Abstract
Active materials are characterized by continuous injection of energy at the microscopic level and typically cannot be adequately described by equilibrium thermodynamics. Here we study a class of active fluids in which equilibriumlike properties emerge when fluctuating and activated degrees of freedom are statistically decoupled, such that their mutual information is negligible. We analyse three paradigmatic systems: chiral active fluids composed of spinning frictional particles that are free to translate, oscillating granular gases and active Brownian rollers. In all of these systems, a single effective temperature generated by activity parameterizes both the equation of state and the emergent Boltzmann statistics. The same effective temperature, renormalized by velocity correlations, relates viscosities to steadystate stress fluctuations via a Green–Kubo relation. To rationalize these observations, we develop a theory for the fluctuating hydrodynamics of these nonequilibrium fluids and validate it through largescale molecular dynamics simulations. Our work sheds light on the microscopic origin of odd viscosities and stress fluctuations characteristic of parityviolating fluids, in which mirror symmetry and detailed balance are broken.
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Acknowledgements
The authors thank A. G. Abanov, S. Atis, D. Bartolo, W. T. M. Irvine, C. Nardini, H. C. Öttinger and D. T. Son for helpful discussions and suggestions. S.V., J.J.d.P. and V.V. acknowledge primary support through the Chicago MRSEC, funded by the NSF through grant no. DMR1420709. S.V. acknowledges support from the National Science Foundation under grant no. DMR1848306. V.V. acknowledges support from the Complex Dynamics and Systems Program of the Army Research Office under grant no. W911NF1910268, the Simons Foundation and the KITP program on Symmetry, Thermodynamics and Topology in Active Matter via NSF grant no. PHY1748958. M.H. and M.F. acknowledge support from the University of Chicago MRSEC through Kadanoff–Rice postdoctoral fellowships. C.S. acknowledges support by the National Science Foundation Graduate Research Fellowship under grant no. 1746045. M.H. acknowledges use of the GM4 cluster supported by the National Science Foundation’s Division of Materials Research under the Major Research Instrumentation (MRI) program award no. 1828629. This work was completed in part with resources provided by the University of Chicago’s Research Computing Center. Calculations presented here were performed on the GPU cluster supported by the NSF under grant DMR1828629.
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M.H. performed the simulations and analysed the data. All authors conducted theoretical research. M.H., M.F., C.S. and V.V. wrote the paper. All authors contributed to discussions, interpretation of the results and manuscript revision.
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Extended data
Extended Data Fig. 1 Thermodynamics of an oscillating granular gas.
a. Schematic of the system setup. We simulate a quasi2D granular gas composed of frictional particles, which are forced to oscillate vertically at a constant frequency f but free to move horizontally. Interparticle collision between two oscillating particles could lead to their translational motions in the xyplane. In the middle is a zoomedin, top view of this manybody system. Horizontal translation of a particle is denoted by its tail, whereas its vertical oscillation is colorcoded in the tail: gradient from a dark end to a bright front means the particle is moving towards the xyplane, vice versa; purple denotes z < 0 whereas red denotes z > 0. Δt denotes the averaged collision duration. b. Maxwell distribution. The xcomponent of translational velocity displays a Gaussian distribution P(v_{x}) at various oscillating frequency f. An effective temperature T_{eff} is defined using the halfwidth of P(v_{x}). Dependence of T_{eff} on f is shown on the right. c. Boltzmann distribution. We put the system in a potential well \(U({{{{{\bf{r}}}}}})=0.5{k}_{{{{{\mathrm{B}}}}}}{T}_{{{{{\mathrm{eff}}}}}}\ \left[1+\,{{{{\mathrm{cos}}}}}\,(\pi r/R)\right]\) for r < R, where r denotes the distance from the center of the system. The resultant spatial distribution of the particles turns out to follow the Boltzmann statistics \(n(r)\propto \,{{{{\mathrm{exp}}}}}\,\left[U(r)/{k}_{{{{{\mathrm{B}}}}}}{T}_{{{{{\mathrm{eff}}}}}}\right]\) (purple curve) as well. d. Green–Kubo relation. Shear viscosity of this manybody system can be either directly measured using linear response towards an applied shear or indirectly inferred from the Green–Kubo relation by calculating the integral of the stress–stress correlation function, known as the Green–Kubo relation. The predicted and measured shear viscosity η is compared at a wide range of frequencies f. The Kubo predictions with T_{eff} and renormalized \({T}_{\,{{{{\mathrm{eff}}}}}\,}^{* }\) are marked as the dashed and solid lines, respectively. We have defined f_{0} = 1/Δt.
Extended Data Fig. 2 Thermodynamics of an active Brownian system.
a. Schematic of the system setup. We simulate a 2D system composed of active Brownian rollers. Each particle contains a core (in green) that selfpropels nearly at a constant speed v meanwhile undergoes rotational diffusion as well as a dumbbell (in blue) that is hinged at the core center and free to rotate about it. In particular, the core of particle i is powered by an active force \({{{{{{\bf{F}}}}}}}_{i}^{\,{{{{\mathrm{a}}}}}}={c}_{{{{{\mathrm{d}}}}}}v{\hat{{{{{{\bf{n}}}}}}}}_{i}\) (\({\hat{{{{{{\bf{n}}}}}}}}_{i}\) is the orientation of the core) meanwhile experiences a drag force by the substrate \({{{{{{\bf{F}}}}}}}_{i}^{\,{{{{\mathrm{d}}}}}}={c}_{{{{{\mathrm{d}}}}}}{{{{{{\bf{v}}}}}}}_{i}\), where ζ denotes the substrate friction coefficient. Note that the particle dumbbell is lifted away from the substrate thus does not experience any friction; moreover, the dumbbell rotation does not reorient the selfpropulsion of the core. When two particles collide, the translational motion of the cores could result in the rotational motion of the dumbbells. Δt denotes the averaged collision duration. b. Maxwell distribution. The angular velocity of the dumbbells displays a Gaussian distribution P(Ω) at various selfpropulsion speed v. An effective temperature T_{eff} is defined using the halfwidth of P(Ω). Dependence of T_{eff} on Ω is shown on the right. cd. GreenKubo relation. The rotational drag coefficient of the dumbbell can be either measured through linear response by measuring the terminal angular velocity under an applied torque, or predicted using the Green–Kubo relation by evaluating the integral of the torque–torque correlation function. The measured and predicted drag coefficient γ_{rot} is compared at a wide range of selfpropulsion speed v (c) as well as substrate friction coefficient c_{d} (d). However, when either selfpropulsion speed v or substrate friction coefficient c_{d} is increased, the relative significance of particle interaction compared to selfpropulsion gets reduced. As a consequence, we see that the Green–Kubo relation is restored at either large v or c_{d}. We have defined v_{0} = d/Δt and c_{d} = m/Δt.
Extended Data Fig. 3 Microscopic origin of antisymmetric stress.
a. Schematic of orbital angular momentum change during collision. When two frictional active spinners collide, the angular momentum of selfspinning can be interchanged with the angular momentum of orbital motion around their centerofmass, L = mv_{rel}b, where v_{rel} is the relative moving speed of the particles and b is the impact parameter. The resultant change in the orbital angular momentum ΔL = L_{out} − L_{in} gives rise to effective antisymmetric stress exerted onto the chiral active fluid at the macroscopic level. In the Supplementary Sec. III, we provide a simple kinetic theory to derive the linear relation between antisymmetric stress τ and the average orbital angular momentum change \(\overline{{{\Delta }}L}\) during collision, \(\tau =\sqrt{\pi {k}_{{{{{\mathrm{B}}}}}}{T}_{{{{{\mathrm{eff}}}}}}/m}\cdot d{n}^{2}\cdot \overline{{{\Delta }}L}\). b. Validation of our kinetic theory. We measure the average orbital angular momentum change \(\overline{{{\Delta }}L}\) by performing scattering simulations and then use it to predict the antisymmetric stress τ based upon the kinetic theory. The prediction on τ from \(\overline{{{\Delta }}L}\) agrees well with the simulation measurement of a manybody system at the steady state.
Extended Data Fig. 4 Transverse mode in a shock wave.
a. Shock wave. A piston moving at speed U = 1.9d/Δt (faster than the speed of sound c = 1.4d/Δt) generates a shock wave accompanied with transverse flows, which is characterized by the vertical flow velocity u_{y} (gradient coloring). The particles selfspin counterclockwise at speed Ω = 25.3/Δt and have an initial global density n_{0} = 0.125d^{−2}. According to the viscid Burgers’ equation \({\partial }_{t}u+u{\partial }_{x}u=\nu {\partial }_{x}^{2}u\), the width of this shock is approximately λ_{s} = 4ν/U, where ν = η/n_{0}m is the kinematic viscosity. Hydrodynamic profiles are quantified near the wave front. Also see Supplementary Mov. S8. b. Density profile n(x). The simulation results are compared with continuum hydrodynamic theory (solid line), which employs parameters measured in a separate homogeneous microscopic systems of number density n_{0} (dashed line). Thus, theoretical predictions would break down at extreme densities (shaded region). c. Horizontal flow velocity u_{x}(x). d. Vertical flow velocity u_{y}(x). The same color coding as panel A is applied here. Predictions using continuum hydrodynamic theory are plotted as solid lines.
Extended Data Fig. 5 Power spectra of the velocity–velocity correlation functions \(\langle {u}_{a}({{{{{\bf{k}}}}}},\omega ){u}_{b}^{* }({{{{{\bf{k}}}}}},\omega )\rangle\).
Here we compare the measured velocity–velocity correlation functions with the empirical prediction using fluctuating hydrodynamic theory. a. Correlation functions directly measured in the particlebased simulations of our chiral active fluid. b. Correlation functions predicted using the fluctuating hydrodynamic theory with the measured stress–stress correlation functions and viscosity tensor. The empirical prediction matches with the simulation results expect at very high kmodes, where the linear response approximation is no longer valid. Although the comparison is made at a given wave frequency ω_{0} = 0.055π/Δt, the consistency between direct measurements in simulations and predictions using fluctuating hydrodynamic theory generally holds at all wave frequencies.
Supplementary information
Supplementary Information
Supplementary Sects. I–IX and Figs. 1–18.
Supplementary Video 1
Collision between two active spinners. Two particles both spinning at speed Ω = 9.5/Δ t are set to collide with a relative velocity v_{rel} = 0.63d/Δt. Although the particles are perfectly aligned to undergo a headtohead collision, due to the presence of selfspinning and interparticle friction, they gain transverse motion after the collision.
Supplementary Video 2
Chiral active fluid. We show the microscopic dynamics in a fluid of active spinners at its dynamical steady state. The particles selfspin clockwise at speed Ω = 9.5/Δt. The system is of number density n = 0.254d^{−2}.
Supplementary Video 3
Dense chiral active fluid. We show the microscopic dynamics in a fluid of active spinners at its dynamical steady state. The particles selfspin clockwise at speed Ω = 9.5/Δt. The system is of number density n = 0.508d^{−2}
Supplementary Video 4
Collision between two oscillating grains. Two grains both oscillating in z direction at frequency f = 0.33/Δt are set to collide. We map out the threedimensional trajectories of the particles as well as their projections onto the x–y plane.
Supplementary Video 5
Oscillating granular gas. We show the microscopic dynamics in a gas of oscillating grains at its dynamical steady state. The particles oscillate in zdirection at frequency f = 0.33/Δt with random initial phases. The system is of number density n = 0.254d^{−2}.
Supplementary Video 6
Collision between two active Brownian rollers. Two active Brownian rollers selfpropel at speed v = 0.5d/Δt and are set to collide with impact parameter b = 0.5d. The nonzero impact parameter leads to the rotation of the rollers after the collision.
Supplementary Video 7
Active Brownian rollers. We show the microscopic dynamics in a fluid of active Brownian rollers at its dynamical steady state. The core of each roller selfpropels at speed v = 0.5d/Δt. The system is of number density n = 0.076d^{−2}.
Supplementary Video 8
Shock wave. A piston moving at speed U = 1.4c, where c = 1.4d/Δt is the speed of sound, generates a shock wave propagating from left to right in our chiral active fluid. Here the particles spin counterclockwise at speed Ω = 26.7/Δt and have initial global density n_{0} = 0.125d^{−2}. To demonstrate the resultant shear flow in the transverse direction, we colour code the fluid according to the y component of the flow velocity u_{y}. To characterize the shock wave, we further show the density profile ρ(x) as well as the flow profile u_{x}(x) and u_{y}(x).
Supplementary Video 9
Chiral Brownian motion in shear–stress space. To illustrate the steadystate fluctuations, we plot the shear stresses and of an unperturbed system against each other. In the system, all the particles spin counterclockwise at speed Ω = 26.7/Δt and have global density n_{0} = 0.25d^{−2}. Over time, the stress vector \(S={({s}_{1},{s}_{2})}^{{{\mathrm{T}}}}\) exhibits a chiral Brownian motion, confined near the origin (0,0)^{T} and having a tendency towards a clockwise rotation. To illustrate such chiral rotation, we plot the normalized stress \(\hat{S}=S/  S \) and mark its polar angle θ. The winding number nw(t) = [θ(t) − θ(0)]/2π clearly shows the tendency of the stress vector S to perform a clockwise ration.
Source data
Source Data Fig. 1
Effective thermodynamic properties and transport coefficients
Source Data Fig. 2
Statistical properties of interparticle collisions
Source Data Fig. 3
Spectra of stress–stress correlation functions and velocity–velocity correlation functions
Source Data Fig. 4
Stress–stress correlation functions, Green–Kubo relations, time evolution of shear stress
Source Data Fig. 5
Boltzmann distribution and Green–Kubo relation for three different active systems
Source Data Fig. 6
Mutual information, probability distribution of active and fluctuating degrees of freedoms, deviation of Green–Kubo relation
Source Data Extended Data Fig. 1
Boltzmann distributions, Green–Kubo relation
Source Data Extended Data Fig. 2
Boltzmann distributions, Green–Kubo relation
Source Data Extended Data Fig. 3
Antisymmetric stress
Source Data Extended Data Fig. 4
Density and velocity profiles of a shock wave
Source Data Extended Data Fig. 5
Spectra of all the velocity–velocity correlation functions
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Han, M., Fruchart, M., Scheibner, C. et al. Fluctuating hydrodynamics of chiral active fluids. Nat. Phys. 17, 1260–1269 (2021). https://doi.org/10.1038/s41567021013607
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DOI: https://doi.org/10.1038/s41567021013607
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