Diffusion velocity modulus of self-propelled spherical and circular particles in the generalized Langevin approach

TL;DR

This work presents the accelerated self-propelled diffusive particle (ASPDP) model, which splits dynamics into an internal propulsion driven by independent Ornstein-Uhlenbeck processes and a diffusion stage in a thermal bath under a parabolic field described by a modified generalized Langevin equation. By averaging over colored noise and mapping the 3D OUPs to spherical coordinates, the authors derive the diffusion velocity modulus for both spheres and disks, obtaining a closed-form expression for the root-mean-square diffusion velocity $s_d(t)$ that encapsulates the interplay between propulsion, bath memory, and thermal fluctuations. The results reveal transient, geometry-dependent fluctuations in the diffusion velocity modulus induced by the internal mechanism, which decay at long times, and demonstrate how the ASPDP framework extends the diffusion GLE to self-propelled systems while highlighting limitations for active matter. The methodology provides insights for nanoscale motile systems where internal driving couples to bath-mediated diffusion, with potential applications to nano-motors and related technologies.

Abstract

This research provides a framework for describing the averaged modulus of the velocity reached by an accelerated self-propelled Brownian particle diffusing in a thermal fluid and constrained to a harmonic external potential. The system is immersed in a thermal bath of harmonic oscillators at a constant temperature, where its constituents also interact with the external field. The dynamics is investigated for a sphere and a disk, and is split into two stochastic processes. The first describes the gross-grained inner time-dependent self-velocity generated from a set of independent Ornstein-Uhlenbeck processes without the influence of the external field. This internal mechanism provides the initial velocity for the particle to diffuse in the fluid, which is implemented in a modified generalized Langevin equation as the second process. We find that the system exhibits spontaneous fluctuations in the diffusive velocity modulus due to the inner mechanism; however, as expected, the momentary diffusive velocity fluctuations fade out at large times. The internal propelled velocity module in spherical coordinates is derived, as well as the simulation of the different modules for both the sphere and the already known equations for a disk in polar coordinates.

Figures (5)

• Figure 1: The susceptibility $\chi(t)$ defined by Eq. (\ref{['chivt']}) and previously calculated by the author in Ref. PJPRE6.
• Figure 2: Diffusion VM for the sphere. The curves are identified by the set $\{\epsilon,\kappa,\epsilon_{z},\kappa_{z}\}$, so they correspond to $\{1,1,1,1\}$ (black), $\{0.5,0.5,1,1\}$ (blue), $\{1,1,0.5,0.5\}$ (dashed blue) , $\{2,2,1,1\}$ (red), and $\{1,1,2,2\}$ (dashed red). The inset is an amplification at low times.
• Figure 3: Diffusion VM for the disk. The curves are identified by the set $\{\epsilon,\kappa\}$. Panel (a) corresponds to the blue and red curves of Fig. \ref{['Fig2']} for vanishing $\epsilon_{z}$ and $\kappa_{z}$, and with the same color identification, that is, $\{1,1\}$ black, $\{0.5,0.5\}$ blue and $\{2,2\}$ red. The plots of graph (b) are for $\{1,1\}$ black, $\{1,0.25\}$ (cyan), $\{0.25,1\}$ (dashed cyan), $\{1,1.75\}$ (puple), and and $\{1.75,1\}$ (dashed purple). See the text for details.
• Figure 4: Simulated averaged propelled VM of the sphere for various combinations of the set $\{\epsilon_{x},\epsilon_{y},\epsilon_{z}, \kappa_{x},\kappa_{y},\kappa_{z}\}$. (a) The curves correspond to those shown in Fig. \ref{['Fig2']}, with the same color identification, namely $\{1,1,1,1,1,1\}$ (black) as reference, $\{1,1,2,1,1,2\}$ (dashed red), $\{2,2,1,2,2,1\}$ (red), $\{0.5,0.5,1,0.5,0.5,1\}$(blue), and $\{1,1,0.5,1,1,0.5\}$ (dashed blue). (b) This figure shows the results for other ar-bi-tra-ry parameter sets. The solid curves for $\{1,1,1,1,1,1\}$ (black) and $\{2,2,2,2,2,2\}$ (green) are shown as references; the remaining dashed curves are for $\{1,2,2,1,2,2\}$ (green), $\{2,2,2,1,1,1\}$ (gray), $\{2,1,1,2,1,1\}$ (purple), and $\{1,1,1,2,2,2\}$ (brown) .
• Figure 5: Simulated averaged propelled VM of the disk for various combinations of the set $\{\epsilon,\kappa\}$. The graphs (a) and (b) correspond to the parameter sets of Fig. \ref{['Fig3']}a and \ref{['Fig3']}b, respectively, with the same color identification.