Collective dynamics of trail-interacting particles

Abstract

Trail interactions occur when past particle trajectories bias future motion, rendering the system out of thermodynamic equilibrium. While such systems are abundant in nature, their understanding is limited to the single-particle level or phenomenological mean-field theories. Here, we introduce a minimal model of many trail-interacting particles that extends this paradigm to the fluctuating collective level. Particles diffuse while depositing long-lasting repelling/attracting trails that act as a shared memory field, coupling their dynamics across time and space. Using stochastic density functional theory, we derive fluctuating hydrodynamic equations and analyze analytically and numerically the resulting behaviors. We show that memory, coupled with fluctuations, fundamentally reshapes collective dynamics; In the repulsive case, the particle density displays superdiffusive spreading characterized by transient clustering and ballistic motion; In the attractive case, the system condensates in finite time into frozen, localized states. Our results establish general principles for trail-interacting systems and reveal how persistent fields generate novel instabilities and self-organization.

Figures (4)

• Figure 1: Schematic illustration of the model. Each particle (in red) lives in a potential landscape $\phi$ (in blue). Inset: two possible dynamics of a single particle: hopping to neighboring sites with rates $\lambda_{i\pm 1}$ and trail deposition with rate $k$. One deposition increase locally $\phi$ by 1 (purple event) . Translucent red dots represents the three possible future state of the particle.
• Figure 2: (Color online) Results in the repulsive case ($h>0$). (a+b) Density- and trail structure factors as a function of the wavevector at different times. Simulation results are plotted as solid lines and the theoretical prediction of Eq. \ref{['eq3']} as dashed lines. The asymptotic value for long times is plotted in dashed black. Inset of (a) shows how the long-time density structure factor scales with $h$, according to the simulations (symbols) and theory (solid line). (c) Kymograph of density as a function of space and time. The texture indicates that low- and high-density fluctuations travel at a velocity $\pm c=\pm\sqrt{hk}$. (d) Characteristic trail profiles at long times for different $h$ values, according to the simulations.
• Figure 3: Rescaled profiles of (a) the particle density and (b) the trail density, compared with the theoretical predictions of Eq. \ref{['eq_scalings']}. Simulations correspond to $N=4096$, $h=0.05$, $k=1$, averaged over 50 realizations. The collapse confirms the superdiffusive scaling $\alpha=2/3$ and the compact support predicted analytically.
• Figure 4: (Color online) Results in the attractive case ($h<0$). (a+b) Density structure factor as a function of the wavevector at different times (a) and different $h$ values at long times (b). Simulation results are plotted as solid lines, theoretical prediction of Eq. \ref{['eq3']} as dashed lines for (a), and sigmoid-type fit as dashed lines for (b). (c+d) Snapshots of the density profile in permanent state for $h = 0.05, k=0.1$ (c) and $h=0.1, k=1$ (d). (e) The characteristic wavevector $q_\ast$ and asymptotic value of the structure factor $S_p$ as a function of $h$ for $k=1$. The theoretical scaling is plotted in dashed black.