@phdthesis{Kruesemann2016,
  author    = {Kr{\"u}semann, Henning},
  title     = {First passage phenomena and single-file motion in ageing continuous time random walks and quenched energy landscapes},
  school      = {Universit{\"a}t Potsdam},
  pages     = {122},
  year      = {2016},
  abstract  = {In der Physik gibt es viele Prozesse, die auf Grund ihrer Komplexit{\"a}t nicht durch physikalische Gleichungen beschrieben werden k{\"o}nnen, beispielsweise die Bewegung eines Staubkorns in der Luft. Durch die vielen St{\"o}ße mit Luftmolek{\"u}len f{\"u}hrt es eine Zufallsbewegung aus, die so genannte Diffusion. Auch Molek{\"u}le in biologischen Zellen diffundieren, jedoch befinden sich in einer solchen Zelle im selben Volumen viel mehr oder viel gr{\"o}ßere Molek{\"u}le. Das beobachtete Teilchen st{\"o}ßt dementsprechend {\"o}fter mit anderen zusammen und die Diffusion wird langsamer, sie wird subdiffusiv. Mit der Zeit kann sich die Charakteristik der Subdiffusion {\"a}ndern; dies wird als (mikroskopisches) Altern bezeichnet. Ich untersuche in der vorliegenden Arbeit zwei mathematische Modelle f{\"u}r eindimensionale Subdiffusion, einmal den continuous time random walk (CTRW) und einmal die Zufallsbewegung in einer eingefrorenen Energielandschaft (QEL=quenched energy landscape). Beide sind Sprungprozesse, das heißt, sie sind Abfolgen von r{\"a}umlichen Spr{\"u}ngen, die durch zufallsverteilte Wartezeiten getrennt sind. Die Wartezeiten in der QEL sind r{\"a}umlich korrelliert, w{\"a}hrend sie im CTRW unkorrelliert sind. Ich untersuche in der vorliegenden Arbeit verschiedene statistische Gr{\"o}ßen in beiden Modellen. Zun{\"a}chst untersuche ich den Einfluss des Alters und den Einfluss der Korrellationen einer QEL auf die Verteilung der Zeiten, die das diffundierendes Teilchen ben{\"o}tigt, um eine (r{\"a}umliche) Schwelle zu {\"u}berqueren. Ausserdem bestimme ich den Effekt des Alters auf Str{\"o}me von (sub)diffundierenden Partikeln, die sich auf eine absorbierende Barriere zubewegen. Zuletzt besch{\"a}ftige ich mich mit der Diffusion einer eindimensionalen Anordnung von Teilchen in einer QEL, in der diese als harte Kugeln miteinander wechselwirken. Dabei vergleiche ich die gemeinsame Bewegung in einer QEL und als individuelle CTRWs miteinander {\"u}ber die Standartabweichung von der Startposition, f{\"u}r die ich das Mittel {\"u}ber mehrere QELs untersuche. Meine Arbeit setzt sich zusammen aus theoretischen {\"U}berlegungen und Berechnungen sowie der Simulation der Zufallsprozesse. Die Ergebnisse der Simulation und, soweit vorhanden, experimentelle Daten werden mit der Theorie verglichen.},
  language  = {en}
}
@misc{GranadoAbadMetzleretal.2020,
  author    = {Granado, Felipe Le Vot and Abad, Enrique and Metzler, Ralf and Yuste, Santos B.},
  title     = {Continuous time random walk in a velocity field},
  series = {Postprints der Universit{\"a}t Potsdam : Mathematisch-Naturwissenschaftliche Reihe},
  journal   = {Postprints der Universit{\"a}t Potsdam : Mathematisch-Naturwissenschaftliche Reihe},
  number    = {1005},
  issn      = {1866-8372},
  doi       = {10.25932/publishup-47999},
  url       = {http://nbn-resolving.de/urn:nbn:de:kobv:517-opus4-479997},
  pages     = {28},
  year      = {2020},
  abstract  = {We consider the emerging dynamics of a separable continuous time random walk (CTRW) in the case when the random walker is biased by a velocity field in a uniformly growing domain. Concrete examples for such domains include growing biological cells or lipid vesicles, biofilms and tissues, but also macroscopic systems such as expanding aquifers during rainy periods, or the expanding Universe. The CTRW in this study can be subdiffusive, normal diffusive or superdiffusive, including the particular case of a L{\´e}vy flight. We first consider the case when the velocity field is absent. In the subdiffusive case, we reveal an interesting time dependence of the kurtosis of the particle probability density function. In particular, for a suitable parameter choice, we find that the propagator, which is fat tailed at short times, may cross over to a Gaussian-like propagator. We subsequently incorporate the effect of the velocity field and derive a bi-fractional diffusion-advection equation encoding the time evolution of the particle distribution. We apply this equation to study the mixing kinetics of two diffusing pulses, whose peaks move towards each other under the action of velocity fields acting in opposite directions. This deterministic motion of the peaks, together with the diffusive spreading of each pulse, tends to increase particle mixing, thereby counteracting the peak separation induced by the domain growth. As a result of this competition, different regimes of mixing arise. In the case of L{\´e}vy flights, apart from the non-mixing regime, one has two different mixing regimes in the long-time limit, depending on the exact parameter choice: in one of these regimes, mixing is mainly driven by diffusive spreading, while in the other mixing is controlled by the velocity fields acting on each pulse. Possible implications for encounter-controlled reactions in real systems are discussed.},
  language  = {en}
}
@article{GranadoAbadMetzleretal.2020,
  author    = {Granado, Felipe Le Vot and Abad, Enrique and Metzler, Ralf and Yuste, Santos B.},
  title     = {Continuous time random walk in a velocity field},
  series = {New Journal of Physics},
  volume    = {22},
  journal   = {New Journal of Physics},
  publisher = {Dt. Physikalische Ges.},
  address   = {Bad Honnef},
  issn      = {1367-2630},
  doi       = {10.1088/1367-2630/ab9ae2},
  pages     = {27},
  year      = {2020},
  abstract  = {We consider the emerging dynamics of a separable continuous time random walk (CTRW) in the case when the random walker is biased by a velocity field in a uniformly growing domain. Concrete examples for such domains include growing biological cells or lipid vesicles, biofilms and tissues, but also macroscopic systems such as expanding aquifers during rainy periods, or the expanding Universe. The CTRW in this study can be subdiffusive, normal diffusive or superdiffusive, including the particular case of a L{\´e}vy flight. We first consider the case when the velocity field is absent. In the subdiffusive case, we reveal an interesting time dependence of the kurtosis of the particle probability density function. In particular, for a suitable parameter choice, we find that the propagator, which is fat tailed at short times, may cross over to a Gaussian-like propagator. We subsequently incorporate the effect of the velocity field and derive a bi-fractional diffusion-advection equation encoding the time evolution of the particle distribution. We apply this equation to study the mixing kinetics of two diffusing pulses, whose peaks move towards each other under the action of velocity fields acting in opposite directions. This deterministic motion of the peaks, together with the diffusive spreading of each pulse, tends to increase particle mixing, thereby counteracting the peak separation induced by the domain growth. As a result of this competition, different regimes of mixing arise. In the case of L{\´e}vy flights, apart from the non-mixing regime, one has two different mixing regimes in the long-time limit, depending on the exact parameter choice: in one of these regimes, mixing is mainly driven by diffusive spreading, while in the other mixing is controlled by the velocity fields acting on each pulse. Possible implications for encounter-controlled reactions in real systems are discussed.},
  language  = {en}
}