic environment around a tagged particle in the many-body driven liquid and by associating the entropy production rate $⟨\dot{\sigma }⟩$ with the rate of work performed $⟨\dot{w}⟩$ in the many-body liquid, Eq. 18 demonstrates how the nonconservative forces can modify the diffusion properties of a particle in the fluid. In particular, the minimal model predicts that the diffusion constant can increase as $P{e}^2$ due to the energy dissipation from nonconservative forces. To probe how the microscopic environment around a tagged particle changes due to the driving forces, we measured the average energy of a driven tracer particle in a fluid, $\int g(\text{𝐫})u(\text{𝐫})d\text{𝐫}$, where $g(\text{𝐫})$ is the steady-state two-body pair correlation function of a tagged active particle in the driven liquid with 10% active particles (Fig. S5). The average energy increases with $Pe$. Effectively, the active driven particles are sampling configurations characteristic of an equilibrium state with higher temperature. In other words, the variance $⟨{\stackrel{\sim }{U}}^2⟩$, used in our minimal model, should increase with $Pe$. Taken together, these qualitative arguments suggest that the increase in the diffusion constant in our driven liquid is driven mainly by the work performed by the nonconservative forces.

Phase Transitions Due to Enhanced Diffusion

Finally, the increase of the diffusion constant due to the nonconservative forces renders the diffusion constant composition dependent. Specifically, unmixed regions with particles either being all driven or all undriven effectively have equilibrium dynamics and diffusion properties since the nonconservative forces do not induce any collisions in such regions. The diffusion constant in regions with mixed compositions can be enhanced (as described above) due to collisions induced by the nonconservative forces. As we demonstrate in Supporting Information, section 10, such a composition-dependent diffusion constant can drive a transition from the low drive mixed phase (a) to the phase separated region (b) in the phase diagram in Fig. 1 when

$$\lambda =\frac{b(\tau )P{e}^2}{D_{eq}+b(\tau )P{e}^2},$$ [19]

where $b(\tau )$ is defined in Eq. 14 and $\lambda$ is a constant defined in Supporting Information, section 10. Eq. 19 provides us with a lower bound on the value of $Pe$ where phase separation first occurs.

Indeed, we numerically find that the shape of the phase transition curve in Fig. 1 (solid green line) is well described by $b(\tau )P{e}^2=c\approx D_{\text{eq}}$, where $c$ is a constant, in accordance with Eq. 19 (with $\lambda \approx 0.5$). Further, our numerical simulations show $b(\tau )P{e}^2\propto ⟨w⟩$ to be a very good approximation (Fig. S7), and we also find that the shape of the phase transition curve is well-described by a line of constant $⟨w⟩$ (purple line in Fig. 1). The results in Eqs. 17–19 qualitatively show how the increase in diffusion due to energy dissipation can control the phase properties of our nonequilibrium liquid. Numerically, we find that the phase transition properties of our driven liquid are effectively controlled by $⟨w⟩$, the energy injected into it per cycle by the nonequilibrium driving forces.

Conclusions

From the rich physics of nonequilibrium materials, and in particular of active particles with rotating dynamics, it is clear that dissipation plays an important role in modifying the structural and dynamical properties of the steady states of these systems. Here, in the context of a class of systems with rotating dynamics, we have identified how the rate of work done by the external forces renormalizes force fluctuations. Using simplified descriptions of density fluctuations based on our observation of Gaussian density statistics, we were also able to model how the work performed in this many-body system depends on the nonequilibrium forces. Finally, using a minimal model, we explained the observed enhancement of the diffusion due to the nonequilibrium driving forces and proposed a relation between diffusion and dissipation. The renormalization of the diffusion due to dissipation also helped explain the observed dependence of the phase behavior on the magnitude $Pe$ of the driving force. Our results demonstrate how the material properties of nonequilibrium liquids can be tuned simply by violating time reversal symmetry and controlling the amount of energy put into the system.

Materials and Methods

The simulations were performed using a modified version of the LAMMPS simulation package. The density fluctuations in Fig. 2 were computed in randomly placed probe volumes of size $1.75r_0$. The diffusion constant was measured by computing the mean squared displacement (MSD). The work performed was computed by measuring the average steady state conservative force acting on the driven particles. The details of the work calculations and further details of the simulations are provided in Supporting Information, section 1.