## Anomalous Features of Diffusion in Corrugated Potentials with Spatial Correlations: Faster than Normal, and Other Surprises

Normal diffusion in corrugated potentials with spatially uncorrelated Gaussian energy disorder famously explains the origin of non-Arrhenius $\exp [-\sigma^2/(k_{B}T^2)]$ temperature dependence in disordered systems.Here we show that unbiased diffusion remains asymptotically normal also in the presence of spatia
correlations decaying to zero. However, because of a temporal lack of self-averaging, transient subdiffusion emerges on the mesoscale, and it can readily reach macroscale even for moderately strong disorder fluctuations of $\sigma\sim 4 - 5k_{B}T$. Because of its nonergodic origin, such subdiffusion exhibits a large scatter in single-trajectory averages. However, at odds with intuition, it occurs essentially faster than one expects from the normal diffusion in the absence of correlations. We apply these results to diffusion of regulatory proteins on DNA molecules and predict that such diffusion should be anomalous, but much faster than earlier expected on a typical length of genes for a realistic energy disorder of several room $k_{B}T$, or merely 0.05–0.075 eV.

Igor Goychuk, Vasyl O. Kharchenko. Anomalous Features of Diffusion in Corrugated Potentials with Spatial Correlations: Faster than Normal, and Other Surprises, Physical Review Letters, 113, 100601 (2014)

## Subdiffusive rocking ratchets in viscoelastic media: Transport optimization and thermodynamic efficiency in overdamped regime

We study subdiffusive overdamped Brownian ratchets periodically rocked by an external zero-mean force in viscoelastic media within the framework of a non-Markovian generalized Langevin equation approach and associated multidimensional Markovian embedding dynamics. Viscoelastic deformations of the medium caused by the transport particle are modeled by a set of auxiliary Brownian quasiparticles elastically coupled to the transport particle and characterized by a hierarchy of relaxation times which obey a fractal scaling. The most slowly relaxing deformations which cannot immediately followto the moving particle imprint long-range memory about its previous positions and cause subdiffusion and anomalous transport on a sufficiently long time scale. This anomalous behavior is combined with normal diffusion and transport on an initial time scale of overdamped motion. Anomalously slowdirected transport in a periodic ratchet potentialwith broken space inversion symmetry emerges due to a violation of the thermal detailed balance by a zero-mean periodic driving and is optimized with frequency of driving, its amplitude, and temperature. Such optimized anomalous transport can be low dispersive and characterized by a large generalized Peclet number. Moreover, we show that overdamped subdiffusive ratchets can sustain a substantial load and do useful work. The corresponding thermodynamic efficiency decays algebraically in time since the useful work done against a load scales sublinearly with time following to the transport particle position, but the energy pumped by an external force scales with time linearly. Nevertheless, it can be transiently appreciably high and compare well with the thermodynamical efficiency of the normal diffusion overdamped ratchets on sufficiently long temporal and spatial scales.

Vasyl O. Kharchenko, Igor Goychuk Subdiffusive rocking ratchets in viscoelastic media: Transport optimization and thermodynamic efficiency in overdamped regime PHYSICAL REVIEW E 87, 052119 (2013)

## Universality and self-similar behaviour of non-equilibrium systems with non-Fickian diffusion

Analytical approaches describing non-Fickian diffusion in complex systems are presented. The corresponding methods are applied to the study of statistical properties of pyramidal islands formation with interacting adsorbate at epitaxial growth. Using the generalized kinetic approach we consider universality, scaling dynamics and fractal properties of pyramidal islands growth. In the framework of generalized kinetics, we propose a theoretical model to examine the numerically obtained data for averaged islands size, the number of islands and the corresponding universal distribution over the island size.

D.O.Kharchenko, V.O.Kharchenko, S.V.Kokhan. Universality and self-similar behaviour of non-equilibrium systems with non-Fickian diffusion, Condensed Matter Physics, 17(3), art. no. 33004 (2014)

## Statistical properties of nanosized clusters on a surface in overdamped stochastic reaction-Cattaneo systems

In this work we study an overdamped stochastic reaction-Cattaneo model describing nanosized pattern formation on a surface at monolayer deposition. We study and compare an influence of both primary and secondary mechanisms onto pattern formation processes. The primary mechanisms relate to the rates of chemical reactions and interaction strength of adsorbate; the secondary mechanisms are related to finite atomic disturbance propagation speed and stochastic contribution satisfying fluctuation-dissipation relation. Considering statistical properties of surface structures we discuss transitions between homogeneous phases related to low and high density states. We illustrate that these transitions are accompanied by a formation of adsorbate or vacancy islands. It was found that spherical adsorbate and vacancy islands are characterized by different distribution functions over their sizes for different symmetry of substrate lattice. We have shown that depending on system control parameters island size distributions can change their modality. The size of localized nano-clusters can be controlled by both primary and secondary mechanisms of pattern formation.

V.O.Kharchenko, D.O.Kharchenko, A.V.Dvornichenko. Statistical properties of nanosized clusters on a surface in overdamped stochastic reaction-Cattaneo systems, Surface Science, 630, pp.158-167 (2014)

### Our Team

• ##### Dmytro Kharchenko

Head of the dept., Dr.Sci.,  Professor

Dr.Sci.,  Professor

• ##### Serhiy Kokhan

PhD, Research fellow

• ##### Vasyl Kharchenko

Phd, Research fellow

• ##### Iryna Lysenko

PhD, Research fellow

• ##### Olga Shchokotova

PhD, Research fellow

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