From Attraction to Repulsion: Emergent Interactions in Harmonically Coupled Active Binary System

TL;DR

Problem: how do two harmonically coupled active Brownian particles behave in a thermal bath, and can activity induce effective attraction or repulsion between them? Approach: analyze the nonequilibrium stationary state of the separation by solving the relative Langevin dynamics and compute explicit $P(r)$, $P(x)$, and orientation statistics across strong, moderate, and weak coupling limits; also characterize short-time centroid fluctuations. Key contributions: demonstration of emergent short-range repulsion at low $T$ in the strong and moderate coupling regimes, quantified by a threshold temperature $T^*$ and an effective potential $V_{\rm eff}(r)$; identification of distinct radial and marginal distributions in each regime, including special case $v_1=v_2$ that yields attraction; and a detailed account of non-Gaussian short-time centroid dynamics. Significance: reveals how propulsion diversity and coupling tune emergent interactions in active matter and provides analytic benchmarks for experiments and design of active composites.

Abstract

We investigate the emergent interactions between two active Brownian particles coupled by an attractive harmonic potential and in contact with a thermal reservoir. By analyzing the stationary distribution of their separation, we demonstrate that the effective interaction can be either attractive or repulsive, depending on the interplay between activity, coupling strength, and temperature. Notably, we find that an effective short-range repulsion emerges in the strong and moderate-coupling regimes, when the temperature is below some threshold value, which we characterize analytically. In the strong-coupling regime, the repulsion emerges solely due to the difference in the self-propulsion speeds of the particles. We also compute the short-time position distribution of the centroid of the coupled particles, which shows strongly non-Gaussian fluctuations at low temperatures.

Figures (11)

• Figure 1: Stationary distribution $P\left((x,y)\right)$ of the relative coordinate $\bm{r}=(x,y)$, obtained from numerical simulations, in the strong-coupling regime (a), moderate-coupling regime (b), and weak-coupling regime (c). The projection on the $x-y$ plane is shown by the green contour plot. For (a), the parameters are $D_1=D_2=0.01, k=4, T=0.05, v_1=2$ and $v_2=4$. The self-propulsion velocities are fixed at $v_1=4, v_2=1$ for the moderate and weak-coupling regimes. The remaining parameters for (b) are $D_1=0.001, D_2=10, k=0.2, T=1$, and for (c) $D_1=10, D_2=20, k=0.02, T=0.1$.
• Figure 2: Plot of $\langle x^2(t) \rangle$, the second moment of the $x$-component of the radial separation ${\bm r}$, as a function of time for different values of the temperature $T$. The symbols indicate the data obtained from the numerical simulations, whereas the black solid lines correspond to the theoretical prediction Eq. \ref{['2nd_moment_of_x']}. The dashed lines indicate the stationary values \ref{['stationary_x2']}. Here we have taken $k=1$, $D_1=1$, $D_2=4$, $v_1=7$, and $v_2=3$.
• Figure 3: Strong-coupling regime: Plot of the stationary radial distribution $P(r)$ (a) and the corresponding effective potentials (b), for different values of $T$, with $v_1=4$, $v_2=1$, $D_1=D_2=0.01$, and $k=4$. The black solid lines in (a) and (b) correspond to Eqs. \ref{['radial_dist_strong']} and \ref{['eff_pot2']} respectively. The brown dashed line in (b) corresponds to the underlying harmonic potential $k r^2/2$ with $k=4$.
• Figure 4: Strong-coupling regime: Plot of threshold temperature $T^*$ obtained by numerically solving Eq. \ref{['b_eq']} as a function of (a) coupling strength $k$ for different values of $v_1$, and (b) self-propulsion velocity $v_1$ for different values of $k$ with fixed $v_2=1$.
• Figure 5: Strong-coupling regime: Stationary distribution of (a) the $x$-component of the separation ${\bm r}$, and (b) the relative orientation $P(\Delta\theta|r^*)$ for different temperatures $T$ obtained from numerical simulations with $v_1=4$, $v_2=1$, $D_1=D_2=0.01$, and $k=4$. In the inset of (b), $P(x)$ is shown along with the $r^*$ using dashed vertical lines. The black solid lines in (a) and red dashed line in (b) correspond to Eq. \ref{['marg_dist_1']} and uniform distribution $U(0,\pi)$, respectively.