A measurement of disorder in binary sequences

Abstract

We propose a complex quantity, AL, to characterize the degree of disorder of L-length binary symbolic sequences. As examples, we respectively apply it to typical random and deterministic sequences. One kind of random sequences is generated from a periodic binary sequence and the other is generated from the logistic map. The deterministic sequences are the Fibonacci and Thue–Morse sequences. In these analyzed sequences, we find that the modulus of AL, denoted by |AL|, is a (statistically) equivalent quantity to the Boltzmann entropy, the metric entropy, the conditional block entropy and/or other quantities, so it is a useful quantitative measure of disorder. It can be as a fruitful index to discern which sequence is more disordered. Moreover, there is one and only one value of |AL| for the overall disorder characteristics. It needs extremely low computational costs. It can be easily experimentally realized. From all these mentioned, we believe that the proposed measure of disorder is a valuable complement to existing ones in symbolic sequences.