Transfer of active motion from medium to probe via the induced friction and noise

TL;DR

We study how activity from a homogeneous active bath can be transmitted to a heavy Newtonian probe by deriving a reduced probe dynamics in a moving frame via a quasistatic expansion, yielding velocity-dependent friction $f(v)$, noise $B(v)$, and a second-order correction $G(v)$ that depend on fixed-$v$ bath dynamics. The analysis covers 1D run-and-tumble and 2D active Brownian baths, establishing regimes of passive motion and several active regimes where the probe inherits persistent motion, with explicit expressions and quantitative agreement with simulations. The active regimes in 1D feature run-and-tumble or run-and-stop behavior, while 2D allows active Brownian motion or switching between active and passive states, and all results depend critically on the probe–bath coupling and dimensionality. The work demonstrates a universal mechanism for cross-scale transfer of persistence via induced friction and noise, with implications for designing active-environment devices and for fundamental understanding of nonequilibrium bath–probe interactions.

Abstract

Can activity be transmitted from smaller to larger scales? We report on such a transfer from a homogeneous active medium to a Newtonian spherical probe. The active medium consists of faster and dilute self-propelled particles, modeled as run-and-tumble particles in 1D or as active Brownian particles in 2D. We derive the reduced fluctuating dynamics of the probe, valid for arbitrary probe velocity, characterized by velocity-dependent friction and noise. In addition to a standard passive regime, we identify peculiar active regimes where the probe becomes self-propelled with high persistence, and its velocity distribution begets peaks at nonzero values. These features are quantitatively confirmed by numerical simulations of the joint probe-medium system. The emergence of active regimes depends not only on the far-from-equilibrium nature of the medium but also on the probe-medium coupling. Our findings reveal how, solely via the induced friction and noise, persistence can cross different scales to transfer active motion.

Figures (3)

• Figure 1: For 1D run-and-tumble medium with interaction $F(r)=k\sin(\pi r/R)$ within range $r<R$: (a) Friction $f(v)$ and noise intensity $B(v)$ (per medium particle) with different flip rates $\alpha$, showing regimes (R1) and (R2a). Parameters are $L=10$, $R=0.5$, $k=2.4$, $u=3$, $\mu=1$. (b) Same for $k=3.3$, corresponding to regimes (R1) and (R2b). (c) Stationary velocity distribution $\rho^\text{st}(v)$ of the probe, from the simulation (blue), from the nonlinear dynamics \ref{['1dr']} (red dashed), and from the effective active dynamics (green dashed). The upper panel corresponds to the blue line ($k=2.4$, $\alpha=4.5$) in (a) with probe mass $M=30$, and the lower panel corresponds to the blue line ($k=3.3$, $\alpha=1.6$) in (b) with probe mass $M=40$. (d) Diffusion coefficient of the probe for different mass: from the effective active motion (blue line) and the simulation (orange points). The upper panel and lower panel share the same parameters with the blue lines in (a) and (b), respectively.
• Figure 2: For 2D active Brownian medium with Lennard-Jones interaction: (a) Landscapes of $f(v)$, $B_\parallel(v)$, and $B_\perp(v)$ (per medium particle) for different $\alpha$, showing regimes (A1) and (A2a). Other parameters are $k=2.4$, $u=3$, $R=0.5$, $L=10$, $\mu=1$. (b) Same for $k=1.95$, $\alpha=3.0$, corresponding to regime (A2b). (c) Stationary distributions of the probe velocity $\rho^\text{st}(v_x,v_y)$ and speed $\rho^\text{speed}(v)=2\pi v\rho^\text{st}(v_x,v_y)$. The 3D plots of $\rho^\text{st}(v_x,v_y)$ and the red-dashed lines in $\rho^\text{speed}(v)$ are obtained from the reduced dynamics \ref{['2dr']}. The blue line denotes the simulation results of the composite system. The green-dashed line is from the effective active motion. The upper panel corresponds to $k=2.4$, $\alpha=4.5$ in (a) and $M=30$; the lower corresponds to $k=1.95$, $\alpha =3.0$ in (b) and $M=46$. (d) Diffusion coefficient of the probe as a function of mass: from the effective active motion (blue line) and the simulation (orange points). The upper panel and lower panel correspond to $k=2.4$, $\alpha=4.5$ in (a) and $k=1.95$, $\alpha =3.0$ in (b), respectively.
• Figure 3: Mechanism in 1D: The green pentagon represents the medium particles, the acute angles of which denote their propulsion direction. The yellow disk represents the probe, moving rightwards. The blue solid line represents the force (divided by $\mu$) exerted on the medium particles. If medium particles can pass through the probe, the distance (shown in red) between $F(r)/\mu$ and $u+v$ or $-(u-v)$ yields the relative velocity $\abs{\dot r}$ at different position $r$.