Regularity and chaos in interacting two-body systems

We study classical and quantum chaos for two interacting particles on the plane. This is the simplest nontrivial case which sheds light on chaos in interacting many-body systems. The system consists of a confining one-body potential, assumed to be a deformed harmonic oscillator, and a two-body interaction of Coulomb type. In general, the dynamics is mixed with regular and chaotic trajectories. The relative roles of the one-body field and the two-body interaction are investigated. Chaos sets in as the strength of the two-body interaction increases. However, the degree of chaoticity strongly depends on the shape of the one-body potential and, for some shapes of the harmonic oscillator, the dynamics remains regular for all values of the two-body interaction. Scaling properties are found for the classical as well as for the quantum mechanical problem.

Figure 1

The landscape of the effective potential (19) at fixed deformation parameter $\epsilon =0.4$ is displayed in (a). (b) and (c) show the intersections of the surface $V(x,y)$ with the planes $x=0$ and $y=0$.

Figure 2

The equipotential lines of the potential energy surface (19).

Figure 3

The Poincaré surfaces of section (23) for the scaled energy $E=1.5<V_{\mathrm{sad}}$ at $\epsilon =0.4$. Dots represent one chaotic orbit. A few regular orbits labeled by digits 1, 2, 3, and 4 correspond to different initial conditions for the Hamilton equations of motion.

Figure 4

The examples of the Poincaré surfaces of section (23) for the scaled energies $E=3$ and $E=10$ above the saddle point energy (21) at deformation $\epsilon =0.4$. As the energy $E$ increases, the fraction of chaotic parts of phase space decreases.

Figure 5

The dependence of the Poincaré surfaces of section (23) on the deformation parameter $\epsilon$ [Eq. (8)] at fixed scaled energy $E=3$. We notice that $E=3$ is above the saddle point energy (21) for every $\epsilon ∊[0,1]$. In the cases $\epsilon =0$ ($1:1$ deformation), and $\epsilon =0.6$ ($2:1$ deformation), the classical phase space is regular due to the presence of second constants of motion, the square of the angular momentum and Eq. (24), respectively. For the deformation parameter $\epsilon =0.8$ corresponding to the integer frequency ratio ${\omega }_x∕{\omega }_y=3$ ($3:1$ deformation), the classical phase space is mostly regular due to the degeneracy of the effective potential (19) for integer ${\omega }_x∕{\omega }_y$.

Figure 6

The dependence of the chaoticity parameter $q=q(E,\epsilon )$ on the deformation parameter $\epsilon$ at the scaled energy $E=3>V_{\mathrm{sad}}\left(\epsilon \right)$. Local minima at 0.8, 0.88, 0.92, and 0.97 correspond to the integer frequency ratios ${\omega }_x∕{\omega }_y=3$, 4, 5, and 6, respectively. In the limit $\epsilon \to 1$, the chaoticity parameter $q$ approaches the value 0.52. The stars mark the numerically calculated values of $q$; the solid line is a spline interpolation to guide the eye.

Figure 7

The ratio $q=q(E,\epsilon )$ between the area of chaotic regions and the total area of allowed phase space in the Poincaré surfaces of section (23) versus the scaled energy $E$ at fixed deformation parameter $\epsilon =0.4$ (left) and $\epsilon =0.55$ (right). A similar dependence on $E$ is expected for every deformation parameter $\epsilon$. With increasing energy $E$, the relative fraction $q$ of chaotic orbits decreases. The stars mark the numerically calculated values of $q$ the solid line is a spline interpolation to guide the eye.

Figure 8

Counting function $N\left(E_{qm}\right)$ for the first 50 levels with positive parity $\gamma =+1$, on the left hand side for deformation parameter $\epsilon =0.4$, on the right hand side for deformation parameter $\epsilon =0.55$. The solid lines are the corresponding smooth parts of the counting function, obtained by fitting a second order polynomial.

Figure 9

The cumulative spacing distribution $F\left(s\right)$ calculated for the deformation parameters $\epsilon =0.4$ and $\epsilon =0.55$. The fit gives $q\approx 0.9$ for $\epsilon =0.4$ and $q\approx 0.4$ for $\epsilon =0.55$. For comparison, the cumulative Poisson and Wigner-Dyson statistics are also shown as dotted lines.

Figure 10

The nearest neighbor spacing distribution $p\left(s\right)$ for two values of deformation parameter $\epsilon =0.4$ and $\epsilon =0.55$, obtained from the numerics (histograms). The lowest calculated 400 positive-parity and negative-parity energy states are included. The solid lines show the Brody distributions resulting from the fit of the cumulative spacing distributions. For comparison, the Poisson and the Wigner-Dyson spacing distributions are also shown as dotted lines.

Figure 11

Spacing distribution $p\left(s\right)$ for the deformation parameters $\epsilon =0$ and $\epsilon =0.6$ both corresponding to classically regular motion.