Fluctuations, Complexity and Statistical Measures for Particles in a Tight-Binding Lattice

Understanding wave-packet deformation and spreading is central to the description of transport and localization phenomena in both classical and quantum systems. In this work, we introduce information-theoretic and complexity-based indicators. We analyze the evolution of the probability distribution using Shannon and linear entropies, together with the inverse participation ratio, to quantify localization–delocalization processes and their associated fluctuations. These measures naturally motivate the use of statistical complexity definitions, including the López–Mancini–Calbet (CLMC) and Shiner–Davison–Landsberg (CSDL) complexities. In addition, we compute the probability current for both single-particle and two-particle configurations, providing a dynamical description of transport in terms of particle motion. The influence of an external field on entropy production, complexity, and probability currents is also examined, highlighting the interplay between information spreading, localization, and transport in lattice systems.