2.2 Full form factor for τ < 1
2.2.1 Number of orbit pairs
We can treat more complicated orbits along the same lines as described in the previous section. A general orbit can have V encounters, where each encounter α involveslαorbit stretches. The orbit pairs will be described by a certain ‘structure’, which describes the links and encounter stretches of the original orbit, and how they are reconnected to form the partner orbit. The restriction that the partner must remain a single orbit still holds.
If we define a vectorv, where the componentsvl specify how manyl-encounters the orbit has, it is easy to see that
V =X
l≥2
vl L=X
α
lα=X
l≥2
lvl (2.2.5)
whereLis the total number of orbit links, or encounter stretches. For each vectorv there may be many different structures, depending on the symmetry of the dynamics, but we will see later that each structure with the samevgives the same contribution to the form factor, so an important step will be the evaluation of the number of structuresN(v) associated with a given vectorv.
For a particular vector, finding the number of possible structures, which give a permissible periodic orbit pair, is a combinatorial problem. This problem can be simplified by relating structures to permutation matrices, and full details are given by M¨uller (2005). The situation is simpler for systems without time reversal symmetry,
because all encounter stretches traverse each encounter in the same direction, and the orbit links all join a right port to a left port. A permutation matrix represents a structure as long as it satisfies the properties mentioned in the previous section.
Specifically, to represent a structure with a corresponding vectorv, the permutation must consist of vl l-cycles for l ≥ 2, with each cycle corresponding to one of the V encounters. Of course, the partner orbit must be a single complete orbit as described before. Without time reversal symmetry, each orbit link starting on right port j leads to left port j+ 1. For the partner, if we start on left port i we cross the encounter region leaving from right port π(i) and then follow an orbit link to the left portπ(i) + 1. We can define a second permutation matrix that describes in which sequence the left ports of the partner are traversed.
1 2 . . . L
π(1) + 1 π(2) + 1 . . . π(L) + 1
0
(2.2.6)
The prime is to show that this permutation matrix connects left ports to left ports.
This permutation must be a singleL-cycle for the partner to be a complete orbit. The number of structures corresponding tovis then given by the number of permutation matrices that satisfy both of these properties.
From our example in Figure 2.4, the permutation matrix linking one left port to another is
1 2 3 3 1 2
0
(2.2.7)
which is indeed a 3-cycle. In fact the permutation matrix of equation (2.2.2) is the only one that satisfies both properties. Therefore, there is only one structure corresponding to a single 3-encounter for systems without time reversal symmetry.
For systems with time reversal symmetry, the situation is complicated by the fact that orbit links can connect any combination of left and right ports, and the encounter stretches can travel in either direction. M¨uller (2005) reverted to a picture of entrance and exit ports and crucially considered both the orbit and its time reverse
together. Together, the orbit and its time reverse contain all the encounter stretches and orbit links traversed in both directions. The entrance ports of the time reversed orbit are the time reverse of the exit ports of the original orbit. The reconnection of the encounters can be recorded in a double permutation matrix that describes the reconnection of both the partner orbit and its time reverse
1 2 . . . L 1 2 . . . L
π(1) π(2) . . . π(L) π(1) π(2) . . . π(L)
(2.2.8)
where the overbar denotes time reversal of the ports. The result of the permutation π(i) can be either a portj or its time reversalj. The reconnection is subject to the restriction that if an encounter stretch takes the portmtoπ(m) =n, then the time reversed stretch takes the portn back tom=π−1(n), where m andnare any port or its time reversal (ie they are elements of 1, . . . , L,1, . . . , L). Note also that a port obviously cannot be connected to its own time reversal.
To represent a structure with a given vectorv, the permutation must consist ofvl
pairs ofl-cycles. The cycles in each pair are mutually time reversed and correspond to one of theV encounters in both the partner and its time reversal. The orbit links of the original orbit take the portjto the portj+1, while the orbit links of the time reverse take the portj to the portj−1. When we combine the encounter stretches with the following link (traversed in the correct direction), the resulting permutation matrix must consist of two L-cycles corresponding to the partner orbit and its time reversal. The number of permutation matrices that satisfy these properties is the number of structures corresponding to v.
The number of structures for each symmetry class can then be calculated nu-merically by counting the permutation matrices that satisfy the requirements, and these numbers are tabulated in M¨uller (2005). M¨uller et al. (2004, 2005) took this further, and derived a recursion relation for the number of structures for a given vector v. To obtain this relation, they considered the effect of removing one link from an orbit. By establishing the number of possible ways of recovering orbits with
fewer links, the required recursion relation was obtained.
For systems without time reversal symmetry, the number of structures can be calculated explicitly by using the following formula (M¨uller, 2003), re-expressed in terms of our notation
N(v) = 1 L+ 1
X
v0≤v
(−1)L0−V0L0! (L−L0)!
Q
n≥2
nvnv0n! (vn−vn0)! (2.2.9)
The sum here is over all integer vectors v0 whose components satisfy 0≤ v0n ≤ vn for all n. L0 and V0 are the number of links and encounters of the vectorv0 given byL0= P
n≥2
nvn0 and V0 = P
n≥2
v0n.
In Chapter 3, we apply the semiclassical calculation described in this Section (M¨uller et al., 2004, 2005) to parametric correlations, and we compare our results with those obtained from RMT. For systems without time reversal symmetry, the RMT integral can be obtained in closed form. To be able to compare our expansion to all orders in τ, we will later need the number of orbits calculated with this formula. For systems with time reversal symmetry, and for the correlation functions for open systems that we consider in Chapter 4, closed form final RMT results are as yet unknown. Instead we compare terms of a small τ expansion calculated semiclassically with those from the RMT integrals. For this purpose, the number of orbit pairs tabulated in M¨uller (2005) suffices.