# Multiple Current Reversals Using Superimposed Driven Lattices
## 3 Results  
To explore the transport properties of our setup, we initialize \(N=10^{4}\) particles within a square region \(x,y\in[-5,5]\times[-5,5]\) with small random velocities \(v_{x}\), \(v_{y}\in[-0.1,0.1]\times[-0.1,0.1]\). The initial velocities of the particles are chosen randomly within the low velocity regime such that their initial kinetic energies are small compared to the potential heights of the lattices. Subsequently we time evolve our ensemble up to time \(t=t_{f}=4\times 10^{4}\) T by numerical integration of Equation (1) using a Runge-Kutta Dormand Prince integrator [53]. The background lattice is large enough such that the ensemble never leaves the lattice throughout the duration of the simulation; hence no boundary conditions are imposed on the particles. We now discuss the transport properties of our setup characterized by the average velocity \(\bar{v}_{x}\) of the particle ensemble along the \(x\)-direction. In the presence of only the background lattice \(V_{B}\) (setup I in Figure 1a), the particles exhibit directed transport along the positive \(x\)-direction with an asymptotic transport velocity \(\bar{v}_{x}\simeq 1.3\) (Figure 1b). In the setup II we consider a spatially localized lattice of Gaussian barriers \(V_{G1}\) superimposed on the lattice \(V_{B}\) (Figure 1a), such that \(U_{G}(\mathbf{R}_{mn})=5\) for \(5\times 10^{3}<m<10^{4}\) and \(U_{G}(\mathbf{R}_{mn})=0\) everywhere else. We define \(x_{1}=5\times 10^{3}-\epsilon\) and \(x_{2}=10^{4}+\epsilon\) with \(\epsilon=5L\) as the left and right 'edges' of the lattice \(V_{G1}\), since the force on the particles due to the Gaussian barriers is negligible for \(x<x_{1}\) and \(x>x_{2}\). In this case, we observe an initial directed transport along the positive \(x\)-direction with \(\bar{v}_{x}>0\) (Figure 1b). However, the transport velocity starts to decelerate and at \(t\simeq 3.1\times 10^{3}\) T, the transport completely vanishes. Thereafter, the ensemble is transported along the negative \(x\)-direction with \(\bar{v}_{x}<0\) and the transport velocity finally saturates to \(\bar{v}_{x}\simeq-7.1\). Hence, a superimposed spatially localized lattice of Gaussian barriers can trigger a current reversal with the reversal timescale in this case given by \(t_{\tau 1}=3.1\times 10^{3}\) T, i.e., when \(\bar{v}_{x}\) changes its sign. In the third setup (setup III), we consider a second identical lattice of Gaussian barriers \(V_{G2}\) superimposed on the lattice \(V_{B}\) between \(x=-x_{1}\) and \(x=-x_{2}\), such that now \(U_{G}(\mathbf{R}_{mn})=5\) for \(5\times 10^{3}<|m|<10^{4}\) and \(U_{G}(\mathbf{R}_{mn})=0\) elsewhere (Figure 1a). Up to \(t\simeq 4.6\times 10^{3}T\), the transport velocity exhibits a similar behavior as that observed in setup II (Figure 1b). Thereafter, instead of asymptotically attaining a negative value, the velocity increases steadily, exhibiting a second current  
Figure 1: (**a**) Schematic representation of a slice of our 2D setup along the \(x\)-direction and viewed along the \(y\)-direction. The filled dots denote particles and the colors red and black denote diffusive and ballistic motion respectively. The arrows denote the direction of motion of the particles at asymptotic timescales (\(t=t_{f}\)) with the length of the arrow being proportional to the magnitude of the \(x\) component of their velocities, i.e., \(v_{x}\). In the presence of only the driven background lattice \(V_{B}\) depicted in blue (setup I, upper panel), most particles exhibit diffusive/chaotic transport towards right, hence the average transport is along the positive \(x\)-direction. On superimposing a finite lattice of 2D Gaussian barriers \(V_{G1}\) (in green), most particles undergo a conversion from diffusive to ballistic motion leading to a reversal of their average transport direction (setup II, middle panel). Their velocities can be reversed once again due to the superposition of a second identical lattice of Gaussian barriers \(V_{G2}\), thus leading to a second current reversal.