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Anomalous diffusion describes a deviation from the classical Brownian motion, where the mean squared displacement of particles is not linear with time, indicating either subdiffusive or superdiffusive behavior. This phenomenon is crucial in understanding complex systems such as biological environments, disordered media, and financial markets, where traditional diffusion models fail to capture the intricacies of particle movement.
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Fractional Brownian Motion (fBm) is a generalization of classical Brownian motion that incorporates memory and self-similarity, characterized by the Hurst parameter, H, which dictates the roughness of the path. Unlike standard Brownian motion, fBm is neither a semimartingale nor has independent increments, making it useful in modeling phenomena with long-range dependence in fields like finance and telecommunications.
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A continuous-time random walk (CTRW) is a stochastic process that generalizes the concept of a random walk by allowing the waiting times between steps to be random and independently distributed. This approach models systems where events occur sporadically and the time between events follows a specified probability distribution, making it applicable in fields like physics, finance, and ecology.
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Mean Squared Displacement (MSD) is a measure of the average squared distance traveled by particles over time, typically used to analyze diffusion and random motion in physical systems. It provides insight into the mobility and dynamic behavior of particles, with applications in fields such as physics, chemistry, and biology.
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Scaling laws describe how different properties of a system change with size, revealing consistent patterns across diverse domains such as physics, biology, and technology. They provide insights into the efficiency, performance, and limitations of systems as they grow, helping to predict behavior and optimize design.
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Stochastic processes are mathematical objects used to model systems that evolve over time with inherent randomness. They are essential in various fields such as finance, physics, and biology for predicting and understanding complex systems where outcomes are uncertain.
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Random Walk Theory suggests that stock market prices evolve according to a random walk and thus cannot be predicted based on past movements. This implies that the market is efficient, and any attempt to outperform it through analysis or timing is futile, as all known information is already reflected in stock prices.
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Heterogeneous media refers to materials or systems composed of distinct components with differing properties, such as varying phases or compositions, which significantly influence their overall behavior and characteristics. Understanding and modeling Heterogeneous media are crucial in fields like material science, geophysics, and engineering, where predicting the interaction of different materials is essential for designing and optimizing applications.
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Subdiffusion is a type of anomalous diffusion where the mean squared displacement of particles grows slower than linearly with time, often due to obstacles or binding interactions in complex environments. It is characterized by a power-law behavior with an exponent less than one, indicating hindered or constrained particle movement compared to normal diffusion.
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