Quantification of Model-Form Uncertainty in Drift-Diffusion Simulation Using Fractional Derivatives

2015 ◽  
Author(s):  
Yan Wang
Keyword(s):  
Diffusion Processes ◽  
Planck Equation ◽  
Optimization Procedure ◽  
Physical World ◽  
Drift Diffusion ◽  
Long Range Correlations ◽  
Modeling Process ◽  
Model Form ◽  
Generic Description ◽  
Non Gaussian

In modeling and simulation, model-form uncertainty arises from the lack of knowledge and simplification during modeling process and numerical treatment for ease of computation. Traditional uncertainty quantification approaches are based on assumptions of stochasticity in real, reciprocal, or functional spaces to make them computationally tractable. This makes the prediction of important quantities of interest such as rare events difficult. In this paper, a new approach to capture model-form uncertainty is proposed. It is based on fractional calculus, and its flexibility allows us to model a family of non-Gaussian processes, which provides a more generic description of the physical world. A generalized fractional Fokker-Planck equation (fFPE) is proposed to describe the drift-diffusion processes under long-range correlations and memory effects. A new model calibration approach based on the maximum accumulative mutual information is also proposed to reduce model-form uncertainty, where an optimization procedure is taken.

Model-Form Calibration in Drift-Diffusion Simulation Using Fractional Derivatives

2016 ◽  
Vol 2 (3) ◽  
Author(s):  
Yan Wang
Keyword(s):  
Diffusion Processes ◽  
Planck Equation ◽  
Optimization Procedure ◽  
Physical World ◽  
Drift Diffusion ◽  
Long Range Correlations ◽  
Model Form ◽  
Maximum Mutual Information ◽  
Generic Description ◽  
Non Gaussian

In modeling and simulation, model-form uncertainty arises from the lack of knowledge and simplification during the modeling process and numerical treatment for ease of computation. Traditional uncertainty quantification (UQ) approaches are based on assumptions of stochasticity in real, reciprocal, or functional spaces to make them computationally tractable. This makes the prediction of important quantities of interest, such as rare events, difficult. In this paper, a new approach to capture model-form uncertainty is proposed. It is based on fractional calculus, and its flexibility allows us to model a family of non-Gaussian processes, which provides a more generic description of the physical world. A generalized fractional Fokker–Planck equation (fFPE) is used to describe the drift-diffusion processes under long-range correlations and memory effects. A new model-calibration approach based on the maximum mutual information is proposed to reduce model-form uncertainty, where an optimization procedure is taken.

Simulating Drift-Diffusion Processes With Generalized Interval Probability

2012 ◽  
Author(s):  
Yan Wang
Keyword(s):  
Path Integral ◽  
Stochastic Systems ◽  
Diffusion Processes ◽  
Epistemic Uncertainty ◽  
Planck Equation ◽  
Integral Approach ◽  
Drift Diffusion ◽  
Interval Probability ◽  
Fokker Planck Equation ◽  
Fokker Planck

The Fokker-Planck equation is widely used to describe the time evolution of stochastic systems in drift-diffusion processes. Yet, it does not differentiate two types of uncertainties: aleatory uncertainty that is inherent randomness and epistemic uncertainty due to lack of perfect knowledge. In this paper, a generalized Fokker-Planck equation based on a new generalized interval probability theory is proposed to describe drift-diffusion processes under both uncertainties, where epistemic uncertainty is modeled by the generalized interval while the aleatory one is by the probability measure. A path integral approach is developed to numerically solve the generalized Fokker-Planck equation. The resulted interval-valued probability density functions rigorously bound the real-valued ones computed from the classical path integral method. The new approach is demonstrated by numerical examples.

Governing equations for probability densities of Marcus stochastic differential equations with Lévy noise

2016 ◽  
Vol 17 (05) ◽  
pp. 1750033 ◽  
Author(s):  
Xu Sun ◽  
Xiaofan Li ◽  
Yayun Zheng
Keyword(s):  
Differential Equations ◽  
Stochastic Differential Equations ◽  
Probability Density ◽  
Lévy Processes ◽  
Planck Equation ◽  
Levy Processes ◽  
Stochastic Dynamical Systems ◽  
Non Gaussian ◽  
Fokker Planck Equations ◽  
Fokker Planck

Marcus stochastic differential equations (SDEs) often are appropriate models for stochastic dynamical systems driven by non-Gaussian Lévy processes and have wide applications in engineering and physical sciences. The probability density of the solution to an SDE offers complete statistical information on the underlying stochastic process. Explicit formula for the Fokker–Planck equation, the governing equation for the probability density, is well-known when the SDE is driven by a Brownian motion. In this paper, we address the open question of finding the Fokker–Planck equations for Marcus SDEs in arbitrary dimensions driven by non-Gaussian Lévy processes. The equations are given in a simple form that facilitates theoretical analysis and numerical computation. Several examples are presented to illustrate how the theoretical results can be applied to obtain Fokker–Planck equations for Marcus SDEs driven by Lévy processes.

Nonextensive Effects in Hamiltonian Systems

2004 ◽  
Author(s):  
Andrea Rapisarda ◽  
Vito Latora
Keyword(s):  
Statistical Mechanics ◽  
Thermodynamic Formalism ◽  
Mean Field ◽  
Fractal Structures ◽  
Range Interaction ◽  
Equilibrium Statistical Mechanics ◽  
Long Range Correlations ◽  
Equilibrium Dynamics ◽  
Nonextensive Thermodynamics ◽  
Non Gaussian

The Boltzmann-Gibbs formulation of equilibrium statistical mechanics depends crucially on the nature of the Hamiltonian of the JV-body system under study, but this fact is clearly stated only in the introductions of textbooks and, in general, it is very soon neglected. In particular, the very same basic postulate of equilibrium statistical mechanics, the famous Boltzmann principle S = k log W of the microcanonical ensemble, assumes that dynamics can be automatically an easily taken into account, although this is not always justified, as Einstein himself realized [20]. On the other hand, the Boltzmann-Gibbs canonical ensemble is valid only for sufficiently short-range interactions and does not necessarily apply, for example, to gravitational or unscreened Colombian fields for which the usually assumed entropy extensivity postulate is not valid [5]. In 1988, Constantino Tsallis proposed a generalized thermostatistics formalism based on a nonextensive entropic form [24]. Since then, this new theory has been encountering an increasing number of successful applications in different fields (for some recent examples see Abe and Suzuki [1], Baldovin and Robledo [4], Beck et al. [8], Kaniadakis et al. [12], Latora et al. [16], and Tsallis et al. [25]) and seems to be the best candidate for a generalized thermodynamic formalism which should be valid when nonextensivity, long-range correlations, and fractal structures in phase space cannot be neglected: in other words, when the dynamics play a nontrivial role [11] and fluctuations are quite large and non-Gaussian [6, 7, 8, 24, 26]. In this contribution we consider a nonextensive JV-body classical Hamiltonian system, with infinite range interaction, the so-called Hamiltonian mean field (HMF) model, which has been intensively studied in the last several years [3, 13, 14, 15, 17, 18, 19]. The out-of-equilibrium dynamics of the model exhibits a series of anomalies like negative specific heat, metastable states, vanishing Lyapunov exponents, and non-Gaussian velocity distributions. After a brief overview of these anomalies, we show how they can be interpreted in terms of nonextensive thermodynamics according to the present understanding.

On the solution of the Fokker–Planck equation for a Feller process

1990 ◽  
Vol 22 (01) ◽  
pp. 101-110
Author(s):  
L. Sacerdote
Keyword(s):  
Diffusion Processes ◽  
Planck Equation ◽  
Hyperbolic Type ◽  
Diffusion Equations ◽  
Time Varying ◽  
Feller Process ◽  
Parameter Group ◽  
One Dimensional ◽  
Dimensional Diffusion ◽  
Type Boundary

Use of one-parameter group transformations is made to obtain the transition p.d.f. of a Feller process confined between the origin and a hyperbolic-type boundary. Such a procedure, previously used by Bluman and Cole (cf., for instance, [4]), although useful for dealing with one-dimensional diffusion processes restricted between time-varying boundaries, does not appear to have been sufficiently exploited to obtain solutions to the diffusion equations associated to continuous Markov processes.

A resonant test-field model of gravity waves

1983 ◽  
Vol 132 ◽  
pp. 417-430 ◽  
Author(s):  
Bruce J. West
Keyword(s):  
Steady State ◽  
Gravity Waves ◽  
Planck Equation ◽  
Steady State Solution ◽  
Test Field ◽  
Statistical Fluctuations ◽  
Resonant Interactions ◽  
Energy Spectral Density ◽  
Non Gaussian ◽  
Turbulent Wind Field

In this paper we propose an ‘irreversible’ resonant test-field (RTF) model to describe the statistical fluctuations of gravity waves on deep water driven by a turbulent wind field. The non-resonant interactions in the gravity-wave Hamiltonian are replaced by a Markov process in the equation of motion for the resonantly interacting gravity waves, i.e. Hamilton's equations are replaced by a Langevin equation for the RTF waves. The RTF models the irreversible energy-transfer process by a Fokker-Planck equation for the phase-space probability density, the exact steady-state solution of which is determined to be non-Gaussian. An H-theorem for the RTF predicts the monotonic approach to the asymptotic steady state near which the transport properties of the field are studied. The steady-state energy-spectral density is calculated (in some approximation) to be k−4.

Modeling Soft Breakdown of Ultra-Thin Gate Oxide Layers

1999 ◽  
Vol 567 ◽  
Author(s):  
Michel Houssa ◽  
P.W. Mertens ◽  
M.M. Heyns
Keyword(s):  
Gate Voltage ◽  
Dielectric Breakdown ◽  
Percolation Cluster ◽  
Gate Oxide ◽  
Oxide Layers ◽  
Long Range Correlations ◽  
Electrical Stress ◽  
Non Gaussian ◽  
Thin Gate Oxide ◽  
Soft Breakdown

ABSTRACTThe time-dependent dielectric breakdown of MOS capacitors with ultra-thin gate oxide layers is investigated. After the occurrence of soft breakdown, the gate current increases by 3 to 4 orders of magnitudes and behaves like a power law of the applied gate voltage. It is shown that this behavior can be explained by assuming that a percolation path is formed between the electron traps generated in the gate oxide layer during electrical stress of the capacitors. The time dependence of the gate voltage signal after soft breakdown is next analysed. It is shown that the fluctuations in the gate voltage are non-gaussian as well as that long-range correlations exist in the system after soft breakdown. These results can be explained by a dynamic percolation model, taking into account the trapping-detrapping of charges within the percolation cluster formed at soft breakdown.

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