Stochastic Processes and Applications

Pavliotis, Grigorios A.

doi:10.1007/978-1-4939-1323-7

Cited by 475 publications

(385 citation statements)

References 170 publications

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“…As in the original paper [34], we set N = 40. We must note that the right-hand side of (6.1) does not satisfy the requirements imposed on (1.9) in Section 1; observe that the deterministic part of the right-hand side of (6.1) is neither bounded nor even uniformly Lipschitz in R N , which means that the existence of strong solutions to (6.1) is not guaranteed [19,40] • Spin-up time window (time skipped between the initial condition and the beginning of the time averaging window): T skip =10000 time units; • Initial condition: each initial state x i , 1 ≤ i ≤ N, is generated at random using normal distribution with zero mean and unit standard deviation. In Figure 1 we show the histograms of the probability density functions (PDFs), computed by the standard bin-counting, as well as the simplest time-lag autocorrelation functions of the solution with itself, the latter computed numerically as…”

Section: The Rescaled Lorenz 96 Modelmentioning

confidence: 99%

Leading Order Response of Statistical Averages of a Dynamical System to Small Stochastic Perturbations

Abramov

2017

J Stat Phys

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Abstract. The classical fluctuation-dissipation theorem predicts the average response of a dynamical system to an external deterministic perturbation via time-lagged statistical correlation functions of the corresponding unperturbed system. In this work we develop a fluctuation-response theory and test a computational framework for the leading order response of statistical averages of a deterministic or stochastic dynamical system to an external stochastic perturbation. In the case of a stochastic unperturbed dynamical system, we compute the leading order fluctuation-response formulas for two different cases: when the existing stochastic term is perturbed, and when a new, statistically independent, stochastic perturbation is introduced. We numerically investigate the effectiveness of the new response formulas for an appropriately rescaled Lorenz 96 system, in both the deterministic and stochastic unperturbed dynamical regimes.

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Section: The Rescaled Lorenz 96 Modelmentioning

confidence: 99%

Leading Order Response of Statistical Averages of a Dynamical System to Small Stochastic Perturbations

Abramov

2017

J Stat Phys

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“…For exit rate computations it is only needed to know how long it would take until a molecular process starting in a point x = x 0 ∈ M (of some subset M ⊂ Ω) leaves this subset. In particular it is asked for the exit rate, the holding probability, and the mean holding time of the set M. The connection between these values and operator theory is very well explained by Pavliotis in [35]. As an example, M ⊂ Ω could define the 'bound' macro-state of a ligand-receptor-system.…”

Section: Exit Ratesmentioning

confidence: 99%

Implications of PCCA+ in Molecular Simulation

Weber

2018

Computation

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Upon ligand binding or during chemical reactions the state of a molecular system changes in time. Usually we consider a finite set of (macro-) states of the system (e.g., 'bound' vs. 'unbound'), although the process itself takes place in a continuous space. In this context, the formula χ = XA connects the micro-dynamics of the molecular system to its macro-dynamics. χ can be understood as a clustering of micro-states of a molecular system into a few macro-states. X is a basis of an invariant subspace of a transfer operator describing the micro-dynamics of the system. The formula claims that there is an unknown linear relation A between these two objects. With the aid of this formula we can understand rebinding effects, the electron flux in pericyclic reactions, and systematic changes of binding rates in kinetic ITC experiments. We can also analyze sequential spectroscopy experiments and rare event systems more easily. This article provides an explanation of the formula and an overview of some of its consequences.Keywords: Robust Perron Cluster Analysis; membership function; invariant subspace. MSC: 60J22 The Foundation of PCCA+Molecular simulation is used as a tool to estimate thermodynamical properties. Moreover, it could be used for investigation of molecular processes and their time scales. In computational drug design, e.g., binding affinities of ligand-receptor systems are estimated. In this case, trajectories in a 3N-dimensional space Ω of molecular conformations (N is the number of atoms) are generated. They will be denoted as micro-states x ∈ Ω in the following. After simulation the micro-states are classified as "bound" or "unbound". This classification can be achieved by projecting the state space Ω onto a low number n of macro-states of the system. Spectral clustering algorithms are one possible method in this context. Robust Perron Cluster Analysis (PCCA+) turned out to be a commonly used practical tool for this algorithmic step [1].Conformation Dynamics has been established in the late 1990s. From molecular simulations [2] transition matrices P ∈ R m×m were derived. The entries of P represent conditional transition probabilities between pre-defined m subsets of the state space Ω of a molecular process. Today P would be denoted as Markov State Model. According to the transfer operator theory and PCCA [3] the eigenvectors of these matrices were studied in the early 2000s leading to an empirical observation [4] and Figure 1: Given a matrix X ∈ R m×n of the n m leading eigenvectors of the m × m-transition matrix P, the m rows of X -plotted as n-dimensional points -seem to always form an (n − 1)-simplex. Some of those m points are located at the vertices of the simplices, other points are located on the edges. The points which are located at the n vertices of these simplices represent the thermodynamically stable conformations of the molecular systems. Whereas, the points which are located on the edges of the simplices represent transition regions. In order to transform the (n − 1)-simplex σ 1 ...

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“…converges as n f → ∞ to a normal distribution with mean 0 and standard deviation σ(·, x, t)/ √ n f , where σ(·, x, t) denotes the standard deviation of H(Y(x, t), N (·, x, t)) [50,53].…”

Section: Mixing On the Molecular Scalementioning

confidence: 99%

“…On the other hand, for consistency with the Itô interpretation of the stochastic integral, we apply an Euler-Maruyama discretization to the stochastic terms in Eqs. (62) and (65) [50]. The micromixing fractional step (Eq.…”

Section: Numerical Solution Schemementioning

confidence: 99%

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An LES-PBE-PDF approach for modeling particle formation in turbulent reacting flows

Sewerin

Rigopoulos

2017

Physics of Fluids

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Many chemical and environmental processes involve the formation of a polydispersed particulate phase in a turbulent carrier flow. Frequently, the immersed particles are characterized by an intrinsic property such as the particle size and the distribution of this property across a sample population is taken as an indicator for the quality of the particulate product or its environmental impact.In the present article, we propose a comprehensive model and an efficient numerical solution scheme for predicting the evolution of the property distribution associated with a polydispersed particulate phase forming in a turbulent reacting flow. Here, the particulate phase is described in terms of the particle number density whose evolution in both physical and particle property space is governed by the population balance equation (PBE). Based on the concept of large eddy simulation (LES), we augment the existing LES-transported PDF approach for fluid phase scalars by the particle number density and obtain a modelled evolution equation for the filtered probability density function (PDF) associated with the instantaneous fluid composition and particle property distribution. This LES-PBE-PDF approach allows us to predict the LES-filtered fluid composition and particle property distribution at each spatial location and point in time without any restriction on the chemical or particle formation kinetics. In view of a numerical solution, we apply the method of Eulerian stochastic fields, invoking an explicit adaptive grid technique in order to discretize the stochastic field equation for the number density in particle property space. In this way, sharp moving features of the particle property distribution can be accurately resolved at a significantly reduced computational cost.As a test case, we consider the condensation of an aerosol in a developed turbulent mixing layer. Our investigation not only demonstrates the predictive capabilities of the LES-PBE-PDF model, but also indicates the computational efficiency of the numerical solution scheme.

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Stochastic Processes and Applications

Cited by 475 publications

References 170 publications

Leading Order Response of Statistical Averages of a Dynamical System to Small Stochastic Perturbations

Leading Order Response of Statistical Averages of a Dynamical System to Small Stochastic Perturbations

Implications of PCCA+ in Molecular Simulation

An LES-PBE-PDF approach for modeling particle formation in turbulent reacting flows

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