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Superstatistical approaches have played a crucial role in the investigations of mixtures of Gaussian processes. Such approaches look to describe non-Gaussian diffusion emergence in single-particle tracking experiments realized in soft and biological matter. Currently, relevant progress in superstatistics of Gaussian diffusion processes has been investigated by applying χ2-gamma and χ2-gamma inverse superstatistics to systems of particles in a heterogeneous environment whose diffusivities are randomly distributed; such situations imply Brownian yet non-Gaussian diffusion. In this paper, we present how the log-normal superstatistics of diffusivities modify the density distribution function for two types of mixture of Brownian processes. Firstly, we investigate the time evolution of the ensemble of Brownian particles with random diffusivity through the analytical and simulated points of view. Furthermore, we analyzed approximations of the overall probability distribution for log-normal superstatistics of Brownian motion. Secondly, we propose two models for a mixture of scaled Brownian motion and to analyze the log-normal superstatistics associated with them, which admits an anomalous diffusion process. The results found in this work contribute to advances of non-Gaussian diffusion processes and superstatistical theory.

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In this work, we propose to use the distance correlation ([Formula: see text]) and the Euclidian distance ([Formula: see text]) as powerful quantifiers to describe correlation and synchronization in spatially extended systems. As examples, we use the coupled Kuramoto oscillator model and coupled map lattices to study chimera states. Results for [Formula: see text] and [Formula: see text] are compared and discussed in terms of recurrence plots, local synchronization order parameter, local recurrence rate, and local [Formula: see text] rate. The existence of correlations between desynchronized states, not visible in usual recurrence plots, becomes evident when using [Formula: see text] and [Formula: see text]. Since correlation is a broader concept than synchronization, the proposed quantifiers amplify the characterization, description and possible application of spatially extended systems.

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