Chaotic Scattering

The importance of scattering problems in physics has been appreciated for centuries. Yet it has become clear only recently that the vast majority of classical scattering problems exhibit irregular or chaotic behavior.

The purpose of this section is to describe in general terms the classical scattering problem, with an emphasis on the numerical methodology used in its solution and the appearance of chaotic behavior in the scattering functions.

Scattering

Consider an arbitrary potential with which a particle of mass $m$ interacts. A typical scattering experiment consists in launching particles from the asymptotic region and observe the trajectory as it re-enters in the asymptotic region at a later time. Consider for simplicity potentials of finite range; this implies that for $r$ larger that some large distance $R$ the potential can safely be taken as zero. Particles then follow straight line trajectories in the asymptotic region. If this assumption would not be valid (e.g., in Coulomb or gravitational fields), then asymptotic trajectories would have to be calculated. This is not always an easy task.

Two important scattering functions (in 2-dimension) are the scattering angle and the escape time. The scattering angle is defined as the angle between the outgoing straight line trajectory makes with the initial incoming direction.

The escape time or scattering time is defined as the time is takes to go from the initial launching time until the trajectory reaches the asymptotic region again.

Newton's Equation

Given the potential and the initial conditions, solving Newton's equation allows one to find the scattering trajectory. Newton's equation is written as

$$F = m a = m \frac{d v} {d t }$$

where $F$ is the force, $m$ the mass and $v$ the velocity. For many systems, the force itself originates from a potential function, a time-independent function $V(x)$, via

$$F = - \frac{d V}{d x}$$

Newton's equation leads to a system of ODEs to be solved:

$$\frac{d x}{d t} = v$$

$$\frac{d v} {d t } = \frac{F}{m}$$

The total energy of a mechanical system is the sum of the kinetic and potential energy,

$$E = K + V = \frac{m v^2}{2} + V(x)$$

You can interpret the function above as a function of two vector variables, $x$ and $v$; conservative systems are such that $E=constant$.