TY - JOUR A1 - Grebenkov, Denis S. A1 - Sposini, Vittoria A1 - Metzler, Ralf A1 - Oshanin, Gleb A1 - Seno, Flavio T1 - Exact distributions of the maximum and range of random diffusivity processes JF - New Journal of Physics N2 - We study the extremal properties of a stochastic process xt defined by the Langevin equation ẋₜ =√2Dₜ ξₜ, in which ξt is a Gaussian white noise with zero mean and Dₜ is a stochastic‘diffusivity’, defined as a functional of independent Brownian motion Bₜ.We focus on threechoices for the random diffusivity Dₜ: cut-off Brownian motion, Dₜt ∼ Θ(Bₜ), where Θ(x) is the Heaviside step function; geometric Brownian motion, Dₜ ∼ exp(−Bₜ); and a superdiffusive process based on squared Brownian motion, Dₜ ∼ B²ₜ. For these cases we derive exact expressions for the probability density functions of the maximal positive displacement and of the range of the process xₜ on the time interval ₜ ∈ (0, T).We discuss the asymptotic behaviours of the associated probability density functions, compare these against the behaviour of the corresponding properties of standard Brownian motion with constant diffusivity (Dₜ = D0) and also analyse the typical behaviour of the probability density functions which is observed for a majority of realisations of the stochastic diffusivity process. KW - random diffusivity KW - extremal values KW - maximum and range KW - diffusion KW - Brownian motion Y1 - 2021 U6 - https://doi.org/10.1088/1367-2630/abd313 SN - 1367-2630 VL - 23 PB - Dt. Physikalische Ges. CY - Bad Honnef ER - TY - GEN A1 - Grebenkov, Denis S. A1 - Sposini, Vittoria A1 - Metzler, Ralf A1 - Oshanin, Gleb A1 - Seno, Flavio T1 - Exact distributions of the maximum and range of random diffusivity processes T2 - Postprints der Universität Potsdam : Mathematisch-Naturwissenschaftliche Reihe N2 - We study the extremal properties of a stochastic process xt defined by the Langevin equation ẋₜ =√2Dₜ ξₜ, in which ξt is a Gaussian white noise with zero mean and Dₜ is a stochastic‘diffusivity’, defined as a functional of independent Brownian motion Bₜ.We focus on threechoices for the random diffusivity Dₜ: cut-off Brownian motion, Dₜt ∼ Θ(Bₜ), where Θ(x) is the Heaviside step function; geometric Brownian motion, Dₜ ∼ exp(−Bₜ); and a superdiffusive process based on squared Brownian motion, Dₜ ∼ B²ₜ. For these cases we derive exact expressions for the probability density functions of the maximal positive displacement and of the range of the process xₜ on the time interval ₜ ∈ (0, T).We discuss the asymptotic behaviours of the associated probability density functions, compare these against the behaviour of the corresponding properties of standard Brownian motion with constant diffusivity (Dₜ = D0) and also analyse the typical behaviour of the probability density functions which is observed for a majority of realisations of the stochastic diffusivity process. T3 - Zweitveröffentlichungen der Universität Potsdam : Mathematisch-Naturwissenschaftliche Reihe - 1142 KW - random diffusivity KW - extremal values KW - maximum and range KW - diffusion KW - Brownian motion Y1 - 2021 U6 - http://nbn-resolving.de/urn/resolver.pl?urn:nbn:de:kobv:517-opus4-503976 SN - 1866-8372 IS - 1142 ER - TY - THES A1 - Sposini, Vittoria T1 - The random diffusivity approach for diffusion in heterogeneous systems N2 - The two hallmark features of Brownian motion are the linear growth < x2(t)> = 2Ddt of the mean squared displacement (MSD) with diffusion coefficient D in d spatial dimensions, and the Gaussian distribution of displacements. With the increasing complexity of the studied systems deviations from these two central properties have been unveiled over the years. Recently, a large variety of systems have been reported in which the MSD exhibits the linear growth in time of Brownian (Fickian) transport, however, the distribution of displacements is pronouncedly non-Gaussian (Brownian yet non-Gaussian, BNG). A similar behaviour is also observed for viscoelastic-type motion where an anomalous trend of the MSD, i.e., ~ ta, is combined with a priori unexpected non-Gaussian distributions (anomalous yet non-Gaussian, ANG). This kind of behaviour observed in BNG and ANG diffusions has been related to the presence of heterogeneities in the systems and a common approach has been established to address it, that is, the random diffusivity approach. This dissertation explores extensively the field of random diffusivity models. Starting from a chronological description of all the main approaches used as an attempt of describing BNG and ANG diffusion, different mathematical methodologies are defined for the resolution and study of these models. The processes that are reported in this work can be classified in three subcategories, i) randomly-scaled Gaussian processes, ii) superstatistical models and iii) diffusing diffusivity models, all belonging to the more general class of random diffusivity models. Eventually, the study focuses more on BNG diffusion, which is by now well-established and relatively well-understood. Nevertheless, many examples are discussed for the description of ANG diffusion, in order to highlight the possible scenarios which are known so far for the study of this class of processes. The second part of the dissertation deals with the statistical analysis of random diffusivity processes. A general description based on the concept of moment-generating function is initially provided to obtain standard statistical properties of the models. Then, the discussion moves to the study of the power spectral analysis and the first passage statistics for some particular random diffusivity models. A comparison between the results coming from the random diffusivity approach and the ones for standard Brownian motion is discussed. In this way, a deeper physical understanding of the systems described by random diffusivity models is also outlined. To conclude, a discussion based on the possible origins of the heterogeneity is sketched, with the main goal of inferring which kind of systems can actually be described by the random diffusivity approach. N2 - Die zwei grundlegenden Eigenschaften der Brownschen Molekularbewegung sind das lineare Wachstum < x2(t)> = 2Ddt der mittleren quadratischen Verschiebung (mean squared displacement, MSD) mit dem Diffusionskoeffizienten D in Dimension d und die Gauß Verteilung der räumlichen Verschiebung. Durch die zunehmende Komplexität der untersuchten Systeme wurden in den letzten Jahren Abweichungen von diesen zwei grundlegenden Eigenschaften gefunden. Hierbei, wurde über eine große Anzahl von Systemen berichtet, in welchen die MSD das lineare Wachstum der Brownschen Bewegung (Ficksches Gesetzt) zeigt, jedoch die Verteilung der Verschiebung nicht einer Gaußverteilung folgt (Brownian yet non-Gaussian, BNG). Auch in viskoelastischen Systemen Bewegung wurde ein analoges Verhalten beobachtet. Hier ist ein anomales Verhalten des MSD, ~ ta, in Verbindung mit einer a priori unerwarteten nicht gaußchen Verteilung (anomalous yet non-Gaussian, ANG). Dieses Verhalten, welches sowohl in BNG- als auch in ANG-Diffusion beobachtet wird, ist auf eine Heterogenität in den Systemen zurückzuführen. Um diese Systeme zu beschreiben, wurde ein einheitlicher Ansatz, basierend auf den Konzept der zufälligen Diffusivität, entwickelt. Die vorliegende Dissertation widmet sich ausführlich Modellen mit zufälligen Diffusivität. Ausgehend von einem chronologischen Überblick der grundlegenden Ansätze der Beschreibung der BNG- und ANG-Diffusion werden mathematische Methoden entwickelt, um die verschiedenen Modelle zu untersuchen. Die in dieser Arbeit diskutierten Prozesse können in drei Kategorien unterteil werden: i) randomly-scaled Gaussian processes, ii) superstatistical models und iii) diffusing diffusivity models, welche alle zu den allgemeinen Modellen mit zufälligen Diffusivität gehören. Der Hauptteil dieser Arbeit ist die Untersuchung auf die BNG Diffusion, welche inzwischen relativ gut verstanden ist. Dennoch werden auch viele Beispiele für die Beschreibung von ANG-Diffusion diskutiert, um die Möglichkeiten der Analyse solcher Prozesse aufzuzeigen. Der zweite Teil der Dissertation widmet sich der statistischen Analyse von Modellen mit zufälligen Diffusivität. Eine allgemeine Beschreibung basierend auf dem Konzept der momenterzeugenden Funktion wurde zuerst herangezogen, um grundsätzliche statistische Eigenschaften der Modelle zu erhalten. Anschließend konzentriert sich die Diskussion auf die Analyse der spektralen Leistungsdichte und der first passage Statistik für einige spezielle Modelle mit zufälligen Diffusivität. Diese Ergebnisse werden mit jenen der normalen Brownschen Molekularbewegung verglichen. Dadurch wird ein tiefergehendes physikalisches Verständnis über die Systeme erlangt, welche durch ein Modell mit zufälligen Diffusivität beschrieben werden. Abschließend, zeigt eine Diskussion mögliche Ursachen für die Heterogenität auf, mit dem Ziel darzustellen, welche Arten von Systemen durch den Zufalls-Diffusivitäts-Ansatz beschrieben werden können. N2 - Las dos características distintivas del movimiento Browniano son el crecimiento lineal < x2(t)> = 2Ddt del desplazamiento cuadrático medio (mean squared displacement}, MSD) con el coeficiente de difusión D en dimensiones espaciales d, y la distribución Gaussiana de los desplazamientos. Con los continuos avances en tecnologías experimentales y potencia de cálculo, se logra estudiar con mayor detalle sistemas cada vez más complejos y algunos sistemas revelan desviaciones de estas dos propiedades centrales. En los últimos años se ha observado una gran variedad de sistemas en los que el MSD presenta un crecimiento lineal en el tiempo (típico del transporte Browniano), no obstante, la distribución de los desplazamientos es pronunciadamente no Gaussiana (Brownian yet non-Gaussian diffusion}, BNG). Un comportamiento similar se observa asimismo en el caso del movimiento de tipo viscoelástico, en el que se combina una tendencia anómala del MSD, es decir, ~ ta, con a, con distribuciones inesperadamente no Gaussianas (Anomalous yet non-Gaussian diffusion, ANG). Este tipo de comportamiento observado en las difusiones BNG y ANG se ha relacionado con la presencia de heterogeneidades en los sistemas y se ha establecido un enfoque común para abordarlo: el enfoque de difusividad aleatoria. En la primera parte de esta disertación se explora extensamente el área de los modelos de difusividad aleatoria. A través de una descripción cronológica de los principales enfoques utilizados para caracterizar las difusiones BNG y ANG, se definen diferentes metodologías matemáticas para la resolución y el estudio de estos modelos. Los procesos expuestos en este trabajo, pertenecientes a la clase más general de modelos de difusividad aleatoria, pueden clasificarse en tres subcategorías: i) randomly-scaled Gaussian processes, ii) superstatistical models y iii) diffusing diffusivity models. Fundamentalmente el enfoque de este trabajo se centra en la difusión BNG, bien establecida y ampliamente estudiada en los últimos años. No obstante, múltiples ejemplos son examinados para la descripción de la difusión ANG, a fin de remarcar los diferentes modelos de estudio disponibles hasta el momento. En la segunda parte de la disertación se desarolla el análisis estadístico de los procesos de difusividad aleatoria. Inicialmente se expone una descripción general basada en el concepto de la función generadora de momentos para obtener las propiedades estadísticas estándar de los modelos. A continuación, la discusión aborda el estudio de la densidad espectral de potencia y la estadística del tiempo de primer paso para algunos modelos de difusividad aleatoria. Adicionalmente, los resultados del método de difusividad aleatoria se comparan junto a los de movimiento browniano estándar. Como resultado, se obtiene una mayor comprensión física de los sistemas descritos por los modelos de difusividad aleatoria. Para concluir, se presenta una discusión acerca de los posibles orígenes de la heterogeneidad, con el objetivo principal de inferir qué tipo de sistemas pueden describirse apropiadamente según el enfoque de la difusividad aleatoria. KW - diffusion KW - non-gaussianity KW - random diffusivity KW - power spectral analysis KW - first passage KW - Diffusion KW - zufälligen Diffusivität KW - spektrale Leistungsdichte KW - first passage KW - Heterogenität Y1 - 2020 U6 - http://nbn-resolving.de/urn/resolver.pl?urn:nbn:de:kobv:517-opus4-487808 ER - TY - JOUR A1 - Sposini, Vittoria A1 - Grebenkov, Denis S. A1 - Metzler, Ralf A1 - Oshanin, Gleb A1 - Seno, Flavio T1 - Universal spectral features of different classes of random-diffusivity processes JF - New Journal of Physics N2 - Stochastic models based on random diffusivities, such as the diffusing-diffusivity approach, are popular concepts for the description of non-Gaussian diffusion in heterogeneous media. Studies of these models typically focus on the moments and the displacement probability density function. Here we develop the complementary power spectral description for a broad class of random-diffusivity processes. In our approach we cater for typical single particle tracking data in which a small number of trajectories with finite duration are garnered. Apart from the diffusing-diffusivity model we study a range of previously unconsidered random-diffusivity processes, for which we obtain exact forms of the probability density function. These new processes are different versions of jump processes as well as functionals of Brownian motion. The resulting behaviour subtly depends on the specific model details. Thus, the central part of the probability density function may be Gaussian or non-Gaussian, and the tails may assume Gaussian, exponential, log-normal, or even power-law forms. For all these models we derive analytically the moment-generating function for the single-trajectory power spectral density. We establish the generic 1/f²-scaling of the power spectral density as function of frequency in all cases. Moreover, we establish the probability density for the amplitudes of the random power spectral density of individual trajectories. The latter functions reflect the very specific properties of the different random-diffusivity models considered here. Our exact results are in excellent agreement with extensive numerical simulations. KW - diffusion KW - power spectrum KW - random diffusivity KW - single trajectories Y1 - 2020 U6 - https://doi.org/10.1088/1367-2630/ab9200 SN - 1367-2630 VL - 22 IS - 6 PB - Dt. Physikalische Ges. CY - Bad Honnef ER - TY - GEN A1 - Sposini, Vittoria A1 - Grebenkov, Denis S. A1 - Metzler, Ralf A1 - Oshanin, Gleb A1 - Seno, Flavio T1 - Universal spectral features of different classes of random-diffusivity processes T2 - Postprints der Universität Potsdam : Mathematisch-Naturwissenschaftliche Reihe N2 - Stochastic models based on random diffusivities, such as the diffusing-diffusivity approach, are popular concepts for the description of non-Gaussian diffusion in heterogeneous media. Studies of these models typically focus on the moments and the displacement probability density function. Here we develop the complementary power spectral description for a broad class of random-diffusivity processes. In our approach we cater for typical single particle tracking data in which a small number of trajectories with finite duration are garnered. Apart from the diffusing-diffusivity model we study a range of previously unconsidered random-diffusivity processes, for which we obtain exact forms of the probability density function. These new processes are different versions of jump processes as well as functionals of Brownian motion. The resulting behaviour subtly depends on the specific model details. Thus, the central part of the probability density function may be Gaussian or non-Gaussian, and the tails may assume Gaussian, exponential, log-normal, or even power-law forms. For all these models we derive analytically the moment-generating function for the single-trajectory power spectral density. We establish the generic 1/f²-scaling of the power spectral density as function of frequency in all cases. Moreover, we establish the probability density for the amplitudes of the random power spectral density of individual trajectories. The latter functions reflect the very specific properties of the different random-diffusivity models considered here. Our exact results are in excellent agreement with extensive numerical simulations. T3 - Zweitveröffentlichungen der Universität Potsdam : Mathematisch-Naturwissenschaftliche Reihe - 999 KW - diffusion KW - power spectrum KW - random diffusivity KW - single trajectories Y1 - 2020 U6 - http://nbn-resolving.de/urn/resolver.pl?urn:nbn:de:kobv:517-opus4-476960 SN - 1866-8372 IS - 999 ER -