Short note on the emergence of fractional kinetics

In the present Short Note an idea is proposed to explain the emergence and the observation of processes in complex media that are driven by fractional non-Markovian master equations. Particle trajectories are assumed to be solely Markovian and described by the Continuous Time Random Walk model. But, as a consequence of the complexity of the medium, each trajectory is supposed to scale in time according to a particular random timescale. The link from this framework to microscopic dynamics is discussed and the distribution of timescales is computed. In particular, when a stationary distribution is considered, the timescale distribution is uniquely determined as a function related to the fundamental solution of the space–time fractional diffusion equation. In contrast, when the non-stationary case is considered, the timescale distribution is no longer unique. Two distributions are here computed: one related to the M-Wright/Mainardi function, which is Green’s function of the time-fractional diffusion equation, and another related to the Mittag-Leffler function, which is the solution of the fractional-relaxation equation.

[1]  J. Rasmussen,et al.  Superdiffusion and viscoelastic vortex flows in a two-dimensional complex plasma. , 2006, Physical review letters.

[2]  F. Mainardi,et al.  The fundamental solution of the space-time fractional diffusion equation , 2007, cond-mat/0702419.

[3]  Arak M. Mathai,et al.  Mittag-Leffler Functions and Their Applications , 2009, J. Appl. Math..

[4]  R. Gorenflo,et al.  Fractional calculus and continuous-time finance II: the waiting-time distribution , 2000, cond-mat/0006454.

[5]  Enrico Scalas,et al.  Uncoupled continuous-time random walks: Solution and limiting behavior of the master equation. , 2004, Physical review. E, Statistical, nonlinear, and soft matter physics.

[6]  W. Dorland,et al.  Finite Larmor radius effects on non-diffusive tracer transport in a zonal flow , 2008, 0807.4516.

[7]  Riccardo Mannella,et al.  Noise in Nonlinear Dynamical Systems , 2009 .

[8]  D. del-Castillo-Negrete,et al.  Non-diffusive, non-local transport in fluids and plasmas , 2010, 1009.2820.

[9]  Paolo Grigolini,et al.  Complexity and the Fractional Calculus , 2013 .

[10]  H. ThalerRichard,et al.  Theoretical foundations , 2007, Interpersonal Meaning in Multimodal English Textbooks.

[11]  Yuri Luchko.  Models of the neutral-fractional anomalous diffusion and their analysis , 2012 .

[12]  West,et al.  Dynamical approach to Lévy processes. , 1996, Physical review. E, Statistical physics, plasmas, fluids, and related interdisciplinary topics.

[13]  F. Mainardi.  Fractional Relaxation-Oscillation and Fractional Diffusion-Wave Phenomena , 1996 .

[14]  M. Shlesinger,et al.  Stochastic pathway to anomalous diffusion. , 1987, Physical review. A, General physics.

[15]  Francesco Mainardi,et al.  The Wright functions as solutions of the time-fractional diffusion equation , 2003, Appl. Math. Comput..

[16]  R. Gorenflo,et al.  Fractional calculus and continuous-time finance , 2000, cond-mat/0001120.

[17]  Martine Chevrollier,et al.  Lévy flights of photons in hot atomic vapours , 2009, 0904.2454.

[18]  Goldenfeld,et al.  Simple lessons from complexity , 1999, Science.

[19]  Christian Beck.  Dynamical Foundations of Nonextensive Statistical Mechanics , 2001 .

[20]  R Balesc,et al.  Aspects of anomalous transport in plasmas , 2005 .

[21]  E. Scalas,et al.  Monte Carlo simulation of uncoupled continuous-time random walks yielding a stochastic solution of the space-time fractional diffusion equation. , 2007, Physical review. E, Statistical, nonlinear, and soft matter physics.

[22]  Christian Beck,et al.  Dynamical Foundations of Nonextensive Statistical Mechanics , 2001, cond-mat/0105374.

[23]  G. Pagnini.  Erdélyi-Kober fractional diffusion , 2011, 1112.0890.

[24]  P. Grigolini,et al.  Fractional calculus as a macroscopic manifestation of randomness , 1999 .

[25]  Francesco Mainardi,et al.  The -Wright Function in Time-Fractional Diffusion Processes: A Tutorial Survey , 2010, 1004.2950.

[26]  Igor M. Sokolov,et al.  Physics of Fractal Operators , 2003 .

[27]  R. Gorenflo,et al.  Some recent advances in theory and simulation of fractional diffusion processes , 2007, 0801.0146.

[28]  Gianni Pagnini,et al.  Characterizations and simulations of a class of stochastic processes to model anomalous diffusion , 2008, 0801.4879.

[29]  E. Montroll,et al.  Random Walks on Lattices. II , 1965 .

[30]  G. Pagnini.  The evolution equation for the radius of a premixed flame ball in fractional diffusive media , 2011 .

[31]  Radu Balescu,et al.  V-Langevin equations, continuous time random walks and fractional diffusion , 2007, 0704.2517.

[32]  Miss A.O. Penney.  (b) , 1974, The New Yale Book of Quotations.

[33]  F. Mainardi.  Fractional Calculus and Waves in Linear Viscoelasticity: An Introduction to Mathematical Models , 2010 .

[34]  R. Gorenflo,et al.  Fractional Calculus: Integral and Differential Equations of Fractional Order , 2008, 0805.3823.

[35]  Gianni Pagnini,et al.  The M-Wright function as a generalization of the Gaussian density for fractional diffusion processes , 2013 .

[36]  E. Scalas,et al.  On the convergence of quadratic variation for compound fractional Poisson processes , 2012 .

[37]  E. Montroll.  Random walks on lattices , 1969 .

[38]  Francesco Mainardi,et al.  THE SPACE-TIME FRACTIONAL DIFFUSION EQUATION , 2001 .

[39]  Hilfer,et al.  Fractional master equations and fractal time random walks. , 1995, Physical review. E, Statistical physics, plasmas, fluids, and related interdisciplinary topics.