Current reversals and current suppression in an open two-degree-of-freedom system

We explore the scattering of particles evolving in a two-degree-of-freedom Hamiltonian system, in which both degrees of freedom are open. Particles, initially having all kinetic energy, are sent into a so-called “interaction region,” where there will be an exchange of energy with particles that are initially at rest. The open nature of both components of this system eliminates any restrictions on which particles can escape from the interaction region. Notably, it is shown that two particles can cooperate in a mutual exchange of energy allowing both particles to escape and travel large distances. It is also shown that this level of cooperation is highly sensitive to the coupling strength between both components of the system. Indeed, large fluctuations of the magnitude and direction of the current are observed for small changes of this coupling parameter. Further, it is seen that current reversals are a prominent feature of this model. Another interesting observation is that even with the presence of chaotic scattering, it is possible that the system, for certain parameter regimes, will express a vanishing current, suggesting that there is a restoration of symmetry which, due to the initial setup, is broken. For an explanation of the different features of particle motion, we relate the phase-space dynamics to the various regimes of particle current.

Figure 1

Plot of the effective potential ($D=0.581\hspace{0.5em}69$). The main diagonal contains the regions of lowest potential energy.

Figure 2

Initial conditions $(q_1,p_1)$ for $D=0.3$ and $D=0.581\hspace{0.5em}69$, respectively.

Figure 3

Example trajectories using a range of different $D$ values. The red (solid) line shows the temporal evolution of particle $A$, while the green (dashed) line shows the time evolution of particle $B$. The initial conditions for each trajectory are chosen as $q_1(0)=-25.5$ and $q_2(0)=0$.

Figure 4

Partial energies corresponding to the trajectories in Fig. 3. Again, the temporal evolution of particle $A$ is shown by the red (solid) line and particle $B$ by the green (dashed) line.

Figure 5

Current as a function of $D$. The inset displays the current for the full range of $D$ values, namely $0⩽D\lesssim 0.5817$. The main figure displays, in detail, the sensitivity of the current to changes in $D$. This corresponds to the bottom right corner of the inset.

Figure 6

Sojourn time of an ensemble of particles in the interaction region. Left: The green (scattered) points show the time for the ensemble when $D=0.5617$. Similarly, red (lower line) is for the particles when $D=0.5672$. Right: Same as left with $D=0.581\hspace{0.5em}69$.

Figure 7

Histograms displaying the final partial energies, again using the five $D$ values from the table in Sec. 2, of particles $A$ [blue (dark gray)] and $(q_1,p_1)$ [yellow (light gray)] for an ensemble of initial conditions.

Figure 8

Manifolds for $D=0.5617$ and $D=0.581\hspace{0.5em}69$. The inset shows the manifold extended in configuration space.

Figure 9

Locations of the $(i,j)$th equilibrium point in configuration space for $-20⩽i⩽20$ and $-1⩽j⩽1$, for $D=0.581\hspace{0.5em}69$. Center-center points are indicated by a star, saddle-center points by a cross, and saddle-saddle points by a plus sign.

Figure 10

Effective potential as a function of $D$ shown at the saddle points $x_{(0,2i+1)}$ (left panel) and $x_{(1,2i+1)}$ (right panel). The horizontal line indicates the total simulation energy, i.e., $E=0.9$. The two vertical lines located at $D=0.3$ and $D=0.581\hspace{0.5em}69$ were used in the simulations. The green line (negative slope) shows the initial energy of particle $A$ as a function of $D$.

Figure 11

Shown for $D=0.581\hspace{0.5em}69$ is the effective potential. The energetically inaccessible regions are shown in black.

Figure 12

Surfaces displaying the phase-space dynamics of particle $A$ (panels on the left) and particle $B$ (panels on the right) for three different $D$ values. From top to bottom, these are $D=0.3$, $D=0.5672$, and $D=0.581\hspace{0.5em}69$. The coordinates $q_2$ and $q_2$ are presented mod(1).