# Multiple Current Reversals Using Superimposed Driven Lattices
## 4 Discussion
### Second Current Reversal  
We now discuss why the transport can be reversed once again by superimposing a second identical lattice of Gaussian barriers \(V_{\text{C2}}\) between \(x=-x_{1}\) and \(x=-x_{2}\) as in the setup III. Initially since the particles are initialized near the origin \((0,0)\), the ensemble drifts towards the positive \(x\)-direction and exhibit the same dynamics as in the setup II. As a result, the transport velocity is initially positive till the first current reversal occurs at \(t_{r1}=3.1\times 10^{3}\) T and then continues to be negative until \(t\simeq 4.6\times 10^{3}\)\(T\). Thereafter, the particle dynamics undergo another conversion process due to which the transport velocity is reversed again. Unlike the situation in setup II, the particles in the setup III moving ballistically with negative \(v_{x}\) after \(t=t_{r1}\) cannot keep moving through the lattice \(V_{B}\) for all longer timescales. Instead at some point, they interact with the lattice \(V_{\text{G2}}\) in the region \(-x_{2}<x<-x_{1}\). Due to the high kinetic energy of the particles (since \(|v_{x}|\gtrsim 6\)), some of them can pass through the region and continue their ballistic flights for longer timescales through the lattice \(V_{B}\). This can be seen from the asymptotic ensemble distribution projected on to the 2D PSOS in Figure 3b, showing that even at \(t=t_{f}\) a considerable fraction of the ensemble moves with \(v_{x}\lesssim-6\). However, once a particle enters the region \(-x_{2}<x<-x_{1}\), its dynamics is no longer confined to the region \(v_{x}\lesssim-6\) of the 2D PSOS and can explore the different regions of the 5D phase space. Hence most of the particles attain \(v_{x}>0\) which in turn allow them to cross the right edge of the lattice \(V_{\text{G2}}\) at \(x=-x_{1}\) back into the region \(-x_{1}<x<x_{1}\) where only the lattice \(V_{B}\) is present. After crossing to this region, these particles can either belong to the chaotic sea or to the invariant spanning curves with velocity higher than the FISC at \(v_{x}\simeq 10\) of the 2D PSOS in Figure 2a. The particles with \(v_{x}\gtrsim 10\) perform unidirectional ballistic flights in the positive \(x\)-direction. Due to their significantly higher kinetic energy, most of these particles are not'reflected' further by the lattice \(V_{\text{G1}}\) in the region \(x_{1}<x<x_{2}\); instead, after crossing this region, they continue moving ballistically through the lattice \(V_{B}\) with \(v_{x}\gtrsim 10\). This can be observed from the significant distribution of particles with \(v_{x}\gtrsim 10\) in Figure 3b at \(t=t_{f}\). As the velocities of more and more particles undergo the conversion from \(v_{x}\lesssim-6\) to \(v_{x}\gtrsim 10\), the transport velocity increases steadily after \(t\simeq 4.6\times 10^{3}\) T, leading to a second current reversal at \(t=t_{r2}=5.6\times 10^{3}\) T (Figure 1b). On the other hand, the particles in the chaotic layer would eventually again encounter the lattices \(V_{\text{G1}}\) or \(V_{\text{G2}}\) so that their chaotic dynamics is eventually converted to ballistic motion either with \(v_{x}\gtrsim 10\) or with \(v_{x}\lesssim-6\). Due to the overall higher number of particles moving asymptotically with \(v_{x}\gtrsim 10\) compared to those with \(v_{x}\lesssim-6\) (see Figure 3b), the asymptotic transport velocity is \(\sigma_{x}\simeq 4.7\). It is worth stressing that the conversion from chaotic to ballistic dynamics of the particles is possible solely due to the 2D nature of the potential generated by the superimposed Gaussian barriers. This ensures that the particle dynamics is coupled in the \(x\) and \(y\) directions upon entering the regions \(x_{1}<|x|<x_{2}\) (where the Gaussian barriers are present) whereas it is uncoupled in all other regions. Figure 3: The asymptotic particle distribution as a function of position \(x\) mod \(L\) and \(v_{x}\) (in colormap) of all the \(N=10^{4}\) particles propagating in the (**a**) setup II and (**b**) setup III projected on to the PSOS shown in Figure 2a.