@article{GuggenbergerChechkinMetzler2022,
  author    = {Guggenberger, Tobias and Chechkin, Aleksei and Metzler, Ralf},
  title     = {Absence of stationary states and non-Boltzmann distributions of fractional Brownian motion in shallow external potentials},
  series = {New journal of physics : the open-access journal for physics},
  volume    = {24},
  journal   = {New journal of physics : the open-access journal for physics},
  number    = {7},
  publisher = {Dt. Physikalische Ges.},
  address   = {[Bad Honnef]},
  issn      = {1367-2630},
  doi       = {10.1088/1367-2630/ac7b3c},
  pages     = {18},
  year      = {2022},
  abstract  = {We study the diffusive motion of a particle in a subharmonic potential of the form U(x) = |x|( c ) (0 < c < 2) driven by long-range correlated, stationary fractional Gaussian noise xi ( alpha )(t) with 0 < alpha <= 2. In the absence of the potential the particle exhibits free fractional Brownian motion with anomalous diffusion exponent alpha. While for an harmonic external potential the dynamics converges to a Gaussian stationary state, from extensive numerical analysis we here demonstrate that stationary states for shallower than harmonic potentials exist only as long as the relation c > 2(1 - 1/alpha) holds. We analyse the motion in terms of the mean squared displacement and (when it exists) the stationary probability density function. Moreover we discuss analogies of non-stationarity of Levy flights in shallow external potentials.},
  language  = {en}
}
@article{HerrmannMetzler2017,
  author    = {Herrmann, Carl J. J. and Metzler, Ralf},
  title     = {A self-avoiding walk with neural delays as a model of fixational eye movements},
  series = {Scientific reports},
  volume    = {7},
  journal   = {Scientific reports},
  publisher = {Springer Nature},
  address   = {London},
  issn      = {2045-2322},
  doi       = {10.1038/s41598-017-13489-8},
  pages     = {1 -- 17},
  year      = {2017},
  abstract  = {Fixational eye movements show scaling behaviour of the positional mean-squared displacement with a characteristic transition from persistence to antipersistence for increasing time-lag. These statistical patterns were found to be mainly shaped by microsaccades (fast, small-amplitude movements). However, our re-analysis of fixational eye-movement data provides evidence that the slow component (physiological drift) of the eyes exhibits scaling behaviour of the mean-squared displacement that varies across human participants. These results suggest that drift is a correlated movement that interacts with microsaccades. Moreover, on the long time scale, the mean-squared displacement of the drift shows oscillations, which is also present in the displacement auto-correlation function. This finding lends support to the presence of time-delayed feedback in the control of drift movements. Based on an earlier non-linear delayed feedback model of fixational eye movements, we propose and discuss different versions of a new model that combines a self-avoiding walk with time delay. As a result, we identify a model that reproduces oscillatory correlation functions, the transition from persistence to antipersistence, and microsaccades.},
  language  = {en}
}
@article{GrebenkovMetzlerOshanin2021,
  author    = {Grebenkov, Denis S. and Metzler, Ralf and Oshanin, Gleb},
  title     = {A molecular relay race: sequential first-passage events to the terminal reaction centre in a cascade of diffusion controlled processes},
  series = {New Journal of Physics (NJP)},
  volume    = {23},
  journal   = {New Journal of Physics (NJP)},
  publisher = {IOP - Institute of Physics Publishing},
  address   = {Bristol},
  issn      = {1367-2630},
  doi       = {10.1088/1367-2630/ac1e42},
  pages     = {18},
  year      = {2021},
  abstract  = {We consider a sequential cascade of molecular first-reaction events towards a terminal reaction centre in which each reaction step is controlled by diffusive motion of the particles. The model studied here represents a typical reaction setting encountered in diverse molecular biology systems, in which, e.g. a signal transduction proceeds via a series of consecutive 'messengers': the first messenger has to find its respective immobile target site triggering a launch of the second messenger, the second messenger seeks its own target site and provokes a launch of the third messenger and so on, resembling a relay race in human competitions. For such a molecular relay race taking place in infinite one-, two- and three-dimensional systems, we find exact expressions for the probability density function of the time instant of the terminal reaction event, conditioned on preceding successful reaction events on an ordered array of target sites. The obtained expressions pertain to the most general conditions: number of intermediate stages and the corresponding diffusion coefficients, the sizes of the target sites, the distances between them, as well as their reactivities are arbitrary.},
  language  = {en}
}