Active bound states arising from transiently nonreciprocal pair interactions

Static nonreciprocal forces between particles generically drive persistent motion reminiscent of self-propulsion. Here, we demonstrate that reciprocity-breaking fluctuations about a reciprocal mean coupling strength are sufficient to generate this behavior in a minimal two-particle model, with the velocity of the ensuing active bound state being modulated in time according to the nature of these fluctuations. To characterize the ensuing nonequilibrium dynamics, we derive exact results for the time-dependent center of mass mean-square displacement and average rate of entropy production for two simple examples of discrete- and continuous-state fluctuations in one dimension. We find that the resulting dimer can exhibit unbiased persistent motion akin to that of an active particle, leading to a significantly enhanced effective diffusivity.

Figure 1

MSD and entropy production rate for correlated continuous fluctuations. (a) MSD for the active bound states as given by Eq. (16), where we observe transient ballistic scaling implying persistent motion and an effective diffusion coefficient larger than that of an isolated particle. Here, $D_{\kappa }=5$ and $D_x=\overline{k}=\mu =1.$ (b) Rate of entropy production for the active bound state, made up of two contributions as identified in Eq. (13), for $\overline{k}=5, D_x=1$, and $D_{\kappa }=\mu =10.$ Symbols are from numerical simulations (see Appendix pp6 for details), and solid lines are evaluated using Eqs. (13) and (14).

Figure 2

MSD and entropy production rate for synchronized discrete fluctuations. (a) MSD for the active bound states as given by Eq. (23) again displaying transient ballistic scaling and enhanced diffusion. Here, ${\kappa }_0=5$ and $D_x=\overline{k}=\lambda =1.$ (b) Rate of entropy production for the active bound state where we have fixed $D_x=1$. Symbols are measured from numerical simulations, and solid lines are the results of Eq. (25).

Figure 3

Steric repulsion enhances persistent motion. We compare our results on the dimer's dynamics above to simulations which include a repulsive force between the two particles through a Weeks-Chandler-Anderson potential. For both (a) continuous and (b) discrete fluctuations, this additional interaction leads to a significantly enhanced effective diffusion coefficient. Parameters are the same as in Figs. 1 and 2.

Figure 4

Allowed transitions for the Markov jump process controlling the stiffness asymmetry $\psi \left(t\right)$ and total stiffness fluctuation $\phi \left(t\right)$ in the case of discrete fluctuations in the coupling strength. Here, $\lambda$ denotes the switching rate, while $\chi \in [0,1]$ is a correlation parameter controlling the degree of synchronization of the single potential switching events.

Figure 5

Entropy production rate when $U_r\ne 0$ for continuous fluctuations. We here use a repulsive potential of Weeks-Chandler-Anderson form. While the functional dependence on the correlation coefficient $\theta$ remains similar, the magnitude of all contributions is significantly larger. This agrees with the more persistent motion that we observe in Fig. 3 in the main text.

Figure 6

Entropy production rate when $U_r\ne 0$ for discrete fluctuations. (a) For an initialization which sets $\psi =0$ in the presence of nonzero $U_r$. We observe a similar functional dependence on the fluctuation strength ${\kappa }_0$ as in Fig. 2, but the overall amount of entropy produced is significantly larger, as was observed in the continuous case. In (b), the same trend is observed for the initialization which sets $\phi =0$.