Superdiffusive trajectories in Brownian motion

The Brownian motion of a microscopic particle in a fluid is one of the cornerstones of statistical physics and the paradigm of a random process. One of the most powerful tools to quantify it was provided by Langevin, who explicitly accounted for a short-time correlated “thermal” force. The Langevin picture predicts ballistic motion, $\langle x^2\rangle \sim t^2$ at short-time scales, and diffusive motion $\langle x^2\rangle \sim t$ at long-time scales, where $x$ is the displacement of the particle during time $t$, and the average is taken over the thermal distribution of initial conditions. The Langevin equation also predicts a superdiffusive regime, where $\langle x^2\rangle \sim t^3$, under the condition that the initial velocity is fixed rather than distributed thermally. We analyze the motion of an optically trapped particle in air and indeed find $t^3$ dispersion. This observation is a direct proof of the existence of the random, rapidly varying force imagined by Langevin.

Figure 1

Top: average position of subsets of particles having a given initial velocity $v_0$. Bottom: distribution of the position of particles at $\omega t=0.97$ for subsets of particles having a given initial position $x_0=0$ and its decomposition in subset of fixed initial velocity $v_0$ (with $v_0/\sqrt{{\langle v^2\rangle }_{\mathrm{eq}}}=-3,-2,-1,0,1,2,3$). Data from the high pressure experiment of Ref. [13].

Figure 2

Experimentally measured particle dispersion as a function of time conditioned to the initial conditions $x_0$ and $v_0$ at $t=0$. Dispersion curves are calculated for 2500 sets of trajectories with unique initial conditions and then averaged to create the curve shown in the figure. The time is normalized by the trap frequency, ${\omega }_0$, while the dispersion is normalized by the long-time thermal particle distribution, ${\langle x^2\rangle }_{\mathrm{eq}}$. The power law at short time (black line) is $t^3$, as anticipated from Eq. (6). The solid lines are the exact prediction of the Langevin model in a harmonic potential [Eq. (11)]. The red is for the low-pressure case and the blue is for the high-pressure case in the original data [13].

Figure 3

Left: Particle dispersion uniquely due to the distribution of its initial velocity $v_0$: trajectories are conditioned to initial position $x_0$ at $t=0$ and averaged over multiple realizations of noise $f\left(t\right)$ and the thermal distribution of initial velocities. Right: Particle dispersion uniquely due to the distribution of its initial position $x_0$: trajectories are conditioned to initial velocity $v_0$ at $t=0$, and averaged over all realization of noise $f\left(t\right)$. Dispersion curves are calculated for 50 unique initial conditions and then averaged to create the curve shown in the figure. The time is normalized by the trap frequency, ${\omega }_0$, while the dispersion is normalized by the long-time thermal particle distribution, ${\langle x^2\rangle }_{\mathrm{eq}}$. The black line is the expected short-time power law behavior in both cases. The solid lines are the exact prediction of the Langevin model in a harmonic potential. The red curves and points correspond to the low-pressure case and the blue curves correspond to the high-pressure case [13].