## Final Activity Report Summary - THEORMETHODS (The Development of New Theoretical Methods for Analysis of Driven Systems)

The project aimed to develop the analytical methods for the study of physical systems driven by regular and stochastic fields. The systems driven by regular fields included:

1. a chain of charged particles that interacted with each other with a ratchet potential and were also subject to an alternating electric field; and

2. assemblies of magnetic nanoparticles driven by a rotating magnetic field.

For the former system, an analytical approach for studying the chain dynamics was developed, a drift criterion was derived and two types of the overdamped chain transport, namely integer and fractional, were detected. For the latter system, the stability criterion for the steady-state precession of the nanoparticle magnetic moments was found and the switching of the magnetic moments under the action of the rotating field was discovered.

The stochastic systems considered in the project included:

1. finite Ising chains;

2. assemblies of magnetic nanoparticles at nonzero temperatures driven by a rotating magnetic field; and

3. model systems driven by the cross-correlated Gaussian white noises.

Via the use of extensive combinatorics, the domain statistics in a finite Ising chain was thoroughly investigated. Specifically, the exact probability distribution functions for the number of domain walls, number of up domains and number of spins in an up domain were derived and the corresponding averages and variances were calculated. In addition, a criterion that a finite Ising chain exhibited the ferromagnetic-like behaviour was derived. An analytical method for the calculation of the mean first-passage times for the nanoparticle magnetic moment driven by a rapidly rotating magnetic field was also developed. It was based on the equations for the mean first-passage times derived from the two-dimensional backward Fokker-Planck equation in the rotating frame. These equations were solved in the high-frequency limit and the precise numerical simulations which verified the analytical findings were performed.

Furthermore, the obtained results were applied for the description of the features of magnetic relaxation in assemblies of magnetic nanoparticles. It was shown, in particular, that the rapidly rotating magnetic field decreased the relaxation time and magnetised the nanoparticle system. The induced magnetisation exhibited a resonant character as a function of the field frequency, possessing a well-pronounced maximum.

Finally, the statistical properties of overdamped particles driven by two cross-correlated multiplicative Gaussian white noises in a time-dependent environment were studied. Within the Langevin and Fokker-Planck approaches, the exact probability distribution function for the particle positions was derived and its moments and corresponding long-time asymptotics were calculated. The generally anomalous diffusive regimes of the particles were classified and their dependence on the friction coefficient and the noises characteristics was analysed in detail.

1. a chain of charged particles that interacted with each other with a ratchet potential and were also subject to an alternating electric field; and

2. assemblies of magnetic nanoparticles driven by a rotating magnetic field.

For the former system, an analytical approach for studying the chain dynamics was developed, a drift criterion was derived and two types of the overdamped chain transport, namely integer and fractional, were detected. For the latter system, the stability criterion for the steady-state precession of the nanoparticle magnetic moments was found and the switching of the magnetic moments under the action of the rotating field was discovered.

The stochastic systems considered in the project included:

1. finite Ising chains;

2. assemblies of magnetic nanoparticles at nonzero temperatures driven by a rotating magnetic field; and

3. model systems driven by the cross-correlated Gaussian white noises.

Via the use of extensive combinatorics, the domain statistics in a finite Ising chain was thoroughly investigated. Specifically, the exact probability distribution functions for the number of domain walls, number of up domains and number of spins in an up domain were derived and the corresponding averages and variances were calculated. In addition, a criterion that a finite Ising chain exhibited the ferromagnetic-like behaviour was derived. An analytical method for the calculation of the mean first-passage times for the nanoparticle magnetic moment driven by a rapidly rotating magnetic field was also developed. It was based on the equations for the mean first-passage times derived from the two-dimensional backward Fokker-Planck equation in the rotating frame. These equations were solved in the high-frequency limit and the precise numerical simulations which verified the analytical findings were performed.

Furthermore, the obtained results were applied for the description of the features of magnetic relaxation in assemblies of magnetic nanoparticles. It was shown, in particular, that the rapidly rotating magnetic field decreased the relaxation time and magnetised the nanoparticle system. The induced magnetisation exhibited a resonant character as a function of the field frequency, possessing a well-pronounced maximum.

Finally, the statistical properties of overdamped particles driven by two cross-correlated multiplicative Gaussian white noises in a time-dependent environment were studied. Within the Langevin and Fokker-Planck approaches, the exact probability distribution function for the particle positions was derived and its moments and corresponding long-time asymptotics were calculated. The generally anomalous diffusive regimes of the particles were classified and their dependence on the friction coefficient and the noises characteristics was analysed in detail.