Nonlinear Response of Inertial Tracers in Steady Laminar Flows: Differential and Absolute Negative Mobility

We study the mobility and the diffusion coefficient of an inertial tracer advected by a two-dimensional incompressible laminar flow, in the presence of thermal noise and under the action of an external force. We show, with extensive numerical simulations, that the force-velocity relation for the tracer, in the nonlinear regime, displays complex and rich behaviors, including negative differential and absolute mobility. These effects rely upon a subtle coupling between inertia and applied force that induces the tracer to persist in particular regions of phase space with a velocity opposite to the force. The relevance of this coupling is revisited in the framework of nonequilibrium response theory, applying a generalized Einstein relation to our system. The possibility of experimental observation of these results is also discussed.

Figure 1

(a) Force-velocity relation $⟨v_x⟩(F)$ and mobility $\mu$ (inset) for different values of $D_0$, in the case $\tau =10$. NDM is observable for $D_0={10}^{-5}$ and $D_0=2\times {10}^{-5}$, around $F\sim 4\times {10}^{-3}$. (b) Force-velocity relation $⟨v_x⟩(F)$ in the case $\tau =1$ for $D_0={10}^{-5}$. Notice the negative peak, corresponding to ANM, observed in a range of forces near $F\sim 6.5\times {10}^{-2}$, which is magnified in the inset.

Figure 2

Samples of the tracer’s trajectories for $\tau =1$, $F=0.065$, $D_0=0$. (a) History of the particle’s position $(x,y)$, recorded for a time length $760{\tau }^{*}$, starting near (0,0) (marked as a black spot); (b) the black arrows represent the same trajectory for a time length $\sim 10{\tau }^{*}$ (folded into a single cell, for the purpose of visualization), the green (dark gray) and cyan (light gray) arrows start with the same initial condition but are realized with $F=0.04$ and $F=0.09$, respectively, and the red arrows illustrate the underlying velocity field.

Figure 3

(a) Tracer diffusivity $D_x(F)$ for different values of $D_0$ [see legend of Fig. 1] with $\tau =10$. The continuous lines represent the values ${\tau }^2{D}_0$. In the inset it is plotted $D_x(F=0)$ as a function of the microscopic diffusivity $D_0$. (b) Mobility $\mu$ over ${\mu }_0=D_x/T$, for $\tau =10$. The dot-dashed lines represent the predictions of the GER in two cases.