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The nonlinear dependence of the mean-squared displacement (MSD) on time is a common characteristic of particle transport in complex environments. Frequently, this anomalous behavior only occurs transiently before the particle reaches a terminal Fickian diffusion. This study shows that a system of hierarchically coupled Ornstein–Uhlenbeck equations is able to describe both transient subdiffusion and transient superdiffusion dynamics, as well as their sequential combinations. To validate the model, five distinct experimental, molecular dynamics simulation, and theoretical studies are successfully described by the model. The comparison includes the transport of particles in random optical fields, supercooled liquids, bedrock, soft colloidal suspensions, and phonons in solids. The model’s broad applicability makes it a convenient tool for interpreting the MSD profiles of particles exhibiting transient anomalous diffusion. 

Brownian Motion, with some persistence in the direction of motion, typically known as active Brownian Motion, has been observed in many significant chemical and biological transport processes. Here, we present a model of drifted Brownian Motion that considers a nonlinear stochastic drift with constant or fluctuating diffusivity. The interplay between nonlinearity and structural heterogeneity of the environment can explain three essential features of active transport. These features, which are commonly observed in experiments and molecular dynamics simulations, include transient superdiffusion, ephemeral non-Gaussian displacement distribution, and non-monotonic evolution of non-Gaussian parameter. Our results compare qualitatively well with experiments of self-propelled particles in simple hydrogen peroxide solutions and molecular dynamics simulations of self-propelled particles in more complex settings such as viscoelastic polymeric media. 

Early embryonic development involves forming all specialized cells from a fluid-like mass of identical stem cells. The differentiation process consists of a series of symmetry-breaking events, starting from a high-symmetry state (stem cells) to a low-symmetry state (specialized cells). This scenario closely resembles phase transitions in statistical mechanics. To theoretically study this hypothesis, we model embryonic stem cell (ESC) populations through a coupled Boolean network (BN) model. The interaction is applied using a multilayer Ising model that considers paracrine and autocrine signaling, along with external interventions. It is demonstrated that cell-to-cell variability can be interpreted as a mixture of steady-state probability distributions. Simulations have revealed that such models can undergo a series of first- and second-order phase transitions as a function of the system parameters that describe gene expression noise and interaction strengths. These phase transitions result in spontaneous symmetry-breaking events that generate new types of cells characterized by various steady-state distributions. Coupled BNs have also been shown to self-organize in states that allow spontaneous cell differentiation.