scholarly journals Stochastic theory of polarized light in nonlinear birefringent media: An application to optical rotation

2018 ◽  
Vol 32 (12) ◽  
pp. 1850147 ◽  
Author(s):  
Satoshi Tsuchida ◽  
Hiroshi Kuratsuji
Keyword(s):  
Langevin Equation ◽  
Optical Rotation ◽  
Gaussian White Noise ◽  
Stochastic Theory ◽  
Functional Integral ◽  
Stokes Parameters ◽  
Random Fluctuation ◽  
Starting Point ◽  
Linear And Nonlinear ◽  
Nonlinear Birefringence

A stochastic theory is developed for the light transmitting the optical media exhibiting linear and nonlinear birefringence. The starting point is the two-component nonlinear Schrödinger equation (NLSE). On the basis of the ansatz of “soliton” solution for the NLSE, the evolution equation for the Stokes parameters is derived, which turns out to be the Langevin equation by taking account of randomness and dissipation inherent in the birefringent media. The Langevin equation is converted to the Fokker–Planck (FP) equation for the probability distribution by employing the technique of functional integral on the assumption of the Gaussian white noise for the random fluctuation. The specific application is considered for the optical rotation, which is described by the ellipticity (third component of the Stokes parameters) alone: (i) The asymptotic analysis is given for the functional integral, which leads to the transition rate on the Poincaré sphere. (ii) The FP equation is analyzed in the strong coupling approximation, by which the diffusive behavior is obtained for the linear and nonlinear birefringence. These would provide with a basis of statistical analysis for the polarization phenomena in nonlinear birefringent media.

Generalized Langevin equation for relative turbulent dispersion

1998 ◽  
Vol 357 ◽  
pp. 167-198 ◽  
Author(s):  
BURKHARD M. O. HEPPE
Keyword(s):  
Langevin Equation ◽  
Stokes Equations ◽  
Gaussian White Noise ◽  
Random Force ◽  
Turbulent Dispersion ◽  
Relative Dispersion ◽  
Generalized Langevin Equation ◽  
Exact Equation ◽  
Velocity Statistics ◽  
Moderate Reynolds Number

The relative velocity of two fluid particles in homogeneous and stationary turbulence is considered. Looking for reduced dynamics of turbulent dispersion, we apply the nonlinear Mori–Zwanzig projector method to the Navier–Stokes equations. The projector method decomposes the Lagrangian acceleration into a conditionally averaged part and a random force. The result is an exact generalized Langevin equation for the Lagrangian velocity differences accounting for the exact equation of the Eulerian probability density. From the generalized Langevin equation, we obtain a stochastic model of relative dispersion by stochastic estimation of conditional averages and by assuming the random force to be Gaussian white noise. This new approach to dispersion modelling generalizes and unifies stochastic models based on the well-mixed condition and the moments approximation. Furthermore, we incorporate viscous effects in a systematic way. At a moderate Reynolds number, the model agrees qualitatively with direct numerical simulations showing highly non-Gaussian separation and velocity statistics for particle pairs initially close together. At very large Reynolds numbers, the mean-square separation obeys a Richardson law with coefficient of the order of 0.1.

scholarly journals Nonlinear stochastic modelling with Langevin regression

2021 ◽  
Vol 477 (2250) ◽  
pp. 20210092
Author(s):  
J. L. Callaham ◽  
J.-C. Loiseau ◽  
G. Rigas ◽  
S. L. Brunton
Keyword(s):  
Langevin Equation ◽  
Degrees Of Freedom ◽  
Bluff Body ◽  
Gaussian White Noise ◽  
Stochastic Modelling ◽  
State Dependent ◽  
Nonlinear Langevin Equation ◽  
Statistical Consistency ◽  
Fokker Planck Equations ◽  
Random Fluctuations

Many physical systems characterized by nonlinear multiscale interactions can be modelled by treating unresolved degrees of freedom as random fluctuations. However, even when the microscopic governing equations and qualitative macroscopic behaviour are known, it is often difficult to derive a stochastic model that is consistent with observations. This is especially true for systems such as turbulence where the perturbations do not behave like Gaussian white noise, introducing non-Markovian behaviour to the dynamics. We address these challenges with a framework for identifying interpretable stochastic nonlinear dynamics from experimental data, using forward and adjoint Fokker–Planck equations to enforce statistical consistency. If the form of the Langevin equation is unknown, a simple sparsifying procedure can provide an appropriate functional form. We demonstrate that this method can learn stochastic models in two artificial examples: recovering a nonlinear Langevin equation forced by coloured noise and approximating the second-order dynamics of a particle in a double-well potential with the corresponding first-order bifurcation normal form. Finally, we apply Langevin regression to experimental measurements of a turbulent bluff body wake and show that the statistical behaviour of the centre of pressure can be described by the dynamics of the corresponding laminar flow driven by nonlinear state-dependent noise.

scholarly journals Probing and predicting ganglion cell responses to smooth electrical stimulation in healthy and blind mouse retina

2019 ◽  
Author(s):  
Larissa Höfling ◽  
Philipp Berens ◽  
Günther Zeck
Keyword(s):  
Microelectrode Array ◽  
Ganglion Cells ◽  
Gaussian White Noise ◽  
Electrical Stimulus ◽  
Mouse Retina ◽  
Linear Filters ◽  
Large Space ◽  
Retinal Pathways ◽  
Starting Point ◽  
Electrical Stimuli

ABSTRACTRetinal implants are used to replace lost photoreceptors in blind patients suffering from retinopathies such as retinitis pigmentosa. Patients wearing implants regain some rudimentary visual function. However, it is severely limited compared to normal vision because non-physiological stimulation strategies fail to selectively activate different retinal pathways at sufficient spatial and temporal resolution. The development of improved stimulation strategies is rendered difficult by the large space of potential stimuli. Here we systematically explore a subspace of potential stimuli by electrically stimulating healthy and blind mouse retina in epiretinal configuration using smooth Gaussian white noise delivered by a high-density CMOS-based microelectrode array. We identify linear filters of retinal ganglion cells (RGCs) by fitting a linear-nonlinear-Poisson (LNP) model. Our stimulus evokes fast, reliable, and spatially confined spiking responses in RGC which are accurately predicted by the LNP model. Furthermore, we find diverse shapes of linear filters in the linear stage of the model, suggesting diverse preferred electrical stimuli of RGCs. Our smooth electrical stimulus could provide a starting point of a model-guided search for improved stimuli for retinal prosthetics.

Probabilistic Solutions of Stochastic Oscillators Excited by Correlated External and Parametric White Noises

2014 ◽  
Vol 136 (3) ◽  
Author(s):  
Siu-Siu Guo
Keyword(s):  
Gaussian White Noise ◽  
Nonlinear Oscillators ◽  
Stationary Probability Density Function ◽  
Fpk Equation ◽  
Linear And Nonlinear ◽  
Polynomial Closure ◽  
Stochastic Oscillators ◽  
Random Oscillators ◽  
White Noises ◽  
Probability Density Function Pdf

The stationary probability density function (PDF) solution of random oscillators with correlated additive and multiplicative Gaussian excitations is investigated in this paper. The correlation between additive and multiplicative Gaussian excitations is taken into account. As a result, the generalized Fokker-Planck-Kolmogorov (FPK) equation is expressed with the independent part and the correlated part, which can be solved by the exponential-polynomial closure (EPC) method. The linear and nonlinear oscillators under correlated additive and multiplicative Gaussian white noise excitations are investigated. Two cases of different correlated additive and multiplicative excitations are considered. Compared with the results in the case of independent external and parametric excitations, unsymmetrical PDFs and nonzero means of system responses can be obtained.

Linear and Nonlinear Signatures in the Planetary Wave Dynamics of an AGCM: Phase Space Tendencies

2005 ◽  
Vol 62 (6) ◽  
pp. 1792-1811 ◽  
Author(s):  
Grant Branstator ◽  
Judith Berner
Keyword(s):  
Phase Space ◽  
General Circulation ◽  
Planetary Wave ◽  
Gaussian White Noise ◽  
Circulation Model ◽  
Lag Time ◽  
Empirical Orthogonal Functions ◽  
Linear Component ◽  
Least Squares Fit ◽  
Linear And Nonlinear

Abstract To identify and quantify indications of linear and nonlinear planetary wave behavior, characteristics of a very long integration of an atmospheric general circulation model in a four-dimensional phase space are examined. The phase space is defined by the leading four empirical orthogonal functions of 500-hPa geopotential heights, and the primary investigated characteristic is the state dependence of mean phase space tendencies. Defining the linear component of planetary wave tendencies as that part which can be captured by a least squares fit linear operator driven by additive Gaussian white noise, the study finds that there are distinct linear and nonlinear signatures. These signatures are especially easy to see in plots of mean tendencies projected onto phase space planes. For some planes the mean tendencies are highly linear, while for others there are strong departures from linearity. The results of the analysis are found to depend strongly on the lag time used to estimate tendencies with the linear component monotonically increasing with lag time. This is shown to result from the ergodicity of the system. Using the theory of Markov models it is possible to remove the lag-dependent component of the tendencies from the results. When this is done the projected mean dynamics in some planes is found to be almost exclusively nonlinear, while in others it is nearly linear. In the four-dimensional space the linear component of the dynamics is largely a reflection of a westward propagating Northern Hemisphere pattern concentrated over the Pacific and North America. The nonlinear signature can be approximated by two linear functions, each operating in a different region of phase space. One region is centered around a Pacific blocking pattern while the other is centered on a state with enhanced zonal symmetry. It is concluded that reduced models of the planetary waves should strive to include these state-dependent dynamics.

scholarly journals Discovery of the Environmental Factors Affecting Urban Dwellers’ Mental Health: A Data-Driven Approach

2020 ◽  
Vol 17 (21) ◽  
pp. 8167
Author(s):  
Chao Wu ◽  
Pei Zheng ◽  
Xinyuan Xu ◽  
Shuhan Chen ◽  
Nasi Wang ◽  
...  
Keyword(s):  
Mental Health ◽  
Environmental Factors ◽  
Mental States ◽  
Data Driven ◽  
Neural Network Modeling ◽  
Starting Point ◽  
Data Driven Approach ◽  
Linear And Nonlinear ◽  
The Impact ◽  
Maximal Information Coefficient

Mental health is the foundation of health and happiness as well as the basis for an individual’s meaningful life. The environmental and social health of a city can measure the mental state of people living in a certain areas, and exploring urban dwellers’ mental states is an important factor in understanding and better managing cities. New dynamic and granular urban data provide us with a way to determine the environmental factors that affect the mental states of urban dwellers. The characteristics of the maximal information coefficient can identify the linear and nonlinear relationships so that we can fully identify the physical and social environmental factors that affect urban dwellers’ mental states and further test these relationships through linear and nonlinear modeling. Taking the Greater London as an example, we used data from the London Datastore to discover the environmental factors that had the highest correlation with urban mental health from 2015 to 2017 and to prove that they had a high nonlinear correlation through neural network modeling. This paper aimed to use a data-driven approach to find environmental factors that had not yet received enough attention and to provide a starting point for research by establishing hypotheses for further exploration of the impact of environmental factors on mental health.

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