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First passage phenomena and single-file motion in ageing continuous time random walks and quenched energy landscapes

  • In the first part of my work I have investigated the ageing properties of the first passage time distributions in a one-dimensional subdiffusive continuous time random walk with power law distributed waiting times of the form $\psi(\tau) \sim \tau^{-1-\alpha}$ with $0<\alpha<1$ and $1<\alpha<2$. The age or ageing time $t_a$ is the time span from the start of the stochastic process to the start of the observation of this process (at $t=0$). I have calculated the results for a single target and two targets, also including the biased case, where the walker is driven towards the boundary by a constant force. I have furthermore refined the previously derived results for the non-ageing case and investigated the changes that occur when the walk is performed in a discrete quenched energy landscape, where the waiting times are fixed for every site. The results include the exact Laplace space densities and infinite (converging) series as exact results in the time space. The main results are the dominating long time power law behavior regimes,In the first part of my work I have investigated the ageing properties of the first passage time distributions in a one-dimensional subdiffusive continuous time random walk with power law distributed waiting times of the form $\psi(\tau) \sim \tau^{-1-\alpha}$ with $0<\alpha<1$ and $1<\alpha<2$. The age or ageing time $t_a$ is the time span from the start of the stochastic process to the start of the observation of this process (at $t=0$). I have calculated the results for a single target and two targets, also including the biased case, where the walker is driven towards the boundary by a constant force. I have furthermore refined the previously derived results for the non-ageing case and investigated the changes that occur when the walk is performed in a discrete quenched energy landscape, where the waiting times are fixed for every site. The results include the exact Laplace space densities and infinite (converging) series as exact results in the time space. The main results are the dominating long time power law behavior regimes, which depend on the ageing time. For the case of unbiased subdiffusion ($\alpha < 1$) in the presence of one target, I find three different dominant terms for ranges of $t$ separated by $t_a$ and another crossover time $t^{\star}$, which depends on $t_a$ as well as on the anomalous exponent $\alpha$ and the anomalous diffusion coefficient $K_{\alpha}$. In all three regimes ($t \ll t_a$, $t_a \ll t \ll t^{\star}$, $t \gg t^{\star}$) one finds power law decay with exponents depending on $\alpha$. The middle regime only exists for $t_a \ll t^{\star}$. The dominant terms in the first two regimes (ageing regimes) come from the probability distribution of the forward waiting time, the time one has to wait for the stochastic process to make the first step during the observation. When the observation time is larger than the second crossover time $t^{\star}$, the first passage time density does not show ageing and the non-ageing first passage time dominates. The power law exponents in the respective regimes are $-\alpha$ for strong ageing, $-1-\alpha$ in the intermediate regime, and $-1-\alpha/2$ in the final non-ageing regime. A similar split into three regimes can be found for $1<\alpha<2$, only with a different second crossover time $t^*$. In this regime the diffusion is normal but also age-dependent. For the diffusion in quenched energy landscapes one cannot detect ageing. The first passage time density shows a quenched power law $^\sim t^{-(1+2\alpha)/(1+\alpha)}$. For diffusion between two target sites and the biased diffusion towards a target only two scaling regimes emerge, separated by the ageing time. In the ageing case $t \ll t_a$ the forward waiting time is again dominant with power law exponent $-\alpha$, while the non-ageing power law $-1-\alpha$ is found for all times $t \gg t_a$. An intermediate regime does not exist. The bias and the confinement have similar effects on the first passage time density. For quenched diffusion, the biased case is interesting, as the bias reduces correlations due to revisiting of the same waiting time. As a result, CTRW like behavior is observed, including ageing. Extensive computer simulations support my findings. The second part of my research was done on the subject of ageing Scher-Montroll transport, which is in parts closely related to the first passage densities. It explains the electrical current in an amorphous material. I have investigated the effect of the width of a given initial distribution of charge carriers on the transport coefficients as well as the ageing effect on the emerging power law regimes and a constant initial regime. While a spread out initial distribution has only little impact on the Scher-Montroll current, ageing alters the behavior drastically. Instead of the two classical power laws one finds four current regimes, up to three of which can appear in a single experiment. The dominant power laws differ for $t \ll t_a, t_c$, $t_a \ll t \ll t_c$, $t_c \ll t \ll t_a$, and $t \gg t_a,t_c$. Here, $t_c$ is the crossover time of the non-aged Scher-Montroll current. For strongly aged systems one can observe a constant current in the first regime while the others are dominated by decaying power laws with exponents $\alpha -1$, $-\alpha$, and $-1-\alpha$. The ageing regimes are the 1st and 3rd one, while the classical regimes are the 2nd and the 4th. I have verified the theory using numerical integration of the exact integrals and applied the new results to experimental data. In the third part I considered a single file of subdiffusing particles in an energy landscape. Every occupied site of the landscape acts as a boundary, from which a particle is immediately reflected to its previous site, if it tries to jump there. I have analysed the effects single-file diffusion a quenched landscape compared to an annealed landscape and I have related these results to the number of steps and related quantities. The diffusion changes from ultraslow logarithmic diffusion in the annealed or CTRW case to subdiffusion with an anomalous exponent $\alpha/(1+\alpha)$ in the quenched landscape. The behavior is caused by the forward waiting time, which changes drastically from the quenched to the annealed case. Single-file effects in the quenched landscape are even more complicated to consider in the ensemble average, since the diffusion in individual landscapes shows extremely diverse behavior. Extensive simulations support my theoretical arguments, which consider mainly the long time evolution of the mean square displacement of a bulk particle.show moreshow less
  • In der Physik gibt es viele Prozesse, die auf Grund ihrer Komplexität nicht durch physikalische Gleichungen beschrieben werden können, beispielsweise die Bewegung eines Staubkorns in der Luft. Durch die vielen Stöße mit Luftmolekülen führt es eine Zufallsbewegung aus, die so genannte Diffusion. Auch Moleküle in biologischen Zellen diffundieren, jedoch befinden sich in einer solchen Zelle im selben Volumen viel mehr oder viel größere Moleküle. Das beobachtete Teilchen stößt dementsprechend öfter mit anderen zusammen und die Diffusion wird langsamer, sie wird subdiffusiv. Mit der Zeit kann sich die Charakteristik der Subdiffusion ändern; dies wird als (mikroskopisches) Altern bezeichnet. Ich untersuche in der vorliegenden Arbeit zwei mathematische Modelle für 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 räumlichen Sprüngen, die durchIn der Physik gibt es viele Prozesse, die auf Grund ihrer Komplexität nicht durch physikalische Gleichungen beschrieben werden können, beispielsweise die Bewegung eines Staubkorns in der Luft. Durch die vielen Stöße mit Luftmolekülen führt es eine Zufallsbewegung aus, die so genannte Diffusion. Auch Moleküle in biologischen Zellen diffundieren, jedoch befinden sich in einer solchen Zelle im selben Volumen viel mehr oder viel größere Moleküle. Das beobachtete Teilchen stößt dementsprechend öfter mit anderen zusammen und die Diffusion wird langsamer, sie wird subdiffusiv. Mit der Zeit kann sich die Charakteristik der Subdiffusion ändern; dies wird als (mikroskopisches) Altern bezeichnet. Ich untersuche in der vorliegenden Arbeit zwei mathematische Modelle für 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 räumlichen Sprüngen, die durch zufallsverteilte Wartezeiten getrennt sind. Die Wartezeiten in der QEL sind räumlich korrelliert, während sie im CTRW unkorrelliert sind. Ich untersuche in der vorliegenden Arbeit verschiedene statistische Größen in beiden Modellen. Zunächst untersuche ich den Einfluss des Alters und den Einfluss der Korrellationen einer QEL auf die Verteilung der Zeiten, die das diffundierendes Teilchen benötigt, um eine (räumliche) Schwelle zu überqueren. Ausserdem bestimme ich den Effekt des Alters auf Ströme von (sub)diffundierenden Partikeln, die sich auf eine absorbierende Barriere zubewegen. Zuletzt beschäftige 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 über die Standartabweichung von der Startposition, für die ich das Mittel über mehrere QELs untersuche. Meine Arbeit setzt sich zusammen aus theoretischen Überlegungen und Berechnungen sowie der Simulation der Zufallsprozesse. Die Ergebnisse der Simulation und, soweit vorhanden, experimentelle Daten werden mit der Theorie verglichen.show moreshow less

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Metadaten
Author:Henning Krüsemann
Advisor:Ralf Metzler
Document Type:Doctoral Thesis
Language:English
Year of first Publication:2017
Year of Completion:2016
Publishing Institution:Universität Potsdam
Granting Institution:Universität Potsdam
Date of final exam:2016/12/16
Release Date:2017/03/08
Tag:Diffusion; Scher-Montroll Transport; Wartezeitverteilung; Zufallsbewegung; eingefrorene Energielandschaft
Scher-Montroll transport; continuous time random walk; first passage time; quenched energy landscape; single-file motion
Pagenumber:122
Organizational units:Mathematisch-Naturwissenschaftliche Fakultät / Institut für Physik und Astronomie
Dewey Decimal Classification:5 Naturwissenschaften und Mathematik / 53 Physik / 530 Physik