Stochastic Path Integral for the Active Brownian Particle in a Harmonic Potential

TL;DR

This paper develops a Martin-Siggia-Rose path-integral formulation for stochastic dynamics of an active Brownian particle in a harmonic trap, enabling exact expressions for time-dependent moments and a perturbative expansion of the position distribution in propulsion speed $v_0$. By using the passive Ornstein-Uhlenbeck motion as a solvable reference state, activity is treated perturbatively, yielding exact results for the mean position and MSD that terminate at finite order, and a full position distribution expressed as a series in $v_0$. The authors also demonstrate the method's versatility by analyzing a Brownian circle swimmer with an active torque, obtaining the mean position and MSD in a trap and highlighting the anisotropic-to-isotropic crossover at long times. This path-integral framework offers a powerful and generalizable tool for confined active matter and could be extended to interacting particle systems and collective phenomena such as MIPS and flocking.

Abstract

In this work we develop and apply a path integral formulation for the microscopic degrees of freedom obeying stochastic differential equations to an active Brownian particle (ABP) trapped in a harmonic potential. The formalism allows to derive exact analytic expressions for the time-dependent moments, like the mean position and the mean square displacement, including full dependence on initial conditions. In addition, the probability distribution of the particle's position can be evaluated systematically as a series expansion in the propulsion speed. Compared to previous methods relying on eigenfunction expansions of the equivalent Fokker-Planck equation, our method is easier to generalize to more complex situations: it does not rely on eigenfunctions but on a reference state that can be solved analytically, which in our case is the passive Brownian particle in a harmonic potential. We exemplify this versatility by also briefly treating an ABP with an active torque (Brownian circle swimmer, BCS) in a harmonic potential.

Figures (4)

• Figure 1: a) The mean square displacement MSD$_0(t)$ and b) the effective diffusion coefficient $D_\mathrm{eff}(t)$ for an ABP in an isotropic harmonic potential with initial conditions $\vec{q}_0=0$ and arbitrary direction angle $\vartheta_0\in[0,2\pi)$. The stiffness of the potential $\alpha$ is varied while keeping the Péclet number $\mathrm{Pe}=v_0/\sqrt{D_q\cdot D_\vartheta}$ constant. In a) we also show the MSD for a free ABP (black curve). Parameters are chosen for an experimentally realistic situation for colloidal swimmers, with $v_0 = 15~\mu\mathrm{m/s}$ and diffusion coefficients $D_q=0.27~\mathrm{\mu m^2/s^2}$ and $D_\vartheta=0.42~\mathrm{rad^2/s}$ (corresponding to a sphere of radius $R=0.74~\mu\mathrm{m}$ in water at room temperature obeying the Stokes-Einstein relation).
• Figure 2: a) Shown is the long-time probability distribution up to second order for different values of $\alpha$. The distribution is isotropic and the $x$-direction was chosen for simplicity. Parameters are $v_0=2$, $D_q=1$ and $D_{\vartheta}=0.1$, for varying potential strengths $\alpha$. For these parameters, and to second order in the perturbation, the "Donut" transition occurs at $\alpha_*=3.9$, cf. Eq. (\ref{['crit_donut']}). b), c) Shown are phase diagrams in the plane potential strength $\alpha$ vs. propulsion speed $v_0$ with $D_{\vartheta}=1$. The line indicates the transition between the typical confined shape with a maximum at the center and the "donut" shape. Note that the curvature of the parabola decreases as $D_q$ increases, while its offset is given by $-D_{\vartheta}$.
• Figure 3: Mean positions $\langle{\vec{q}(t)}\rangle$ for a Brownian Circle swimmer (BCS) starting at the origin and initially moving in the $x$-direction. a) A free BCS for varying angular velocity $\Omega_0$. b) A BCS with constant angular velocity $\Omega_0$ trapped in a harmonic potential of varying stiffness $\alpha$. c) Trapped BCS as in b) but now with constant harmonic trap stiffness $\alpha=0.2$ and for varying angular velocity $\Omega_0$. While the free BCS converges to a point that is determined by the parameters and the initial condition, see Eq. (\ref{['freeBCS']}), the trapped BCS always returns to the origin.
• Figure 4: The mean square displacement for free and trapped Brownian Circle Swimmers. Self-propulsion velocity $v_0$ and diffusion coefficients $(D_q,D_\vartheta)$ are chosen as in Fig. \ref{['fig:MSD and D_eff ABP']}. a) The free BCS with varying angular velocities $\Omega_0$ shows oscillations during the transition from the ballistic to the diffusive regime. The angular velocity decreases the amplitude of the MSD at long timescales. b) The trapped BCS for constant angular velocity $\Omega_0=3\pi/4~\mathrm{s}^{-1}$ with varying stiffness $\alpha$ of the trap. c) The trapped BCS for constant trap stiffness $\alpha=2.0$ and varying angular velocity $\Omega_0$.