Motion by stopping: Rectifying Brownian motion of nonspherical particles

We show that Brownian motion is spatially not symmetric for mesoscopic particles embedded in a fluid if the particle is not in thermal equilibrium and its shape is not spherical. In view of applications to molecular motors in biological cells, we sustain nonequilibrium by stopping a nonspherical particle at periodic sites along a filament. Molecular dynamics simulations in a Lennard-Jones fluid demonstrate that directed motion is possible without a ratchet potential or temperature gradients if the asymmetric nonequilibrium relaxation process is hindered by external stopping. Analytical calculations in the ideal gas limit show that motion even against a fluid drift is possible and that the direction of motion can be controlled by the shape of the particle, which is completely characterized by tensorial Minkowski functionals.

Figure 1

(Color online) Asymmetrically shaped motor (here a triangle) built from fluid particles and placed in a Lennard-Jones fluid. Its motion is restricted to a one-dimensional track with periodically spaced binding sites along the $x$ axis. If the motor’s center of mass crosses a binding site, the velocity of the motor is set to zero. To test the importance of the motor’s shape for the momentum transfer during collisions with fluid particles we orient the triangle in two different ways.

Figure 2

Histograms of end positions of a triangular motor. The distance between stopping sites is $\delta s$ (see Fig. 1). The asymmetrically oriented motor moves preferably to positive $X$ values, whereas the symmetric triangle diffuses around $X=0$. A discrete spacing is visible since the motors accumulate around the stopping sites. The histograms show 400 realizations over a simulation time of $t=2000t_0$ with integration step $\delta t=0.001t_0$. With decreasing $\delta s$ between stopping sites, the distribution becomes narrower as expected for a hindered diffusion process.

Figure 3

Average position (400 runs) of an asymmetrically oriented isosceles triangle of height $10\sigma$ and base length $20\sigma$ moving with a constant velocity in the direction opposite to its sharp edge (i.e., in the direction of its base). Thus, directed motion is possible when the triangle is oriented asymmetrically with respect to the direction of motion. Otherwise the motor performs symmetric fluctuations around its starting position. Inset: trajectories of individual motors. Note that the velocity decreases for decreasing distances $\delta s$ between stopping sites if the typical stopping time is smaller than the relaxation time of the motor.

Figure 4

(a) A triangular motor (apex angle $\theta =\pi ∕6$, base length $L\sqrt{\rho }=1$, ${ϵ}^2=m∕M=0.05$) can swim even against a fluid drift velocity $\mathbf{U}=\sqrt{M∕\left(k_{B}T\right)}\mathbf{u}$ during its relaxation in thermal equilibrium. (b) The direction of motion can be inverted solely by changing the shape of the motor. The maximum velocity $V_{\max }$ and the relaxation time $\tau$ of a capped-triangular motor (sketched in the inset) can be regulated by the opening angle $\theta$ which determines the Minkowski functionals in Eqs. (3, 4).

Figure 5

Relaxation of a motor to thermal equilibrium: the molecular dynamics simulations of the LJ fluid (symbols; averages over $4\times {10}^5$ runs) can be well described by the analytic results in the ideal gas limit (lines) if the data are normalized by the maximum velocity $V_{\max }$ and the relaxation time.