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 - JOUR A1 - Teomy, Eial A1 - Metzler, Ralf T1 - Correlations and transport in exclusion processes with general finite memory JF - Journal of statistical mechanics: theory and experiment KW - Brownian motion KW - exclusion processes Y1 - 2019 U6 - https://doi.org/10.1088/1742-5468/ab47fb SN - 1742-5468 VL - 2019 IS - 10 PB - IOP Publ. Ltd. CY - Bristol ER - TY - JOUR A1 - Godec, Aljaz A1 - Metzler, Ralf T1 - Active transport improves the precision of linear long distance molecular signalling JF - Journal of physics : A, Mathematical and theoretical N2 - Molecular signalling in living cells occurs at low copy numbers and is thereby inherently limited by the noise imposed by thermal diffusion. The precision at which biochemical receptors can count signalling molecules is intimately related to the noise correlation time. In addition to passive thermal diffusion, messenger RNA and vesicle-engulfed signalling molecules can transiently bind to molecular motors and are actively transported across biological cells. Active transport is most beneficial when trafficking occurs over large distances, for instance up to the order of 1 metre in neurons. Here we explain how intermittent active transport allows for faster equilibration upon a change in concentration triggered by biochemical stimuli. Moreover, we show how intermittent active excursions induce qualitative changes in the noise in effectively one-dimensional systems such as dendrites. Thereby they allow for significantly improved signalling precision in the sense of a smaller relative deviation in the concentration read-out by the receptor. On the basis of linear response theory we derive the exact mean field precision limit for counting actively transported molecules. We explain how intermittent active excursions disrupt the recurrence in the molecular motion, thereby facilitating improved signalling accuracy. Our results provide a deeper understanding of how recurrence affects molecular signalling precision in biological cells and novel medical-diagnostic devices. KW - noise in biochemical signalling KW - Brownian motion KW - active transport KW - linear response theory KW - fluctuation-dissipation theorem KW - generalised Langevin equation KW - recurrence Y1 - 2016 U6 - https://doi.org/10.1088/1751-8113/49/36/364001 SN - 1751-8113 SN - 1751-8121 VL - 49 PB - IOP Publ. Ltd. CY - Bristol ER - TY - JOUR A1 - Palyulin, Vladimir V. A1 - Chechkin, Aleksei V. A1 - Klages, Rainer A1 - Metzler, Ralf T1 - Search reliability and search efficiency of combined Levy-Brownian motion: long relocations mingled with thorough local exploration JF - Journal of physics : A, Mathematical and theoretical N2 - A combined dynamics consisting of Brownian motion and Levy flights is exhibited by a variety of biological systems performing search processes. Assessing the search reliability of ever locating the target and the search efficiency of doing so economically of such dynamics thus poses an important problem. Here we model this dynamics by a one-dimensional fractional Fokker-Planck equation combining unbiased Brownian motion and Levy flights. By solving this equation both analytically and numerically we show that the superposition of recurrent Brownian motion and Levy flights with stable exponent alpha < 1, by itself implying zero probability of hitting a point on a line, leads to transient motion with finite probability of hitting any point on the line. We present results for the exact dependence of the values of both the search reliability and the search efficiency on the distance between the starting and target positions as well as the choice of the scaling exponent a of the Levy flight component. KW - random search process KW - first passage KW - first arrival KW - Levy flights KW - Brownian motion Y1 - 2016 U6 - https://doi.org/10.1088/1751-8113/49/39/394002 SN - 1751-8113 SN - 1751-8121 VL - 49 SP - 2189 EP - 2193 PB - IOP Publ. Ltd. CY - Bristol ER - TY - INPR A1 - Henkel, Carsten A1 - Pieplow, Gregor T1 - Reply to Comment on 'Fully covariant radiation force on a polarizable particle' T2 - New journal of physics : the open-access journal for physics N2 - We argue that the theories of Volokitin and Persson (2014 New J. Phys. 16 118001), Dedkov and Kyasov (2008 J. Phys.: Condens. Matter 20 354006), and Pieplow and Henkel (2013 New J. Phys. 15 023027) agree on the electromagnetic force on a small, polarizable particle that is moving parallel to a planar, macroscopic body, as far as the contribution of evanescent waves is concerned. The apparent differences are discussed in detail and explained by choices of units and integral transformations. We point out in particular the role of the Lorentz contraction in the procedure used by Volokitin and Persson, where a macroscopic body is 'diluted' to obtain the force on a small particle. Differences that appear in the contribution of propagating photons are briefly mentioned. KW - applied classical electromagnetism KW - fluctuation phenomena KW - random processes KW - noise KW - Brownian motion KW - mechanical effects of light Y1 - 2014 U6 - https://doi.org/10.1088/1367-2630/16/11/118002 SN - 1367-2630 VL - 16 PB - IOP Publ. Ltd. CY - Bristol ER - TY - INPR A1 - Bär, Christian A1 - Pfäffle, Frank T1 - Wiener measures on Riemannian manifolds and the Feynman-Kac formula N2 - This is an introduction to Wiener measure and the Feynman-Kac formula on general Riemannian manifolds for Riemannian geometers with little or no background in stochastics. We explain the construction of Wiener measure based on the heat kernel in full detail and we prove the Feynman-Kac formula for Schrödinger operators with bounded potentials. We also consider normal Riemannian coverings and show that projecting and lifting of paths are inverse operations which respect the Wiener measure. T3 - Preprints des Instituts für Mathematik der Universität Potsdam - 1(2012)17 KW - Wiener measure KW - conditional Wiener measure KW - Brownian motion KW - Brownian bridge KW - Riemannian manifold Y1 - 2012 U6 - http://nbn-resolving.de/urn/resolver.pl?urn:nbn:de:kobv:517-opus-59998 ER - TY - THES A1 - Hernandez Garcia, Hugo Fernando T1 - Multiscale simulation of heterophase polymerization : application to the synthesis of multicomponent colloidal polymer particles T1 - Multidimensionale Simulation von Heterophasenpolymerisationen : Anwendungen zur Synthese von kolloidalen Kompositpolymerteilchen N2 - Heterophase polymerization is a technique widely used for the synthesis of high performance polymeric materials with applications including paints, inks, adhesives, synthetic rubber, biomedical applications and many others. Due to the heterogeneous nature of the process, many different relevant length and time scales can be identified. Each of these scales has a direct influence on the kinetics of polymerization and on the physicochemical and performance properties of the final product. Therefore, from the point of view of product and process design and optimization, the understanding of each of these relevant scales and their integration into one single model is a very promising route for reducing the time-to-market in the development of new products, for increasing the productivity and profitability of existing processes, and for designing products with improved performance or cost/performance ratio. The process considered is the synthesis of structured or composite polymer particles by multi-stage seeded emulsion polymerization. This type of process is used for the preparation of high performance materials where a synergistic behavior of two or more different types of polymers is obtained. Some examples include the synthesis of core-shell or multilayered particles for improved impact strength materials and for high resistance coatings and adhesives. The kinetics of the most relevant events taking place in an emulsion polymerization process has been investigated using suitable numerical simulation techniques at their corresponding time and length scales. These methods, which include Molecular Dynamics (MD) simulation, Brownian Dynamics (BD) simulation and kinetic Monte Carlo (kMC) simulation, have been found to be very powerful and highly useful for gaining a deeper insight and achieving a better understanding and a more accurate description of all phenomena involved in emulsion polymerization processes, and can be potentially extended to investigate any type of heterogeneous process. The novel approach of using these kinetic-based numerical simulation methods can be regarded as a complement to the traditional thermodynamic-based macroscopic description of emulsion polymerization. The particular events investigated include molecular diffusion, diffusion-controlled polymerization reactions, particle formation, absorption/desorption of radicals and monomer, and the colloidal aggregation of polymer particles. Using BD simulation it was possible to precisely determine the kinetics of absorption/desorption of molecular species by polymer particles, and to simulate the colloidal aggregation of polymer particles. For diluted systems, a very good agreement between BD simulation and the classical theory developed by Smoluchowski was obtained. However, for concentrated systems, significant deviations from the ideal behavior predicted by Smoluchowski were evidenced. BD simulation was found to be a very valuable tool for the investigation of emulsion polymerization processes especially when the spatial and geometrical complexity of the system cannot be neglected, as is the case of concentrated dispersions, non-spherical particles, structured polymer particles, particles with non-uniform monomer concentration, and so on. In addition, BD simulation was used to describe non-equilibrium monomer swelling kinetics, which is not possible using the traditional thermodynamic approach because it is only valid for systems at equilibrium. The description of diffusion-controlled polymerization reactions was successfully achieved using a new stochastic algorithm for the kMC simulation of imperfectly mixed systems (SSA-IM). In contrast to the traditional stochastic simulation algorithm (SSA) and the deterministic rate of reaction equations, instead of assuming perfect mixing in the whole reactor, the new SSA-IM determines the volume perfectly mixed between two consecutive reactions as a function of the diffusion coefficient of the reacting species. Using this approach it was possible to describe, using a single set of kinetic parameters, typical mass transfer limitations effects during a free radical batch polymerization such as the cage effect, the gel effect and the glass effect. Using multiscale integration it was possible to investigate the formation of secondary particles during the seeded emulsion polymerization of vinyl acetate over a polystyrene seed. Three different cases of radical generation were considered: generation of radicals by thermal decomposition of water-soluble initiating compounds, generation of radicals by a redox reaction at the surface of the particles, and generation of radicals by thermal decomposition of surface-active initiators "inisurfs" attached to the surface of the particles. The simulation results demonstrated the satisfactory reduction in secondary particles formation achieved when the locus of radical generation is controlled close to the particles surface. N2 - Eine der industriell am meisten verwendeten Methoden zur Herstellung von Hochleistungspolymeren ist die Heterophasenpolymerisation. Industriell von besonderer Bedeutung ist die sogenannte Saatemulsionspolymerisation bei der kleine Saatteilchen durch die sequentielle Zugabe von weiteren Monomeren gezielt modifiziert werden, um Kompositpolymerteilchen mit den gewünschten mechanischen und chemischen Gebrauchseigenschaften herzustellen. Ein häufig auftretendes Problem während dieser Art der Heterophasenpolymerisation ist die Bildung von neuen, kleinen Teilchen im Polymerisationsverlauf. Diese sogenannte sekundäre Teilchenbildung muss vermieden werden, da sie die Herstellung der gewünschten Teilchen mit den angestrebten Eigenschaften verhindert. Ein spezieller Fall der Saatemulsionspolymerisation ist die Kombination von Vinylacetat als Monomer, das auf Saatteilchen aus Polystyrol polymerisieren soll. Die Unterdrückung der Teilchenneubildung ist in diesem Beispiel besonders schwierig, da Vinylacetat eine sehr hohe Wasserlöslichkeit besitzt. In der vorliegenden Arbeit wurden zur Lösung der Aufgabenstellung verschiedene numerische Simulierungsalgorithmen verwendet, die entsprechend den charakteristischen Längen- und Zeitskalen der im Verlauf der Polymerisation ablaufenden Prozesse ausgewählt wurden, um die passenden Bedingungen für die Unterdrückung der sekundären Teilchenbildung zu finden. Die verwendeten numerischen Methoden umfassen Molekulare Dynamik Simulationen, die benutzt werden, um molekulare Bewegungen zu berechnen; Brownsche Dynamik Simulationen, die benutzt werden, um die zufälligen Bewegungen der kolloidalen Teilchen und der molekularen Spezies zu beschreiben, und kinetische Monte Carlo Simulationen, die das zufällige Auftreten von individuellen physikalischen oder chemischen Ereignissen modellieren. Durch die Kombination dieser Methoden ist es möglich, alle für die Beschreibung der Polymerisation relevanten Phänomene zu berücksichtigen. Damit können nicht nur die Reaktionsgeschwindigkeit und die Produktivität des Prozesses simuliert werden sondern auch Aussagen bezüglich der physikalischen und chemischen Eigenschaften des Produktes sowie den Applikationseigenschaften getroffen werden. In dieser Arbeit wurden zum ersten Mal Modelle für die unterschiedlichen Längen- und Zeitskalen bei Heterophasenpolymerisationen entwickelt und erfolgreich zur Modellierung des Prozesses angewendet. Die Ergebnisse führten zu bedeutenden Verbesserungen der Theorie von Emulsionspolymerisationen insbesondere für die Beschreibung des Massenaustausches zwischen den Phasen (bspw. Radikaleintritt in und Radikalaustritt aus die Polymerteilchen), der Bildung von neuen Teilchen, und der Polymerisationskinetik unter den heterogenen Reaktionsbedingungen mit uneinheitlicher Durchmischung. KW - Heterophase Polymerization KW - Emulsion Polymerization KW - Simulation KW - Kinetics KW - Brownian motion Y1 - 2008 U6 - http://nbn-resolving.de/urn/resolver.pl?urn:nbn:de:kobv:517-opus-25036 ER -