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 self-propels 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 self-propulsion 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 self-propulsion speed v. An effective temperature Teff is defined using the halfwidth of P(Ω). Dependence of Teff on Ω is shown on the right. c-d. Green-Kubo 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 γrot is compared at a wide range of self-propulsion speed v (c) as well as substrate friction coefficient cd (d). However, when either self-propulsion speed v or substrate friction coefficient cd is increased, the relative significance of particle interaction compared to self-propulsion gets reduced. As a consequence, we see that the Green–Kubo relation is restored at either large v or cd. We have defined v0 = d/Δt and cd = m/Δt.
