LEFRAC · Levy Random Motion and Fractional Calculus in the Kinetic Theory of Systems Far from Equilibrium

Levy Random Motion and Fractional Calculus in the Kinetic Theory of Systems Far from Equilibrium

The project focused on the development of new probabilistic and physical kinetics methods based on fractional calculus, for the studies of anomalous transport and relaxation phenomena in complex non-equilibrium systems. The following novel analytical methods and tools have been developed to solve the problems of modern non-equilibrium physics: - the two fractional Fokker-Planck equations with fractional derivatives of the distributed orders; - the generalisation of the continuous time random walk with correlated waiting times; - the theory of bulk mediated surface diffusion on a cylinder; - the theory of the Levy ratchet and the conditions for the existence of steady states under the action of Levy-stable noises; - the fractional Langevin equation for a broad class of elastic systems (membranes, polymers, fluctuation interfaces etc.); - the theory of escape from a potential well driven by fractional Gaussian noise. As part of the 'modelling anomalous dynamics' numerical toolbox the following have been elaborated for mathematical modelling of anomalous diffusion: - various extensive simulations based on the Monte Carlo methods have been provided to simulate the Levy ratchet, nonlinear Levy-noise driven systems, to show the validity of the fractional Langevin equation description, and to determine the laws of survival probability of fractional Brownian motion confined to a two-dimensional open wedge domain; - a new method based on matrix approach to discrete fractional calculus was developed, and a set of Matlab routines have been elaborated. As part of the 'fractional kinetics in action', the following were analytical and numerical tools have been applied to describe the following diverse phenomena demonstrating common features of anomalous behaviour: - correlation properties of polymers, fluid membranes and growing interfaces; fluctuation properties of the donor-acceptor distance in a protein; unusual response to localised and periodic perturbations applied to growing surfaces; - peculiarities of anomalous diffusion of proteins along the DNA. - description of rare events in single molecule spectroscopy. As part of 'advanced data analysis' the Levy random motions have been identified by analysing high frequency data samples: - During the L-H like transition in the edge plasma of thermonuclear device stellarator 'Uragan 3M'. - In meteorological wind field data. - The preliminary analysis has been performed of the available database on the underground water pollution in the Chernobyl exclusion zone. The work was done in accordance with the project planning. Minor deviations from the planned research within the particular subtasks arose to ensure the priority in achieving the challenging goals in a rapidly evolving scientific area of anomalous transport and fractional kinetics. All the financial costs have been spent in accordance with the planned financial table. Actual financial table is attached. The project LEFRAC has made an essential contribution for establishing European excellence and competitiveness in the theory of anomalous transport phenomena and fractional kinetic equations. A wide intra-European collaboration within the framework of the project has not only led to impressive results in the field of non-equilibrium statistical physics, but also offered a good opportunity to a number of early-staged researchers to attain synthetic scientific skill. The results of the project have a big potential to provide an impact on wind energy planning and designing, and monitoring of the Chernobyl zone. The project has laid a basis for long-term collaboration with physicists-experimentalists, biologists and technicians, thus strengthening cooperation and advancing knowledge in the fields which give socio-economic impact not only within the EC countries but also on the development of Ukraine.

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