122
4 Chaotic Scattering
ψ
(6)
1 = (0.441, 0.290, 0.290, 0.381, 0.381, 0.441)
φ
(6)
1 = (0.441, 0.671, 0.671, 0.290, 0.290, 0.441)
T
(4.19)
and
ψ
(5)
1 = (0.441, 0.290, 0.290, 0.763, 0.441)
φ
(5)
1 = (0.441, 0.671, 0.671, 0.580, 0.441)
T ,
(4.20)
respectively. Note that, since the transfer matrices are known exactly, the accuracy
of these numbers is only limited by the accuracy of the matrix solver.
We can now reproduce the number of each of the symbols “A”, “B”, “C”,
“+”, “−”, and “ ¯
A” at level n in the branching tree for scattering from the
left by allowing the transfer matrix to act n − 1 times on the initial partition
S L
1 = (1, 1, 1, 0, 0, 0) T of the unstable manifold of F L (see Fig. 4.15a). We obtain
T
n−1
6 ·S L
1 = 1.021(2.315) n−1 φ
(6)
1 . Similarly, if we allow the transfer matrix to
act n − 1 times on the initial partition S R
1 = (0, 0, 1, 1, 0, 0) T of the unstable
manifold of F R (see Fig. 4.15a) for scattering from the right, we obtain T
n−1
6 ·S R
1 =
0.671(2.315) n−1 φ
(6)
1 (a similar analysis can be performed on T 5 ).
It only takes a few steps up the branching tree for scattering from the left, or from
the right, for the fractional distribution of symbols “A”, “B”, “C”, “+”, “−”, and “ ¯
A”
to be determined by the fractional distribution of symbols in the right eigenvector
φ
(6)
1 . Thus, for both incident directions, the branching trees, as regards the fraction
of each type of symbol present in φ
(6)
1 , become the same. The fractions are f A =
0.157, f B = 0.239, f C = 0.239, f + = 0.103, f − = 0.103, and f ¯
A = 0.157.
We now can use this information, and the information in Fig. 4.16b, to obtain a
rough estimate of the likelihood that an incident particle, caught up in the fractal
structure, gets transmitted or reflected. First, remember that
√
(◦) indicates a gap
containing trajectories that ultimately travel to −∞ and are of S L type (those that
travel to +∞ and are of S R type). From Fig. 4.16b, we see that intervals of type “A”
and “ ¯
A” in an unstable manifold, in the next iteration of the map, will contain one
S L gap and one S R gap. Intervals of type “B” and “C”, in the next iteration of the
map, will contain two S L gaps. Intervals of type “+” and “−”, in the next iteration
of the map, will contain two S R gaps. The fraction of S L gaps, in the next iteration
of the map, is 0.636 and the fraction of S R gaps is 0.364. Thus, for particles incident
from the left, 36.4% of the gaps get transmitted and 63.6% of the gaps get reflected.
For particles incident from the right, 63.6% of the gaps get transmitted and 36.4%
get reflected.
The percentage of gaps transmitted or reflected gives an indication of the
asymmetry of the overall scattering process. However, for a given line of initial
points, the stepping time data, like that shown in Fig. 4.14c, can be used to
distinguish which initial points are S R -type and which are S L -type. In Lin et al.
4 Chaotic Scattering
ψ
(6)
1 = (0.441, 0.290, 0.290, 0.381, 0.381, 0.441)
φ
(6)
1 = (0.441, 0.671, 0.671, 0.290, 0.290, 0.441)
T
(4.19)
and
ψ
(5)
1 = (0.441, 0.290, 0.290, 0.763, 0.441)
φ
(5)
1 = (0.441, 0.671, 0.671, 0.580, 0.441)
T ,
(4.20)
respectively. Note that, since the transfer matrices are known exactly, the accuracy
of these numbers is only limited by the accuracy of the matrix solver.
We can now reproduce the number of each of the symbols “A”, “B”, “C”,
“+”, “−”, and “ ¯
A” at level n in the branching tree for scattering from the
left by allowing the transfer matrix to act n − 1 times on the initial partition
S L
1 = (1, 1, 1, 0, 0, 0) T of the unstable manifold of F L (see Fig. 4.15a). We obtain
T
n−1
6 ·S L
1 = 1.021(2.315) n−1 φ
(6)
1 . Similarly, if we allow the transfer matrix to
act n − 1 times on the initial partition S R
1 = (0, 0, 1, 1, 0, 0) T of the unstable
manifold of F R (see Fig. 4.15a) for scattering from the right, we obtain T
n−1
6 ·S R
1 =
0.671(2.315) n−1 φ
(6)
1 (a similar analysis can be performed on T 5 ).
It only takes a few steps up the branching tree for scattering from the left, or from
the right, for the fractional distribution of symbols “A”, “B”, “C”, “+”, “−”, and “ ¯
A”
to be determined by the fractional distribution of symbols in the right eigenvector
φ
(6)
1 . Thus, for both incident directions, the branching trees, as regards the fraction
of each type of symbol present in φ
(6)
1 , become the same. The fractions are f A =
0.157, f B = 0.239, f C = 0.239, f + = 0.103, f − = 0.103, and f ¯
A = 0.157.
We now can use this information, and the information in Fig. 4.16b, to obtain a
rough estimate of the likelihood that an incident particle, caught up in the fractal
structure, gets transmitted or reflected. First, remember that
√
(◦) indicates a gap
containing trajectories that ultimately travel to −∞ and are of S L type (those that
travel to +∞ and are of S R type). From Fig. 4.16b, we see that intervals of type “A”
and “ ¯
A” in an unstable manifold, in the next iteration of the map, will contain one
S L gap and one S R gap. Intervals of type “B” and “C”, in the next iteration of the
map, will contain two S L gaps. Intervals of type “+” and “−”, in the next iteration
of the map, will contain two S R gaps. The fraction of S L gaps, in the next iteration
of the map, is 0.636 and the fraction of S R gaps is 0.364. Thus, for particles incident
from the left, 36.4% of the gaps get transmitted and 63.6% of the gaps get reflected.
For particles incident from the right, 63.6% of the gaps get transmitted and 36.4%
get reflected.
The percentage of gaps transmitted or reflected gives an indication of the
asymmetry of the overall scattering process. However, for a given line of initial
points, the stepping time data, like that shown in Fig. 4.14c, can be used to
distinguish which initial points are S R -type and which are S L -type. In Lin et al.