Loading...

Anomalous Stochastic Processes and Stochastic Löwner Evolution

Ghasemi Nezhad Haghighi, Mohsen | 2014

530 Viewed
  1. Type of Document: Ph.D. Dissertation
  2. Language: Farsi
  3. Document No: 47848 (04)
  4. University: Sharif University of Technology
  5. Department: Computer Engineering
  6. Advisor(s): Rouhani, Shahin
  7. Abstract:
  8. 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 walk, harmonic explorer, and percolation front) are derived. We propose natural parametrized subordinated Schramm-Löwner evolution (NS-SLE) as a mathematical model for anomalous transport in the self similar curves. We numerically compute the MFPT and MSD for NS-SLE. The agreement between NS-SLE and the corresponding lattice models is quite good. Continuous scale invariance has played a key role in the context of statistical fractal curves and Schramm-Löwner evolution. Here, we introduce a large class of fractal curves with discrete scale invariance (DSI). We consider the Weierstrass-Mandelbrot (WM) function as the drift of the Löwner equation. We show that the fractal dimension of the curves can be derived from the diffusion coefficient of the variance of the WM function. Our study opens a way to classify all the fractal curves with DSI. We also investigate the contour lines of two-dimensional WM function as a physical candidate for the stochastic curves with DSI. Here we also propose a general model which allows us to extract generic scale invariant curves in two dimensions. In order to produce such scaling curves we used Löwner equation with drift which is now just scale invariant, i.e. fractional Brownian motion. In the fractional SLE model, scaling symmetry only occurs if we use from fractional time drivative instead of ordinary differential operator. We also numerically investigate the subdiffusive properties of the tagged tracer particle in a sochastic systems i.e. labeled molecule in a membrane or polymer chain which they are described by the generalized elastic model (GEM). We investigate the MSD and the first passage time density of the process which follow a simple power-law. We also analyzed the ergodic properties of the process and demonstrated the ergodicity of the process by measuring the amplitude scatter of individual trajectories and the ergodicity breaking parameter. Finally, we study the driven GEM to extract the characteristics of the average drift for a tagged probe
  9. Keywords:
  10. Schramm-Loewner Evolution (SLE) ; Anomalous Stochastic Processes ; Subdiffusion and Superdiffusion ; Discrete Scale Invariance

 Digital Object List

 Bookmark

No TOC