Ratchet transport for a chain of interacting charged particles

We study analytically and numerically the overdamped, deterministic dynamics of a chain of charged, interacting particles driven by a longitudinal alternating electric field and additionally interacting with a smooth ratchet potential. We derive the equations of motion, analyze the general properties of their solutions and find the drift criterion for chain motion. For ratchet potentials of the form of a double-sine and a phase-modulated sine it is demonstrated that both, a so-called integer and fractional transport of the chain, can occur. Explicit results for the directed chain transport for these two classes of ratchet potentials are presented.

Figure 1

Schematic representation of the ionic chain.

Figure 2

Form of the reduced ratchet potential $W\left(\overline{x}\right)$, part (a), and the corresponding force field $G\left(\overline{x}\right)$, part (b), that are under consideration in this work.

Figure 3

3-D plot of ${\varphi }_{\mathrm{cr}}$ as a function of $\mu$ and ${\eta }_{\mathrm{d}}$.

Figure 4

Plots of $\Delta w$, part (a), and $\overline{v}$, part (b), vs $\chi$ for $\varphi =3$ and $\mu =2$. The intervals $({\chi }_1^s,{\chi }_2^s)$ $(s=1,\dots ,4)$ are not visible on this scale.

Figure 5

Plots of the particle displacements $w\left(\tau \right)$ and $u\left(\tau \right)$ for $\varphi =3$, $\mu =2$, and $\chi ={10}^{-4}$.

Figure 6

Phase trajectory of the chain motion for the same parameters as in Fig. 5.

Figure 7

Plots of the particle displacements $w\left(\tau \right)$ and $u\left(\tau \right)$ for $\varphi =3$, $\mu =2$, and $\chi =0.473$.

Figure 8

Phase trajectory of the chain motion for the same parameters as in Fig. 7.