Superstatistical approach of the anomalous exponent for scaled Brownian motion

Abstract

Anomalous diffusion phenomenon is an intriguing process that tracer diffusion presents in numerous complex systems. Current experimental and theoretical investigations have reported the emergence of random diffusivity scenarios accompanied by the heterogeneity of the α-anomalous diffusion exponents. In this framework, we investigate a heterogeneous ensemble of tracers governed by scaled Brownian motion (sBm). The heterogeneous features are considered on anomalous diffusion exponent and diffusivity. We introduce two superstatistics of anomalous exponent, the truncated-Gaussian and truncated χ2-Gamma distributions. In this way, we show how the statistics of anomalous exponent affect the spreading of a mixture of particles governed by sBm. We also analyse the role of different temporal scales of sBm in superstatistics. Furthermore, we investigate the effects of coupling between diffusivity and anomalous exponent on superstatistics of sBm. The investigation provides a thorough analysis of simulation and analytical approach. The results imply rich classes of anomalous diffusion processes accompanied by non-Gaussian diffusion.