Relevant bibliographies by topics / Continuous Time Random Walk (CTRW)

Academic literature on the topic 'Continuous Time Random Walk (CTRW)'

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Published: 6 September 2023

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Journal articles on the topic "Continuous Time Random Walk (CTRW)"

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FA, KWOK SAU, and K. G. WANG. "INTEGRO-DIFFERENTIAL EQUATIONS ASSOCIATED WITH CONTINUOUS-TIME RANDOM WALK." International Journal of Modern Physics B 27, no. 12 (April 29, 2013): 1330006. http://dx.doi.org/10.1142/s0217979213300065.

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The continuous-time random walk (CTRW) model is a useful tool for the description of diffusion in nonequilibrium systems, which is broadly applied in nature and life sciences, e.g., from biophysics to geosciences. In particular, the integro-differential equations for diffusion and diffusion-advection are derived asymptotically from the decoupled CTRW model and a generalized Chapmann–Kolmogorov equation, with generic waiting time probability density function (PDF) and external force. The advantage of the integro-differential equations is that they can be used to investigate the entire diffusion process i.e., covering initial-, intermediate- and long-time ranges of the process. Therefore, this method can distinguish the evolution detail for a system having the same behavior in the long-time limit but with different initial- and intermediate-time behaviors. An integro-differential equation for diffusion-advection is also presented for the description of the subdiffusive and superdiffusive regime. Moreover, the methods of solving the integro-differential equations are developed, and the analytic solutions for PDFs are obtained for the cases of force-free and linear force.
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FA, KWOK SAU. "CONTINUOUS-TIME FINANCE AND THE WAITING TIME DISTRIBUTION: MULTIPLE CHARACTERISTIC TIMES." Modern Physics Letters B 26, no. 23 (August 13, 2012): 1250151. http://dx.doi.org/10.1142/s0217984912501515.

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In this paper, we model the tick-by-tick dynamics of markets by using the continuous-time random walk (CTRW) model. We employ a sum of products of power law and stretched exponential functions for the waiting time probability distribution function; this function can fit well the waiting time distribution for BUND futures traded at LIFFE in 1997.
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Jara, M., and T. Komorowski. "Limit theorems for some continuous-time random walks." Advances in Applied Probability 43, no. 3 (September 2011): 782–813. http://dx.doi.org/10.1239/aap/1316792670.

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In this paper we consider the scaled limit of a continuous-time random walk (CTRW) based on a Markov chain {Xn,n≥ 0} and two observables, τ(∙) andV(∙), corresponding to the renewal times and jump sizes. Assuming that these observables belong to the domains of attraction of some stable laws, we give sufficient conditions on the chain that guarantee the existence of the scaled limits for CTRWs. An application of the results to a process that arises in quantum transport theory is provided. The results obtained in this paper generalize earlier results contained in Becker-Kern, Meerschaert and Scheffler (2004) and Meerschaert and Scheffler (2008), and the recent results of Henry and Straka (2011) and Jurlewicz, Kern, Meerschaert and Scheffler (2010), where {Xn,n≥ 0} is a sequence of independent and identically distributed random variables.
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Jara, M., and T. Komorowski. "Limit theorems for some continuous-time random walks." Advances in Applied Probability 43, no. 03 (September 2011): 782–813. http://dx.doi.org/10.1017/s0001867800005140.

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In this paper we consider the scaled limit of a continuous-time random walk (CTRW) based on a Markov chain {X n , n ≥ 0} and two observables, τ(∙) and V(∙), corresponding to the renewal times and jump sizes. Assuming that these observables belong to the domains of attraction of some stable laws, we give sufficient conditions on the chain that guarantee the existence of the scaled limits for CTRWs. An application of the results to a process that arises in quantum transport theory is provided. The results obtained in this paper generalize earlier results contained in Becker-Kern, Meerschaert and Scheffler (2004) and Meerschaert and Scheffler (2008), and the recent results of Henry and Straka (2011) and Jurlewicz, Kern, Meerschaert and Scheffler (2010), where {X n , n ≥ 0} is a sequence of independent and identically distributed random variables.
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AGLIARI, ELENA, OLIVER MÜLKEN, and ALEXANDER BLUMEN. "CONTINUOUS-TIME QUANTUM WALKS AND TRAPPING." International Journal of Bifurcation and Chaos 20, no. 02 (February 2010): 271–79. http://dx.doi.org/10.1142/s0218127410025715.

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Recent findings suggest that processes such as the excitonic energy transfer through the photosynthetic antenna display quantal features, aspects known from the dynamics of charge carriers along polymer backbones. Hence, in modeling energy transfer one has to leave the classical, master-equation-type formalism and advance towards an increasingly quantum-mechanical picture, while still retaining a local description of the complex network of molecules involved in the transport, say through a tight-binding approach. Interestingly, the continuous time random walk (CTRW) picture, widely employed in describing transport in random environments, can be mathematically reformulated to yield a quantum-mechanical Hamiltonian of tight-binding type; the procedure uses the mathematical analogies between time-evolution operators in statistical and in quantum mechanics: The result are continuous-time quantum walks (CTQWs). However, beyond these formal analogies, CTRWs and CTQWs display vastly different physical properties. In particular, here we focus on trapping processes on a ring and show, both analytically and numerically, that distinct configurations of traps (ranging from periodical to random) yield strongly different behaviors for the quantal mean survival probability, while classically (under ordered conditions) we always find an exponential decay at long times.
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Weron, Karina, Aleksander Stanislavsky, Agnieszka Jurlewicz, Mark M. Meerschaert, and Hans-Peter Scheffler. "Clustered continuous-time random walks: diffusion and relaxation consequences." Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences 468, no. 2142 (February 2012): 1615–28. http://dx.doi.org/10.1098/rspa.2011.0697.

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We present a class of continuous-time random walks (CTRWs), in which random jumps are separated by random waiting times. The novel feature of these CTRWs is that the jumps are clustered. This introduces a coupled effect, with longer waiting times separating larger jump clusters. We show that the CTRW scaling limits are time-changed processes. Their densities solve two different fractional diffusion equations, depending on whether the waiting time is coupled to the preceding jump, or the following one. These fractional diffusion equations can be used to model all types of experimentally observed two power-law relaxation patterns. The parameters of the scaling limit process determine the power-law exponents and loss peak frequencies.
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MAINARDI, FRANCESCO, ALESSANDRO VIVOLI, and RUDOLF GORENFLO. "CONTINUOUS TIME RANDOM WALK AND TIME FRACTIONAL DIFFUSION: A NUMERICAL COMPARISON BETWEEN THE FUNDAMENTAL SOLUTIONS." Fluctuation and Noise Letters 05, no. 02 (June 2005): L291—L297. http://dx.doi.org/10.1142/s0219477505002677.

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We consider the basic models for anomalous transport provided by the integral equation for continuous time random walk (CTRW) and by the time fractional diffusion equation to which the previous equation is known to reduce in the diffusion limit. We compare the corresponding fundamental solutions of these equations, in order to investigate numerically the increasing quality of approximation with advancing time.
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Kolokoltsov, Vassili. "CTRW modeling of quantum measurement and fractional equations of quantum stochastic filtering and control." Fractional Calculus and Applied Analysis 25, no. 1 (February 2022): 128–65. http://dx.doi.org/10.1007/s13540-021-00002-2.

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AbstractInitially developed in the framework of quantum stochastic calculus, the main equations of quantum stochastic filtering were later on derived as the limits of Markov models of discrete measurements under appropriate scaling. In many branches of modern physics it became popular to extend random walk modeling to the continuous time random walk (CTRW) modeling, where the time between discrete events is taken to be non-exponential. In the present paper we apply the CTRW modeling to the continuous quantum measurements yielding the new fractional in time evolution equations of quantum filtering and thus new fractional equations of quantum mechanics of open systems. The related quantum control problems and games turn out to be described by the fractional Hamilton-Jacobi-Bellman (HJB) equations on Riemannian manifolds. By-passing we provide a full derivation of the standard quantum filtering equations, in a modified way as compared with existing texts, which (i) provides explicit rates of convergence (that are not available via the tightness of martingales approach developed previously) and (ii) allows for the direct applications of the basic results of CTRWs to deduce the final fractional filtering equations.
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Abdel-Rehim, Enstar A. "From power laws to fractional diffusion processes with and without external forces, the non direct way." Fractional Calculus and Applied Analysis 22, no. 1 (February 25, 2019): 60–77. http://dx.doi.org/10.1515/fca-2019-0004.

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Abstract In this paper, a wide view on the theory of the continuous time random walk (CTRW) and its relations to the space–time fractional diffusion process is given. We begin from the basic model of CTRW (Montroll and Weiss [19], 1965) that also can be considered as a compound renewal process. We are interested in studying the random walks in which the probability distributions of the waiting times and jumps have fat tails characterized by power laws with exponent between 0 and 1 for the waiting times, between 0 and 2 for the jumps. We prove the relation between the integral equation of the CTRW having the above fat tails waiting and jump width distributions and the space–time fractional diffusion equations in the Laplace–Fourier domain. The space–time fractional Fokker–Planck equation could also be driven from the discrete Ehren–Fest model and is represented by the theory of CTRW. These space–time fractional diffusion processes are getting increasing popularity in applications in physics, chemistry, finance, biology, medicine and many other fields. The asymptotic behavior of the Mittag–Leffler function plays a significant rule on simulating these models. The behaviors of the studied CTRW models are well approximated and visualized by simulating various types of random walks by using the Monte Carlo method.
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Klamut, Jarosław, and Tomasz Gubiec. "Continuous Time Random Walk with Correlated Waiting Times. The Crucial Role of Inter-Trade Times in Volatility Clustering." Entropy 23, no. 12 (November 26, 2021): 1576. http://dx.doi.org/10.3390/e23121576.

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In many physical, social, and economic phenomena, we observe changes in a studied quantity only in discrete, irregularly distributed points in time. The stochastic process usually applied to describe this kind of variable is the continuous-time random walk (CTRW). Despite the popularity of these types of stochastic processes and strong empirical motivation, models with a long-term memory within the sequence of time intervals between observations are rare in the physics literature. Here, we fill this gap by introducing a new family of CTRWs. The memory is introduced to the model by assuming that many consecutive time intervals can be the same. Surprisingly, in this process we can observe a slowly decaying nonlinear autocorrelation function without a fat-tailed distribution of time intervals. Our model, applied to high-frequency stock market data, can successfully describe the slope of decay of the nonlinear autocorrelation function of stock market returns. We achieve this result without imposing any dependence between consecutive price changes. This proves the crucial role of inter-event times in the volatility clustering phenomenon observed in all stock markets.
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Dissertations / Theses on the topic "Continuous Time Random Walk (CTRW)"

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Gubiec, Tomasz, and Ryszard Kutner. "Two-step memory within Continuous Time Random Walk." Universitätsbibliothek Leipzig, 2015. http://nbn-resolving.de/urn:nbn:de:bsz:15-qucosa-183316.

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Gubiec, Tomasz, and Ryszard Kutner. "Two-step memory within Continuous Time Random Walk." Diffusion fundamentals 20 (2013) 64, S. 1, 2013. https://ul.qucosa.de/id/qucosa%3A13643.

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Chang, Qiang. "Continuous-time random-walk simulation of surface kinetics." The Ohio State University, 2007. http://rave.ohiolink.edu/etdc/view?acc_num=osu1166592142.

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Li, Chao. "Option pricing with generalized continuous time random walk models." Thesis, Queen Mary, University of London, 2016. http://qmro.qmul.ac.uk/xmlui/handle/123456789/23202.

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The pricing of options is one of the key problems in mathematical finance. In recent years, pricing models that are based on the continuous time random walk (CTRW), an anomalous diffusive random walk model widely used in physics, have been introduced. In this thesis, we investigate the pricing of European call options with CTRW and generalized CTRW models within the Black-Scholes framework. Here, the non-Markovian character of the underlying pricing model is manifest in Black-Scholes PDEs with fractional time derivatives containing memory terms. The inclusion of non-zero interest rates leads to a distinction between different types of \forward" and \backward" options, which are easily mapped onto each other in the standard Markovian framework, but exhibit significant dfferences in the non-Markovian case. The backward-type options require us in particular to include the multi-point statistics of the non-Markovian pricing model. Using a representation of the CTRW in terms of a subordination (time change) of a normal diffusive process with an inverse L evy-stable process, analytical results can be obtained. The extension of the formalism to arbitrary waiting time distributions and general payoff functions is discussed. The pricing of path-dependent Asian options leads to further distinctions between different variants of the subordination. We obtain analytical results that relate the option price to the solution of generalized Feynman-Kac equations containing non-local time derivatives such as the fractional substantial derivative. Results for L evy-stable and tempered L evy-stable subordinators, power options, arithmetic and geometric Asian options are presented.
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Niemann, Markus. "From Anomalous Deterministic Diffusion to the Continuous-Time Random Walk." Wuppertal Universitätsbibliothek Wuppertal, 2010. http://d-nb.info/1000127621/34.

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Niemann, Markus [Verfasser]. "From Anomalous Deterministic Diffusion to the Continuous-Time Random Walk / Markus Niemann." Wuppertal : Universitätsbibliothek Wuppertal, 2010. http://d-nb.info/1000127621/34.

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Helfferich, Julian [Verfasser], and Alexander [Akademischer Betreuer] Blumen. "Glass dynamics in the continuous-time random walk framework = Glasdynamik als Zufallsprozess." Freiburg : Universität, 2015. http://d-nb.info/1125885513/34.

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Allen, Andrew. "A Random Walk Version of Robbins' Problem." Thesis, University of North Texas, 2018. https://digital.library.unt.edu/ark:/67531/metadc1404568/.

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Robbins' problem is an optimal stopping problem where one seeks to minimize the expected rank of their observations among all observations. We examine random walk analogs to Robbins' problem in both discrete and continuous time. In discrete time, we consider full information and relative ranks versions of this problem. For three step walks, we give the optimal stopping rule and the expected rank for both versions. We also give asymptotic upper bounds for the expected rank in discrete time. Finally, we give upper and lower bounds for the expected rank in continuous time, and we show that the expected rank in the continuous time problem is at least as large as the normalized asymptotic expected rank in the full information discrete time version.
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Pöschke, Patrick. "Influence of Molecular Diffusion on the Transport of Passive Tracers in 2D Laminar Flows." Doctoral thesis, Humboldt-Universität zu Berlin, 2018. http://dx.doi.org/10.18452/19526.

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In dieser Arbeit betrachten wir das Strömungs-Diffusions-(Reaktions)-Problem für passive Markerteilchen, die in zweidimensionalen laminaren Strömungsmustern mit geringem thermischem Rauschen gelöst sind. Der deterministische Fluss umfasst Zellen in Form von Quadraten oder Katzenaugen. In ihnen tritt Rotationsbewegung auf. Einige der Strömungen bestehen aus wellenförmigen Bereichen mit gerader Vorwärtsbewegung. Alle Systeme sind entweder periodisch oder durch Wände begrenzt. Eine untersuchte Familie von Strömungen interpoliert kontinuierlich zwischen Reihen von Wirbeln und Scherflüssen. Wir analysieren zahlreiche numerische Simulationen, die bisherige theoretische Vorhersagen bestätigen und neue Phänomene offenbaren. Ohne Rauschen sind die Teilchen in einzelnen Bestandteilen des Flusses für immer gefangen. Durch Hinzufügen von schwachem thermischen Rauschen wird die normale Diffusion für lange Zeiten stark verstärkt und führt zu verschiedenen Diffusionsarten für mittlere Zeiten. Mit Continuous-Time-Random-Walk-Modellen leiten wir analytische Ausdrücke in Übereinstimmung mit den numerischen Ergebnissen her, die je nach Parametern, Anfangsbedingungen und Alterungszeiten von subdiffusiver bis superballistischer anomaler Diffusion für mittlere Zeiten reichen. Wir sehen deutlich, dass einige der früheren Vorhersagen nur für Teilchen gelten, die an der Separatrix des Flusses starten - der einzige Fall, der in der Vergangenheit ausführlich betrachtet wurde - und dass das System zu vollkommen anderem Verhalten in anderen Situationen führen kann, einschließlich einem Schwingenden beim Start im Zentrum einesWirbels nach einer gewissen Alterungszeit. Darüber hinaus enthüllen die Simulationen, dass Teilchenreaktionen dort häufiger auftreten, wo sich die Geschwindigkeit der Strömung stark ändert, was dazu führt, dass langsame Teilchen von schnelleren getroffen werden, die ihnen folgen. Die umfangreichen numerischen Simulationen, die für diese Arbeit durchgeführt wurden, mussten jetzt durchgeführt werden, da wir die Rechenleistung dafür besitzen.
In this thesis, we consider the advection-diffusion-(reaction) problem for passive tracer particles suspended in two-dimensional laminar flow patterns with small thermal noise. The deterministic flow comprises cells in the shape of either squares or cat’s eyes. Rotational motion occurs inside them. Some of the flows consist of sinusoidal regions of straight forward motion. All systems are either periodic or are bounded by walls. One examined family of flows continuously interpolates between arrays of eddies and shear flows. We analyse extensive numerical simulations, which confirm previous theoretical predictions as well as reveal new phenomena. Without noise, particles are trapped forever in single building blocks of the flow. Adding small thermal noise, leads to largely enhanced normal diffusion for long times and several kinds of diffusion for intermediate times. Using continuous time random walk models, we derive analytical expressions in accordance with numerical results, ranging from subdiffusive to superballistic anomalous diffusion for intermediate times depending on parameters, initial conditions and aging time. We clearly see, that some of the previous predictions are only true for particles starting at the separatrix of the flow - the only case considered in depth in the past - and that the system might show a vastly different behavior in other situations, including an oscillatory one, when starting in the center of an eddy after a certain aging time. Furthermore, simulations reveal that particle reactions occur more frequently at positions where the velocity of the flow changes the most, resulting in slow particles being hit by faster ones following them. The extensive numerical simulations performed for this thesis had to be done now that we have the computational means to do so. Machines are powerful tools in order to gain a deeper and more detailed insight into the dynamics of many complicated dynamical and stochastic systems.
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Puyguiraud, Alexandre. "Upscaling transport in heterogeneous media : from pore to Darcy scale through Continuous Time Random Walks." Thesis, Montpellier, 2019. http://www.theses.fr/2019MONTG016/document.

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Les mécanismes responsables du transport hydrodynamique anormal (non-Fickéen) peuvent être rattachés à la complexité de la géométrie du milieu à l'échelle des pores. Dans cette thèse, nous étudions la dynamique des vitesses de particules à l'échelle des pores. À l'aide de simulations de suivi de particules effectuées sur un échantillon numérisé de grès de Berea, nous présentons une analyse détaillée de l'évolution Lagrangienne et Eulérienne et de leur dépendance aux conditions initiales. Le long de leur ligne de courant, la vitesse des particules montre un signal intermittent complexe, alors que leur sériede vitesses spatiales présente des fluctuations régulières. La distribution spatiale des vitesses des particules converge rapidementvers l'état stationnaire. Ces résultats dénotent un processus Markovienqui permet de prédire les fluctuations de vitesse dans le réseau poral.Ces processus, associés à la tortuosité et à la distance de corrélation de vitesse permettent de paramétrer un modèle de marche aléatoire dans le temps (CTRW) et de réaliser le changement d’échelle pour simuler le transport à l’échelle de Darcy. Le modèle, comme tout modèle issu d’un changement d'échelle, repose sur la définition d'un volume élémentaire représentatif (VER). Nous montrons qu’un VER basé sur les statistiques de vitesse permet de définir un support pertinent pour la modélisation du transport hydrodynamique pré-asymptotique à asymptotique, et ainsi d’éviter les limitations associées à l’équation d’advection-dispersionFickéenne. Cette approche est utilisée pour étudier l’impact de l’hétérogénéité du réseau poral sur le volume de mélange et la masse du produit d’une réaction bimoléculaire
The mechanisms responsible for anomalous (non-Fickian) hydrodynamictransport can be traced back to the complexity of the medium geometry atthe pore-scale. In this thesis, we investigate the dynamics of pore-scaleparticle velocities. Using particle tracking simulations performed on adigitized Berea sandstone sample, we present a detailed analysis of theevolution of the Lagrangian and Eulerian evolution and their dependenceon the initial conditions. The particles experience a complexintermittent temporal velocity signal along their streamline while theirspatial velocity series exhibit regular fluctuations. The spatialvelocity distribution of the particles converges quickly to thesteady-state. These results lead naturally to Markov processes for theprediction of these velocity series.These processes, together with the tortuosity and the velocitycorrelation distance that are properties of the medium, allow theparameterization of a continuous time random walk (CTRW) for theupscaling of the transport. The model, like any upscaled model, relieson the definition of a representative elementary volume (REV). We showthat an REV based on the velocity statistics allows defining a pertinentsupport for modeling pre-asymptotic to asymptotic hydrodynamictransport at Darcy scale using, for instance, CTRW, thus overcomingthe limitations associated with the Fickian advection dispersionequation. Finally, we investigate the impact of pore-scale heterogeneityon a bimolecular reaction and explore a methodology for the predictionof the mixing volume and the chemical mass produced
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Book chapters on the topic "Continuous Time Random Walk (CTRW)"

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Jin, Bangti. "Continuous Time Random Walk." In Fractional Differential Equations, 3–18. Cham: Springer International Publishing, 2021. http://dx.doi.org/10.1007/978-3-030-76043-4_1.

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Schinazi, Rinaldo B. "Continuous Time Branching Random Walk." In Classical and Spatial Stochastic Processes, 135–52. Boston, MA: Birkhäuser Boston, 1999. http://dx.doi.org/10.1007/978-1-4612-1582-0_6.

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Grigolini, Paolo. "The Continuous-Time Random Walk Versus the Generalized Master Equation." In Fractals, Diffusion, and Relaxation in Disordered Complex Systems, 357–474. Hoboken, NJ, USA: John Wiley & Sons, Inc., 2005. http://dx.doi.org/10.1002/0471790265.ch5.

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Gorenflo, Rudolf, and Francesco Mainardi. "Fractional diffusion Processes: Probability Distributions and Continuous Time Random Walk." In Processes with Long-Range Correlations, 148–66. Berlin, Heidelberg: Springer Berlin Heidelberg, 2003. http://dx.doi.org/10.1007/3-540-44832-2_8.

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Hadian Rasanan, Amir Hosein, Mohammad Mahdi Moayeri, Jamal Amani Rad, and Kourosh Parand. "From Continuous Time Random Walk Models to Human Decision-Making Modelling." In Mathematical Methods in Dynamical Systems, 239–72. Boca Raton: CRC Press, 2023. http://dx.doi.org/10.1201/9781003328032-9.

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Sposini, Vittoria, Silvia Vitali, Paolo Paradisi, and Gianni Pagnini. "Fractional Diffusion and Medium Heterogeneity: The Case of the Continuous Time Random Walk." In SEMA SIMAI Springer Series, 275–86. Cham: Springer International Publishing, 2021. http://dx.doi.org/10.1007/978-3-030-69236-0_14.

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Caffarelli, Luis, and Luis Silvestre. "Hölder Regularity for Generalized Master Equations with Rough Kernels." In Advances in Analysis, edited by Charles Fefferman, Alexandru D. Ionescu, D. H. Phong, Stephen Wainger, Charles Fefferman, Alexandru D. Ionescu, D. H. Phong, and Stephen Wainger. Princeton University Press, 2014. http://dx.doi.org/10.23943/princeton/9780691159416.003.0004.

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This chapter studies evolution problems that are related to continuous time random walks (CTRW), having a discontinuous path for which both the jumps and the time elapsed in between them are random. These processes are governed by a generalized master equation which is nonlocal both in space and time. To illustrate, the chapter considers kernels K(t, x, s, y) in a particular function. Here, studying correlated kernels provides a more flexible framework where more interesting physical phenomena can be observed, and more subtle mathematical questions appear. The regularity estimates are in fact more interesting (harder mathematically) when the jumps in space and the waiting times are strongly correlated.
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"Continuous Time Random Walk model." In Langevin and Fokker–Planck Equations and their Generalizations, 117–35. WORLD SCIENTIFIC, 2018. http://dx.doi.org/10.1142/9789813228412_0006.

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"Quantum Continuous Time Random Walk Model." In Diffusion, 243–54. CRC Press, 2013. http://dx.doi.org/10.1201/b16008-18.

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Fallahgoul, Hasan A., Sergio M. Focardi, and Frank J. Fabozzi. "Continuous-Time Random Walk and Fractional Calculus." In Fractional Calculus and Fractional Processes with Applications to Financial Economics, 81–90. Elsevier, 2017. http://dx.doi.org/10.1016/b978-0-12-804248-9.50007-3.

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Conference papers on the topic "Continuous Time Random Walk (CTRW)"

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Cui, Jie, HongGuang Sun, Ailian Chang, and Xu Zhang. "A Matlab Toolbox for Particle Transport Simulation in Fractal Media." In ASME 2015 International Design Engineering Technical Conferences and Computers and Information in Engineering Conference. American Society of Mechanical Engineers, 2015. http://dx.doi.org/10.1115/detc2015-47186.

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Fractal property has been verified in the research of many kinds of complex media, such as soil, aquifer and concrete. Particle transport in fractal media often does not obey the classical Fickian law, and exhibits anomalous diffusion feature. Several promising physical models have been proposed to describe this kind of anomalous transport in the recent decades. This study will introduce a new Matlab toolbox to investigate three approaches including fractal theory, continuous time random walk (CTRW) model and fractional derivative diffusion equation model on characterizing anomalous transport.
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Capes, H., M. Christova, D. Boland, A. Bouzaher, F. Catoire, L. Godbert-Mouret, M. Koubiti, et al. "Modeling of Line Shapes using Continuous Time Random Walk Theory." In APPLICATION OF MATHEMATICS IN TECHNICAL AND NATURAL SCIENCES: Proceedings of the 2nd International Conference. AIP, 2010. http://dx.doi.org/10.1063/1.3526606.

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GORENFLO, R., F. MAINARDI, and A. VIVOLI. "SUBORDINATION IN FRACTIONAL DIFFUSION PROCESSES VIA CONTINUOUS TIME RANDOM WALK." In Proceedings of the 5th International ISAAC Congress. WORLD SCIENTIFIC, 2009. http://dx.doi.org/10.1142/9789812835635_0043.

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Kang, Kang, Elsayed Abdelfatah, Maysam Pournik, Bor Jier Shiau, and Jeffrey Harwell. "Multiscale Modeling of Carbonate Acidizing Using Continuous Time Random Walk Approach." In SPE Kuwait Oil & Gas Show and Conference. Society of Petroleum Engineers, 2017. http://dx.doi.org/10.2118/187541-ms.

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Capes, H., M. Christova, D. Boland, F. Catoire, L. Godbert-Mouret, M. Koubiti, A. Mekkaoui, et al. "Revisiting the Stark Broadening by fluctuating electric fields using the Continuous Time Random Walk Theory." In 20TH INTERNATIONAL CONFERENCE ON SPECTRAL LINE SHAPES. AIP, 2010. http://dx.doi.org/10.1063/1.3517538.

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Paekivi, S., R. Mankin, and A. Rekker. "Interspike interval distribution for a continuous-time random walk model of neurons in the diffusion limit." In APPLICATION OF MATHEMATICS IN TECHNICAL AND NATURAL SCIENCES: 10th International Conference for Promoting the Application of Mathematics in Technical and Natural Sciences - AMiTaNS’18. Author(s), 2018. http://dx.doi.org/10.1063/1.5064927.

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Packwood, Daniel M. "Phase relaxation in slowly changing environments: Evaluation of the Kubo-Anderson model for a continuous-time random walk." In 4TH INTERNATIONAL SYMPOSIUM ON SLOW DYNAMICS IN COMPLEX SYSTEMS: Keep Going Tohoku. American Institute of Physics, 2013. http://dx.doi.org/10.1063/1.4794620.

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Wang, Yan. "Accelerating Stochastic Dynamics Simulation With Continuous-Time Quantum Walks." In ASME 2016 International Design Engineering Technical Conferences and Computers and Information in Engineering Conference. American Society of Mechanical Engineers, 2016. http://dx.doi.org/10.1115/detc2016-59420.

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Abstract:
Stochastic diffusion is a general phenomenon observed in various national and engineering systems. It is typically modeled by either stochastic differential equation (SDE) or Fokker-Planck equation (FPE), which are equivalent approaches. Path integral is an accurate and effective method to solve FPEs. Yet, computational efficiency is the common challenge for path integral and other numerical methods, include time and space complexities. Previously, one-dimensional continuous-time quantum walk was used to simulate diffusion. By combining quantum diffusion and random diffusion, the new approach can accelerate the simulation with longer time steps than those in path integral. It was demonstrated that simulation can be dozens or even hundreds of times faster. In this paper, a new generic quantum operator is proposed to simulate drift-diffusion processes in high-dimensional space, which combines quantum walks on graphs with traditional path integral approaches. Probability amplitudes are computed efficiently by spectral analysis. The efficiency of the new method is demonstrated with stochastic resonance problems.
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Vadgama, Nikul, Marios Kapsis, Peter Forsyth, Matthew McGilvray, and David R. H. Gillespie. "Development and Validation of a Continuous Random Walk Model for Particle Tracking in Accelerating Flows." In ASME Turbo Expo 2020: Turbomachinery Technical Conference and Exposition. American Society of Mechanical Engineers, 2020. http://dx.doi.org/10.1115/gt2020-16026.

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Abstract Stochastic particle tracking models coupled to RANS fluid simulations are frequently used to simulate particulate transport and hence predict component damage in gas turbines. In simple flows the Continuous Random Walk (CRW) model has been shown to model particulate motion in the diffusion-impaction regime significantly more accurately than Discrete Random Walk implementations. To date, the CRW model has used turbulent flow statistics determined from DNS in channels and experiments in pipes. Robust extension of the CRW model to accelerating flows modelled using RANS is important to enable its use in design studies of rotating engine-realistic geometries of complex curvature. This paper builds on previous work by the authors to use turbulent statistics in the CRW model directly from Reynolds Stress Models (RSM) in RANS simulations. Further improvements are made to this technique to account for strong gradients in Reynolds Stresses in all directions; improve the robustness of the model to the chosen time-step; and to eliminate the need for DNS/experimentally derived statistical flow properties. The effect of these changes were studied using a commercial CFD solver for a simple pipe flow, for which integral deposition prediction accuracy equal to that using the original CRW was achieved. These changes enable the CRW to be applied to more complex flow cases. To demonstrate why this development is important, in a more complex flow case with acceleration, deposition in a turbulent 90° bend was investigated. Critical differences in the predicted deposition are apparent when the results are compared to the alternative tracking models suitable for RANS solutions. The modified CRW model was the only model which captured the more complex deposition distribution, as predicted by published LES studies. Particle tracking models need to be accurate in the spatial distribution of deposition they predict in order to enable more sophisticated engineering design studies.
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Forsyth, Peter, David R. H. Gillespie, Matthew McGilvray, and Vincent Galoul. "Validation and Assessment of the Continuous Random Walk Model for Particle Deposition in Gas Turbine Engines." In ASME Turbo Expo 2016: Turbomachinery Technical Conference and Exposition. American Society of Mechanical Engineers, 2016. http://dx.doi.org/10.1115/gt2016-57332.

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Threats to engine integrity and life from deposition of environmental particulates that can reach the turbine cooling systems (i.e. <10 micron) have become increasing important within the aero-engine industry, with an increase of flight paths crossing sandy, tropical storm-infested, or polluted airspaces. This has led to studies in the turbomachinery community investigating environmental particulate deposition, largely applying the Discrete Random Walk (DRW) model in CFD simulations of air paths. However, this model was conceived to model droplet dispersion in bulk flow regimes, and therefore has fundamental limitations for deposition studies. One significant limitation is an insensitivity to particle size in the turbulent deposition size regime, where deposition is strongly linked to particle size. This is highlighted within this study through comparisons to published experimental data. Progress made within the wider particulate deposition community has recently led to the development and application of the Continuous Random Walk (CRW) model. This new model provides significantly improved predictions of particle deposition seen experimentally in comparison to the DRW for low temperature pipe flow experiments. However, the CRW model is not without its difficulties. This paper highlights the sensitivities within the CRW model and actions taken to alleviate them where possible. For validation of the model at gas turbine conditions, it should be assessed at engine-representative conditions. These include high-temperature and swirling flows, with thermophoretic and wall-roughness effects. Thermophoresis is a particle force experienced in the negative direction of the temperature gradient, and can strongly effect deposition efficiency from certain flows. Previous validation of the model has centred on low temperatures and pipe flow conditions. Presented here is the validation process which is currently being undertaken to assess the model at gas turbine-relevant conditions. Discussion centres on the underlying principles of the model, how to apply this model appropriately to gas turbine flows and initial assessment for flows seen in secondary air systems. Verification of model assumptions is undertaken, including demonstrating that the effect of boundary layer modelling of anisotropic turbulence is shown to be Reynolds-independent. The integration time step for numerical solution of the non-dimensional Langevin equation is redefined, showing improvement against existing definitions for the available low temperature pipe flow data. The grid dependence of particle deposition in numerical simulations is presented and shown to be more significant for particle conditions in the diffusional deposition regime. Finally, the model is applied to an engine-representative geometry to demonstrate the improvement in sensitivity to particle size that the CRW offers over the DRW for wall-bounded flows.
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