Exploring complexity of a Frenkel-Kontorova-Type atomic chain

Spatially ordered patterns that may emerge in various natural phenomena as the result of an evolution cause an increasing order within the system until forming a stationary state. One of special types of such patterns, highly complex cantor sets, arises as a result of successive bifurcations of trajectories of a dynamical system throughout its evolution in the control parameter space. It is well known that evolution towards such ordered structures requires a decrease in the entropy of the system, which can lead to an increase in the degree of complexity of the phase space. Although there are many entropy-based complexity measures in the literature, exploring the appropriate entropy that is capable of explaining such an evolution process and defining the degree of complexity of the structure of the physical system is one of the main research topics in complexity science. We evaluated possible total energy values of Frenkel-Kontorova-type independent chains with a free boundary condition at static equilibrium. We show that the ensemble of the total energies represents period doublings and chaotic band-mergings if one changes the control parameter, which is the value of the amplitude of the substrate potential. The periods of the total energies accumulate at a critical value, causing the emergence of self-similar and spatially organized fractal patterns. Using the energy distributions in the control parameter space, we also calculate entropy-based complexity measures: Shannon, Kullback-Leibler and q-Renormalized entropies. We show that the q-renormalized entropy behaves according to the well-known low-entropy criteria of self-organization, whereas the Shannon and Kullback-Leibler entropies cannot. This implies that the q-renormalized entropy is an appropriate entropy to explore the emergence and disappearance of spatially organized fractal energy patterns and to evaluate their complexities.