Time-delayed feedback control of diffusion in random walkers

Time delay in general leads to instability in some systems, while specific feedback with delay can control fluctuated motion in nonlinear deterministic systems to a stable state. In this paper, we consider a stochastic process, i.e., a random walk, and observe its diffusion phenomenon with time-delayed feedback. As a result, the diffusion coefficient decreases with increasing delay time. We analytically illustrate this suppression of diffusion by using stochastic delay differential equations and justify the feasibility of this suppression by applying time-delayed feedback to a molecular dynamics model.

Figure 1

Suppression of diffusion in random walks of the system (2). The red solid and green dashed lines represent when $l=2$ and $l=20$, respectively. $D=0.001$.

Figure 2

$\mathcal{D}$ with respect to $l$, averaged over 1000 realizations. The dashed line is proportional to $1/{(1+Kl)}^2$. $K=0.5$.

Figure 3

Time evolutions of the positions of two particles without control inputs. Particles flocculate because of the interacting force, and move together diffusively.

Figure 4

Time evolutions of the center position of (a) two particles and (b) three particles. The green and the red lines are the time evolutions without control inputs and with control input $\left(\tau =10\right)$, respectively.

Figure 5

The diffusion coefficient $\mathcal{D}$ of the molecular dynamics model with two particles as a function of $\tau$.