Witnessing the quasiperiodic-ordering transition of one-dimensional k-component Fibonacci sequences

How much disorder in sequences is a fundamental question in many fields of science. A quantity, ZL, is proposed to assess the degree of disorder (DOD) of one-dimensional k-component Fibonacci sequences, where k is an arbitrary integer and L is the sequence length. Hu et al. have proved that such sequences are quasiperiodic when k≤5, while still ordering when k>5 (Hu et al., 1993). It is numerically found that for each k, there is an inflection point in the function of ZL versus L at a certain Lk∗. On one side, ZL∝Lαk when L0 when k≥6. This result is consistent with what found by Hu et al.. Therefore, αk can be as a witness of the quasiperiodic-ordering transition in the studied sequences. On the other hand, ZL∝L2.0139 when L>Lk∗ for all k. Further, the larger the ZL, the more disordered the sequence is. For LLk∗, ZL is almost independent of k, i.e., the DOD is almost same for enough longer sequences. All these provide further understands of disorder properties in the interesting sequences.