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discrete-scale-invariances
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Anomalous Stochastic Processes and Stochastic Löwner Evolution
, Ph.D. Dissertation Sharif University of Technology ; Rouhani, Shahin (Supervisor)
Abstract
In normal diffusion, the mean-square displacement (MSD) of a Brownian particle is proportional to time. However, diffusion in disordered systems, i.e. transport on fractal geometries, does not follow the classical laws of BM, and this leads to many anomalous physical properties. In the anomalous regime, the most famous definition of anomaly is the deviation of MSD, from the ‘normal’ linear dependence on time ⟨r2(t)⟩ ta. Specifically, we study the anomalous diffusion on the self-similar curves in two dimensions. The scaling properties of the mean-square displacement and mean first passage time (MFPT) of two sided and subordinated diffusion on the different fractal curves (loop-erased random...
Discrete scale invariance and stochastic Loewner evolution
, Article Physical Review E - Statistical, Nonlinear, and Soft Matter Physics ; 2010 , Volume 82, Issue 6 ; 15393755 (ISSN) ; Rajabpour, M. A ; Sharif University of Technology
2010
Abstract
In complex systems with fractal properties the scale invariance has an important rule to classify different statistical properties. In two dimensions the Loewner equation can classify all the fractal curves. Using the Weierstrass-Mandelbrot (WM) function as the drift of the Loewner equation we introduce a large class of fractal curves with discrete scale invariance (DSI). We show that the fractal dimension of the curves can be extracted from the diffusion coefficient of the trend of the variance of the WM function. We argue that, up to the fractal dimension calculations, all the WM functions follow the behavior of the corresponding Brownian motion. Our study opens a way to classify all the...