Emergence of continual directed flow in Hamiltonian systems

We propose a minimal model for the emergence of a directed flow in autonomous Hamiltonian systems. It is shown that internal breaking of the spatiotemporal symmetries, via localized initial conditions, which are unbiased with respect to the transporting degree of freedom, and transient chaos conspire to form the physical mechanism for the occurrence of a current. Most importantly, after passage through the transient chaos, trajectories perform solely regular transporting motion so that the resulting current is of continual ballistic nature. This has to be distinguished from the features of transport reported previously for driven Hamiltonian systems with mixed phase space where transport is determined by intermittent behavior exhibiting power-law decay statistics of the duration of regular ballistic periods.

Figure 1

(Color online) Typical time evolutions of the coordinates as indicated in the plot for $D=0.94$, $F=-0.5$, and $\omega =1$. Note the time scale on the inset figure; $0\le t\le {10}^7$. The initial conditions are $p\left(0\right)=q\left(0\right)=0$, $P\left(0\right)=-1.70778$, and $Q\left(0\right)=0.62700$.

Figure 2

(Color online) Poincaré surfaces of section represented in the $\left(p,q\right)$ plane for $D=0.94$, $F=-0.5$, and $\omega =1$.

Figure 3

(Color online) Symmetry lines for $D=0.94$, $F=-0.5$, and $\omega =1$. The total energy is $E=1.625$ and the boundaries of the energetically allowed region are indicated by the horizontal dashed lines. Superimposed is the trajectory shown in Fig. 1.

Figure 4

(Color online) Time evolution of the energy of the washboard particle $E_1$ and the harmonic oscillator $E_2$ for initial conditions given in Fig. 1. The two dashed horizontal lines correspond to the energy levels $1/\pi$ and $E-D-E_a$ bounding the energies $E_1=E_a$ and $E_2=E-D-E_a$ in the asymptotic regime.