Inertia-chirality interplay in active Brownian motion: exact dynamics and phase maps

TL;DR

The paper provides an exact, time-resolved theory for a two-dimensional chiral active Brownian particle with translational inertia, unveiling how inertia and chirality reshape transient kinetics while leaving long-time diffusion in position space mass-independent. Using a Laplace-transform moment hierarchy, it yields closed-form expressions for mean velocity, velocity projections, velocity autocorrelation, MSV, MSD, and the fourth velocity moment, with the VACF decomposing into inertial and chiral envelopes. A kinetic-temperature construct and a modified FDR reveal that equilibrium-like behavior emerges in the limits of large mass or large chirality, pinpointing chirality as an additional route to effective equilibration. The study also maps excess kurtosis in the velocity statistics, showing a Gaussian-like–active–Gaussian-like re-entrant regime and a chirality-driven confinement of activity, with implications for vibrobot and colloid experiments and for future work including rotational inertia and interparticle effects.

Abstract

We present an exact, time-resolved theory for a two-dimensional chiral active Brownian particle (cABP) with translational inertia. Using a Laplace-transform moment hierarchy, we derive closed-form expressions for the mean velocity, velocity-orientation projections, velocity autocorrelation, mean-squared velocity, mean-squared displacement, and the fourth moment of velocity. These results agree quantitatively with simulations over all masses, activities, and chiralities. We show that the velocity autocorrelation factorizes into an inertial envelope and a chiral envelope. Despite rich transients in the velocity sector, the long-time positional diffusion equals the overdamped cABP value, independent of mass. From the steady mean-squared velocity, we define a kinetic temperature and a modified fluctuation-dissipation relation whose violation vanishes in two limits: large mass or large chirality, identifying chirality as an additional route to equilibrium-like behavior. The steady-state velocity excess kurtosis gives a phase map that exhibits a (Gaussian-like)-active(bimodal)-(Gaussian-like) re-entrance with mass; chirality confines activity and shrinks the active sector. A narrow positive-kurtosis window emerges at large mass and intermediate chirality, with analytic boundaries consistent with the heavy-mass asymptote.

Figures (13)

• Figure 1: In this figure we have plotted the trajectory obtained by simulating the Langevin dynamics for fixed values of ${\rm{Pe}}$ and $\Omega$ and with different mass: (a) $M=0.1$, (b) $M=1$ and (c) $M=10$. For all figures (a)-(c) we have fixed ${\rm{Pe}}=100$ and $\Omega=100$.
• Figure 2: Probability density of the speed, $|{\tilde{\bm{v}}}|=\sqrt{{\tilde{\bm{v}}}_x^2+{\tilde{\bm{v}}}_y^2}$, from simulations of $N$ particles. For the overdamped cABP and the underdamped achiral ABP, the peak is at $|{\tilde{\bm{v}}}|\approx{\rm{Pe}}=100$. For an underdamped chiral ABP, inertia and chirality shift the peak to ${\tilde{\bm{v}}}_m \approx {\rm{Pe}}/\sqrt{1+M^2\Omega^2}$.
• Figure 3: Lag function is plotted as a function of time for (a) fixed ${\rm{Pe}}=100$, $\Omega=10$ and different values of effective mass ($M$) and for (b) fixed ${\rm{Pe}}=100$, $M=0.1$ and different chirality. We set the initial velocity ${\tilde{\bm{v}}}= 100\hat{x}$.
• Figure 4: Trajectories at different time intervals for $M=10$, ${\rm{Pe}}=100$, $\Omega=100$. The rotational nature gradually decays as we keep on increasing time.
• Figure 5: Mean squared velocity (MSV) is plotted against for (a) various $\Omega$ and a fixed $M=1$ and (b) various mass $M$ and a fixed $\Omega=100$. We have fixed ${\rm{Pe}}=100$ for both plots. From both (a) and (b) we can conclude that MSV eventually goes to its steady state in the long time limit. In the limit of $t<<1$, MSV first scales as $~{\tilde{t}}$ and then scales as ${\tilde{t}}^2$ in the intermediate time scale. The higher the mass the longer MSV takes to reach the steady state and the steady state value of MSV also gets lower. The lines are plotted from the exact analytical expression given in Eq. \ref{['MSV_ana_in']} and points are obtained by simulating the Langevin equations, Eq. \ref{['Langevin_dimensionless']}.