# Multiple Current Reversals Using Superimposed Driven Lattices
## 4 Discussion
### First Current Reversal  
Next, we discuss why the transport velocity is reversed due to the superposition of a localized lattice of 2D Gaussian barriers \(V_{G1}\) on the background lattice \(V_{B}\) as in the setup II. Here, the particle dynamics is governed by the 2D PSOS (Figure 2a) in the region where only the lattice \(V_{B}\) is present, but by the full 5D phase space in the region \(x_{1}<x<x_{2}\) due to the presence of both the lattices \(V_{B}\) and \(V_{G1}\). Although this 5D phase space cannot be straightforwardly visualized, it turns out that the cause of current reversal can be explained solely on the basis of the ensemble population in the 2D PSOS in Figure 2a. Since the ensemble is initialized near the origin \((0,0)\), the particles initially experience the spatial potential only due to the lattice \(V_{B}\) and hence their initial dynamics is exactly the same as described for setup I in the previous subsection. As a result the initial transport velocity is \(\bar{v}_{x}\simeq 1.3\).  
Since the transport velocity is positive, the particles reach \(x=x_{1}\) in the course of time where they encounter the lattice \(V_{G1}\) in addition to \(V_{B}\). In the region \(x_{1}<x<x_{2}\), since the particle dynamics is no longer governed by the 2D PSOS, the particles are now no longer confined to the central chaotic sea and can access higher velocities beyond the FASC. In fact, the higher potential heights of the Gaussian barriers ensure that most of the particles perform chaotic diffusive motion even at higher velocities corresponding to the full 5D phase space of our setup. This leads to an interesting conversion process between diffusive and ballistic motion of the particles at the left edge of the lattice \(V_{G1}\), i.e., at \(x=x_{1}\), which is the key mechanism behind the current reversal. A diffusive particle close to the left edge but with \(x>x_{1}\) can cross this edge in the course of time back to \(x<x_{1}\) with \(v_{x}<0\). However, its velocity \(v_{x}\) immediately after crossing back can be either \(\gtrsim\!-6\) in which case it lies in the chaotic sea performing diffusive motion or \(\lesssim\!-6\) which means it moves away ballistically from the lattice \(V_{G1}\) towards the negative \(x\)-direction. For the particles with \(v_{x}\lesssim-6\), such a conversion from diffusive to ballistic motion ensures that they perform unidirectional ballistic flights towards the negative \(x\)-direction, thus attaining a permanent negative velocity. On the other hand, since the particles with \(v_{x}\gtrsim-6\) perform diffusive motion they can again enter the region \(x_{1}<x<x_{2}\) in course of time. They would then undergo the same conversion mechanism until all the particles undergo the conversion from diffusive to ballistic motion with \(v_{x}\lesssim-6\). This can be observed in Figure 3a where we plot the asymptotic distribution of the ensemble over the 2D PSOS. Most of the particles are located on the invariant curves \(v_{x}\lesssim-6\) moving ballistically in the negative \(x\)-direction. This results in the asymptotic ensemble transport velocity \(\bar{v}_{x}\simeq-7.1\). The dynamical change in the transport direction leads to the current reversal at \(t_{r1}=3.1\times 10^{3}\) T.