Active Brownian Particle (ABP) Overview

• Active Brownian Particle (ABP) is a stochastic model for self-propelled particles exhibiting persistent ballistic motion disrupted by rotational diffusion.
• ABP dynamics are modeled by overdamped Langevin equations that couple translation and orientation, facilitating studies of diffusion transitions and boundary effects.
• The model reveals rich collective phenomena including motility-induced phase separation and glassy states, with implications for active matter research.

An active Brownian particle (ABP) is a paradigmatic stochastic model for self-propelled microswimmers, such as motile bacteria, synthetic Janus colloids, or cytoskeletal molecular motors. ABPs move via persistent ballistic motion disrupted by angular (rotational) diffusion, yielding non-equilibrium dynamics fundamentally distinct from classical Brownian particles. The ABP model provides a minimal yet versatile framework for the paper of active matter, elucidating single-particle crosses from ballistic to effective diffusion, collective behaviors such as motility-induced phase separation, glassy states, and interactions with complex boundaries or environments.

1. Microscopic Dynamics and Generalized Langevin Description

A canonical ABP in $d=2,3$ dimensions is governed by overdamped Langevin equations coupling translational and orientational degrees of freedom. In two dimensions, for particle $i$ with position $\mathbf{r}_i$ and orientation $\theta_i$, the dynamics reads (Zhang et al., 2022, Basu et al., 2018):

$$\begin{aligned} & \dot{\mathbf{r}}_i = v_0 \, \mathbf{n}_i + \sqrt{2 D_t} \, \boldsymbol{\xi}_i(t) + \mathbf{F}_i/\zeta \ & \dot{\theta}_i = \sqrt{2 D_r} \, \eta_i(t) \end{aligned}$$

where

• $v_0$ is the self-propulsion speed,
• $\mathbf{n}_i = (\cos\theta_i, \sin\theta_i)$ defines the propulsion direction,
• $D_t$ ($D_r$): translational (rotational) diffusion coefficient,
• $\xi$, $\eta$ are independent Gaussian white noises,
• $\mathbf{F}_i$ encodes interparticle or external forces.

The persistence time is $\tau_p = D_r^{-1}$; the persistence length $\ell_p = v_0 / D_r$. For an isolated ABP ($\mathbf{F}_i=0$), the mean-squared displacement (MSD) exhibits a crossover from ballistic to diffusive scaling. Explicit results for $d=2$ (Zhang et al., 2022, Basu et al., 2018):

$$\langle |\mathbf{r}(t) - \mathbf{r}(0)|^2 \rangle = 4 D_t t + \frac{2 v_0^2}{D_r^2}\left[ e^{-D_r t} - 1 + D_r t \right]$$

At $t \ll \tau_p$, MSD $\sim v_0^2 t^2$ (ballistic), while for $t \gg \tau_p$, rotational decorrelation yields $$\langle |\mathbf{r}(t) - \mathbf{r}(0)|^2 \rangle \sim 4 D_{\rm eff} t$$ with $D_{\rm eff} = D_t + v_0^2/(2 D_r)$. In three dimensions $D_{\rm eff} = D_t + v_0^2/(6 D_r)$ (Zhang et al., 2022, Wang et al., 2019).

The Peclet number, $Pe = v_0^2/(D_r D_t)$, and $\ell_p$ control the balance of persistence and diffusion.

2. Short-Time and Large-Deviation Statistics

Short-time statistics of ABP positions, particularly for $t \ll \tau_p$, are governed by the colored, memory-carrying active noise (Majumdar et al., 2020, Basu et al., 2018). The resulting distributions of $x$, $y$ are strongly anisotropic and non-Gaussian:

• $x$ is sharply peaked near $v_0 t$ with non-Gaussian tails,
• $y$ spreads as a random acceleration process, yielding broader (but still non-Brownian) marginals and anomalous first-passage exponents.

In the absence of translational diffusion, the ABP's position distribution has compact support, $x^2 + y^2 \leq v_0^2 t^2$. Optimal fluctuation theory provides explicit rate functions for large deviations, revealing essential singularities as the boundary of this support is approached (Majumdar et al., 2020). These signatures of activity remain robust beyond the Markovian regime.

3. Confinement Effects and Boundary Accumulation

When subject to external potentials or geometric confinement, ABPs display steady-state distributions markedly distinct from passive particles (Zhang et al., 2022, Wang et al., 2019, Wagner et al., 2021). In harmonic traps,

• For weak activity or low persistence ($\tau_r \ll \tau_{\rm trap}$), the stationary distribution is approximately Boltzmann with $T_{\rm eff} = D_{\rm eff}/\mu$.
• As persistence grows, the ABP distribution develops pronounced non-Boltzmann features: spatial plateaus, boundary accumulation, and ultimately bimodal structure reflecting the run length $r_{\rm B} = v_0 / (\mu k)$ (Wang et al., 2019).

Interactions with hard or soft walls induce boundary accumulation and persistent currents. In the zero-noise, steady-state Smoluchowski equation limit, non-equilibrium fluxes are controlled by the persistence length $\ell = v_0 / D_r$ and the nature of the boundary condition. The normal flux at the boundary acts as a source for large-scale density and current patterns, inducing depletion or long-range interactions between obstacles ("active depletion") (Wagner et al., 2021). For boundaries with inhomogeneous or patterned activity profiles, the location-dependent propulsion speed induces steady-state density patterns that "mimic" the substrate shape; typically, the density is low in regions of high activity and peaks at sharp activity gradients (Mishra et al., 2022).

4. Collective Behavior: MIPS, Glassy States, and Mode-Coupling Theory

At finite density, ABPs interact—commonly via short-range repulsion—generating a rich spectrum of collective phenomena:

• Motility-Induced Phase Separation (MIPS): Above a threshold $Pe$ and density, persistent self-propulsion leads to spontaneous liquid-gas coexistence, even in the absence of attractions. Phase boundaries can be derived from spinodal and binodal criteria based on coarse-grained field theory; e.g., effective diffusivity $$D_{\rm eff}(\rho)\propto v_0\ell_p (1-\rho/\rho_l)(1-2\rho/\rho_l)$$ vanishes at the spinodal $\rho = \rho_l/2$ (Perez-Bastías et al., 17 Apr 2025).
• Active Glass Transition: At high density and low persistence, ABPs can form nonergodic glassy states. Mode-Coupling Theory (MCT) captures the kinetic slowing-down and critical behavior. The $\alpha$-relaxation time diverges as

$$\tau_\alpha \sim |\varphi - \varphi_c|^{-\gamma}, \quad \gamma \approx 2.3\ldots2.6$$

and varies nontrivially with both $Pe$ and $\ell_p$ (Liluashvili et al., 2017). The active glass is softer than its passive analog and retains memory of initial orientational fluctuations within positional correlations, encoding the non-equilibrium character of the arrested state.

• Effect of Confinement and Disorder: Quenched disorder, random obstacles, or porous environments localize ABPs, suppress large-scale phase separation, and enhance dynamical heterogeneity. MIPS may be suppressed, pinned, or rendered highly spatially heterogeneous under strong confinement (Moore et al., 2023).

5. ABPs in Complex and Viscoelastic Media: Anomalous Transport

ABPs embedded in complex environments such as polymer solutions or cross-linked gels experience anomalous subdiffusive dynamics due to viscoelastic feedback:

• In star-polymer environments, an ABP exhibits subdiffusive mean-squared displacement scaling $\langle \Delta R^2(t) \rangle \sim t^\alpha$ with $\alpha \leq 1/2$, decreasing as the Péclet number increases. For moderate $Pe$, the motion matches fractional Brownian motion with Hurst exponent $H = \alpha/2$; for large $Pe$, logarithmic subdiffusion and anti-correlated velocities emerge, exactly captured by a fractional Langevin equation with both thermal and athermal noise (Joo et al., 2020).
• For ABPs in polymer solutions, the long-time diffusion coefficient $D$ can be non-monotonic in particle size—larger particles may diffuse faster at high activity—a direct result of the competition between enhanced persistence length and the scale-dependent viscosity of the polymer mesh (Du et al., 2018). The active particle experiences an effective viscosity $\eta_{\rm eff}^{\, a}$ larger than that of a passive probe under identical conditions.

6. Extensions: Anisotropic Particles, Noise-Induced Drifts, and Predictive Control

The minimal ABP model can be extended to include anisotropy and nonthermal noise:

• Fluctuating point-applied propulsion leads to translation-rotation coupling, anisotropic diffusion tensors, and a noise-induced drift term that can persist even in the overdamped regime. These effects alter long-time diffusivity and steady-state properties, and can dominate over thermal contributions depending on particle geometry and noise statistics (Thiffeault et al., 2021).
• In the context of engineering and optimal control, ABP systems can be steered using model-predictive control frameworks and deep-learning-based density forecasting, leveraging the multi-scale linkage between microscopic dynamics, continuum advection-diffusion PDEs, and control protocols acting through modulation of environmental or particle parameters. These approaches bridge single-particle physics to collective, pattern-forming behaviors in real time (Saremi et al., 7 Sep 2025).

7. First Passage and Survival Phenomena

ABPs exhibit fundamentally altered first-passage dynamics compared to passive particles, both in 1D and higher dimensions:

• Asymptotic expansions for the mean first passage time (MFPT) in a 1D interval reveal leading $O(Pe)$ corrections that can either decrease or increase MFPT depending on initial position and orientation bias, with symmetry-breaking appearing for non-uniform initial orientations. For unbiased orientations, the first non-trivial correction is $O(Pe^2)$ and is always a universal reduction in MFPT (Iyaniwura et al., 2023).
• In higher dimensions, partially absorbing or sticky walls induce non-trivial angular first-passage processes for the orientation, and the MFPT is linked to splitting probabilities and angular sojourn statistics. Non-Markovian (encounter-based) killing can be incorporated through higher-order Fokker-Planck equations for joint position–orientation–occupation-time statistics (Bressloff, 2023).

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References

• "Mode-Coupling Theory for Active Brownian Particles" (Liluashvili et al., 2017)
• "Active Brownian Motion in Two Dimensions" (Basu et al., 2018)
• "Anomalous diffusion for active Brownian particles cross-linked to a networked polymer: Langevin dynamics simulation and theory" (Joo et al., 2020)
• "Field Theory of Active Brownian Particles in Potentials" (Zhang et al., 2022)
• "Self-diffusive dynamics of active Brownian particles at moderate densities" (Soto, 2 Jan 2025)
• "Two-field theory for phase coexistence of active Brownian particles" (Perez-Bastías et al., 17 Apr 2025)
• "Active Brownian Particles in Random and Porous Environments" (Moore et al., 2023)
• "Molecular dynamics simulations of active Brownian particles in dilute suspension: diffusion in free space and distribution in confinement" (Wang et al., 2019)
• "Study of Active Brownian Particle Diffusion in Polymer Solutions" (Du et al., 2018)
• "Anisotropic active Brownian particle with a fluctuating propulsion force" (Thiffeault et al., 2021)
• "Steady states of active Brownian particles interacting with boundaries" (Wagner et al., 2021)
• "Active Brownian particles can mimic the pattern of the substrate" (Mishra et al., 2022)
• "Asymptotic analysis and simulation of mean first passage time for active Brownian particles in 1-D" (Iyaniwura et al., 2023)
• "Trapping of an active Brownian particle at a partially absorbing wall" (Bressloff, 2023)
• "Toward the full short-time statistics of an active Brownian particle on the plane" (Majumdar et al., 2020)
• "Understanding Contagion Dynamics through Microscopic Processes in Active Brownian Particles" (Norambuena et al., 2020)
• "Multi-Scale Modeling and Predictive Control of Active Brownian Particles" (Saremi et al., 7 Sep 2025)