## Final Activity Report Summary - THEORMETHODS (The development of theoretical methods for analysis of driven systems)

The main objective of the project was to develop analytical methods for studying the model systems of overdamped particles and classical spins with anisotropy placed in random environments. Depending on the environment, the former model captures all the main features of the dynamics of charge carriers and localised structures in randomly layered media, the directed transport of particles in randomly perturbed channels, the dynamics of prices in financial markets, etc. The latter model describes the magnetic properties of many functional materials including nanostructured ones which provide a variety of present and potential applications.

Within the project, a number of efficient methods were developed for these models. Specifically, the path-integral approach was successfully applied to elaborate a method for finding the statistical properties of particles driving by a constant or alternating force in media with quenched dichotomous disorder. In the case of a constant force, the exact probability distributions for the particle positions and arrival times, which describe the particle behaviour in full detail, were derived for the first time. These distributions were used to find their asymptotic behaviour and to establish the scaling properties of the central moments. Moreover, the method was generalised to account for the influence of the ratchet-type dichotomous disorder in the presence of an alternating force. Within this framework, the main characteristics of the directed transport, including the maximal displacement of particles, the average displacement of particles during one period of an alternating force, and the average velocity of particles in the large-period limit, were calculated analytically.

An original method for studying the magnetic relaxation in nanoparticle systems driven by a rotating magnetic field was developed and applied for the determination of the relaxation law and induced magnetisation in some particular cases. The method is based on the equation for the mean first-passage times, i.e. mean times that the magnetic moments spend in the up and down states, derived from the two-dimensional backward Fokker-Planck equation in the rotating frame. The mean first-passage times describe the rates of escape of the magnetic moments from the metastable domains which in turn determine the magnetic relaxation dynamics. Within this approach, it was shown that the correlation effects arising from the magnetic dipolar interaction essentially influence the relaxation law and induced magnetisation which exhibits as a function of the field frequency a resonant character, possessing a well-pronounced maximum. In the case of non-interacting nanoparticles, a rapidly rotating magnetic field can cause a drastic decrease of the relaxation time and a strong magnetisation of the nanoparticle system.

Finally, to describe the statistical properties of noisy systems with strong fluctuations, a class of Langevin equations with an arbitrary (non-Gaussian) multiplicative white noise was introduced and the corresponding generalised Fokker-Planck equation was derived. The main advantage of this generalised Fokker-Planck equation is that it accounts for the white noise action in a unified way, namely through the characteristic function of the white-noise generating process. This equation was solved for a particle in linear and quadratic potentials driven by an arbitrary additive white noise.

An important feature of these analytical solutions is that they give an opportunity to examine the effects of different white noise sources on the same system. If the increments of the white-noise generating process are distributed with heavy-tailed distributions then the generalised Fokker-Planck equation reduces to the fractional one, which describes a special class of random processes, so-called Levy flights. It was shown that the confined Levy flights, i.e. Levy flights in an infinitely deep potential well, are distributed according to the beta distribution, whose probability density becomes singular at the boundaries of the well.

Within the project, a number of efficient methods were developed for these models. Specifically, the path-integral approach was successfully applied to elaborate a method for finding the statistical properties of particles driving by a constant or alternating force in media with quenched dichotomous disorder. In the case of a constant force, the exact probability distributions for the particle positions and arrival times, which describe the particle behaviour in full detail, were derived for the first time. These distributions were used to find their asymptotic behaviour and to establish the scaling properties of the central moments. Moreover, the method was generalised to account for the influence of the ratchet-type dichotomous disorder in the presence of an alternating force. Within this framework, the main characteristics of the directed transport, including the maximal displacement of particles, the average displacement of particles during one period of an alternating force, and the average velocity of particles in the large-period limit, were calculated analytically.

An original method for studying the magnetic relaxation in nanoparticle systems driven by a rotating magnetic field was developed and applied for the determination of the relaxation law and induced magnetisation in some particular cases. The method is based on the equation for the mean first-passage times, i.e. mean times that the magnetic moments spend in the up and down states, derived from the two-dimensional backward Fokker-Planck equation in the rotating frame. The mean first-passage times describe the rates of escape of the magnetic moments from the metastable domains which in turn determine the magnetic relaxation dynamics. Within this approach, it was shown that the correlation effects arising from the magnetic dipolar interaction essentially influence the relaxation law and induced magnetisation which exhibits as a function of the field frequency a resonant character, possessing a well-pronounced maximum. In the case of non-interacting nanoparticles, a rapidly rotating magnetic field can cause a drastic decrease of the relaxation time and a strong magnetisation of the nanoparticle system.

Finally, to describe the statistical properties of noisy systems with strong fluctuations, a class of Langevin equations with an arbitrary (non-Gaussian) multiplicative white noise was introduced and the corresponding generalised Fokker-Planck equation was derived. The main advantage of this generalised Fokker-Planck equation is that it accounts for the white noise action in a unified way, namely through the characteristic function of the white-noise generating process. This equation was solved for a particle in linear and quadratic potentials driven by an arbitrary additive white noise.

An important feature of these analytical solutions is that they give an opportunity to examine the effects of different white noise sources on the same system. If the increments of the white-noise generating process are distributed with heavy-tailed distributions then the generalised Fokker-Planck equation reduces to the fractional one, which describes a special class of random processes, so-called Levy flights. It was shown that the confined Levy flights, i.e. Levy flights in an infinitely deep potential well, are distributed according to the beta distribution, whose probability density becomes singular at the boundaries of the well.