103
2.6. Stochastic Coulomb scattering
of the optical system, together with the Coulomb scattering with
every other particle. If one could remove the effect of the Coulomb
scattering, the particle of interest would intersect the target plane
at a different position. We call the vector difference between these
two positions the trajectory displacement. It is governed by a probability distribution. In mathematical terms, we wish to find this
probability distribution function.
Because the N -body problem cannot be solved in closed form,
we are led to seek a suitable approximation. To this end, we imagine a second particle, also chosen at random. The second particle
scatters with the first particle, producing a smaller random trajectory displacement. Now we imagine a third particle, chosen at
random, producing a small random trajectory displacement of the
first particle. Similarly, each of the N − 1 particles produces a
random displacement of the first particle. Each of these scattering events is a two-body interaction. As such, each event can be
solved analytically in principle. We now form the vector sum of all
of the N − 1 trajectory displacements of the first particle, making
a resultant trajectory displacement. This is shown schematically
in Figure 2.16. This is essentially the same approximation used as
a starting point by Van Leeuwen and Jansen [89], although the
details of their analysis are quite different from what is presented
here. The vector sum of the two-body displacements is given by
X S , while the N -body displacement is denoted by X N . In general,
these two displacements differ, as they were arrived at by different
means.
At this point we form a key hypothesis, namely, the sum of the
two-body displacements approximates the N -body displacement
to within an error which is small, compared with the displacement.
Mathematically, this is expressed as |X N − X S |/|X N | « 1. This
hypothesis can be tested using Monte Carlo simulation. Two separate Monte Carlo simulations are required, one using the N-body
algorithm described in the previous section, and another using the
vector superposition of two-body interactions described here [4].
The two simulations are run with identical initial conditions for
2.6. Stochastic Coulomb scattering
of the optical system, together with the Coulomb scattering with
every other particle. If one could remove the effect of the Coulomb
scattering, the particle of interest would intersect the target plane
at a different position. We call the vector difference between these
two positions the trajectory displacement. It is governed by a probability distribution. In mathematical terms, we wish to find this
probability distribution function.
Because the N -body problem cannot be solved in closed form,
we are led to seek a suitable approximation. To this end, we imagine a second particle, also chosen at random. The second particle
scatters with the first particle, producing a smaller random trajectory displacement. Now we imagine a third particle, chosen at
random, producing a small random trajectory displacement of the
first particle. Similarly, each of the N − 1 particles produces a
random displacement of the first particle. Each of these scattering events is a two-body interaction. As such, each event can be
solved analytically in principle. We now form the vector sum of all
of the N − 1 trajectory displacements of the first particle, making
a resultant trajectory displacement. This is shown schematically
in Figure 2.16. This is essentially the same approximation used as
a starting point by Van Leeuwen and Jansen [89], although the
details of their analysis are quite different from what is presented
here. The vector sum of the two-body displacements is given by
X S , while the N -body displacement is denoted by X N . In general,
these two displacements differ, as they were arrived at by different
means.
At this point we form a key hypothesis, namely, the sum of the
two-body displacements approximates the N -body displacement
to within an error which is small, compared with the displacement.
Mathematically, this is expressed as |X N − X S |/|X N | « 1. This
hypothesis can be tested using Monte Carlo simulation. Two separate Monte Carlo simulations are required, one using the N-body
algorithm described in the previous section, and another using the
vector superposition of two-body interactions described here [4].
The two simulations are run with identical initial conditions for