Random walk through Anomalous processes: some applications of Fractional calculus
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1 Random walk through Anomalous processes: some applications of Fractional calculus Nickolay Korabel Fractional Calculus UCMerced 12 June 2013
2 Fractional Calculus Mathematics Engineering Physics
3 Levy flights for foraging and knowledge Humphries et al PNAS (2012) R. Metzler, I.M. Sokolov, R. Klages G. Zaslavsky A. Gopinathan KC Huang A. Chechkin E. Barkai, J. Klafter
4 My great teachers
5 Sub-diffusive map CTRW-like behavior NK, R. Klages, Chechkin, Sokolov
6 Time fractional Fokker-Plank equation for sub-diffusion Montroll-Weiss equation jump length pdf waiting time pdf See works of F. Mainardi and R. Gorenflo Caputo fractional derivative
7 Infinite invariant density NK, E. Barkai
8 Paradoxes of sub-diffusion: anomalous infiltration Riemann Liouville operator: NK, E. Barkai
9 Fractional equations with long range interactions Non-linear Schroedinger equations Continuous non-linear Schroedinger equations: Discrete lattice of coupled osccillators: Conserved quantities: Stationary solutions in the form: For large N DNLS is not integrable and chaotic solutions are possible. Osccillators with all-to-all long range interactions: Equations of motion: NK, G. Zaslavsky, V. Tarasov, N. Laskin
10 Fractional equations with long range interactions: Non-linear Schroedinger equation Equations of motion: Transition to continuous equation: Polylogarithmic function NK, G. Zaslavsky, V. Tarasov, N. Laskin
11 Fractional equations with long range interactions: Non-linear Schroedinger equation To visualize numerical results we use: Surface - Power spectum: Phase portrait of the central osccillator: Amplitude of central osccillator: Initial conditions: NK, G. Zaslavsky, V. Tarasov, N. Laskin
12 Fractional equations with long range interactions Non-linear Schroedinger equation To visualize numerical results we use: Surface - M NK, G. Zaslavsky, V. Tarasov, N. Laskin
13 Fractional equations with long range interactions Non-linear Schroedinger equation NK, G. Zaslavsky, V. Tarasov, N. Laskin
14 Fractional equations with long range interactions Sine-Gordon Equation NK, G. Zaslavsky, V. Tarasov, N. Laskin
15 Fractional equations with long range interactions Sine-Gordon Equation NK, G. Zaslavsky, V. Tarasov, N. Laskin
16 Modeling optimality in cytoskeleton transport Ajay Gopinathan Kerwyn Casey Huang
17 Cytoskeleton network is required for structure, organization, and transport Lammelipodium in a neuron
18 A complex cellular transportation system Microtubules are like freeways and actin filaments are like local surface streets. Organelles can move on both types of filaments. Different types of motors work together. How does network architecture influence transport? Are there optimal transport regimes in terms of network residence time, network density, filament length and orientation?
19 Continuum model of a cytoskeletal network Binding rate (probability per unit time) Network residence time
20 There is an optimal network residence time Diffusion constants in the bulk and on the network: Binding rate: Reactivity is defined as the inverse of the mean first passage time G = reactivity/(bulk reactivity) G>1 gain reactivity G<1 lose reactivity
21 There is an optimal network residence time Diffusion constants in the bulk and on the network: Binding rate: Reactivity is defined as the inverse of the mean first passage time G = reactivity/(bulk reactivity) G>1 gain reactivity G<1 lose reactivity
22 How network topology influences transport?
23 Filament orientation affects transport
24
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