Recently, we have shown how a colored-noise Langevin equation can be used in the context of molecular dynamics as a tool to obtain dynamical trajectories whose properties are tailored to display desired sampling features. In the present paper, after having reviewed some analytical results for the stochastic differential equations forming the basis of our approach, we describe in detail the implementation of the generalized Langevin equation thermostat and the fitting procedure used to obtain optimal parameters. We also discuss the simulation of nuclear quantum effects and demonstrate that by carefully choosing parameters one can successfully model strongly anharmonic solids such as neon. For the reader's convenience, a library of thermostat parameters and some demonstrative code can be downloaded from an online repository.
Decoherence is one of the most important obstacles that must be overcome in quantum information processing. It depends on the qubit-environment coupling strength, but also on the spectral composition of the noise generated by the environment. If the spectral density is known, fighting the effect of decoherence can be made more effective. Applying sequences of inversion pulses to the qubit system, we developed a method for dynamical decoupling noise spectroscopy. We generate effective filter functions that probe the environmental spectral density without requiring assumptions about its shape. Comparing different pulse sequences, we recover the complete spectral density function and distinguish different contributions to the overall decoherence.
We construct a model of baryon diffusion which has the desired properties of causality and analyticity. The model also has the desired property of colored noise, meaning that the noise correlation function is not a Dirac delta function in space and time; rather, it depends on multiple time and length constants. The model can readily be incorporated in 3 + 1-dimensional second-order viscous hydrodynamical models of heavy-ion collisions, which is particularly important at beam energies where the baryon density is large.
We present a method, based on a non-Markovian Langevin equation, to include quantum corrections to the classical dynamics of ions in a quasiharmonic system. By properly fitting the correlation function of the noise, one can vary the fluctuations in positions and momenta as a function of the vibrational frequency, and fit them so as to reproduce the quantum-mechanical behavior, with minimal a priori knowledge of the details of the system. We discuss the application of the thermostat to diamond and to ice Ih. We find that results in agreement with path-integral methods can be obtained using only a fraction of the computational effort.
For ARX-like systems, this paper derives a bias compensation based recursive least squares identification algorithm by means of the prefilter idea and bias compensation principle. The proposed algorithm can give the unbiased estimates of the system model parameters in the presence of colored noises, and can be on-line implemented. Finally, the advantages of the proposed bias compensation recursive least squares algorithm are shown by simulation tests.
Langevin equations describe systems driven by internally generated or externally imposed random excitations. If these excitations correspond to Gaussian white noise, it is relatively straightforward to derive a closed form equation for the joint probability density function (PDF) of state variables. Many natural phenomena present however correlated (colored) excitations. For such problems, a full probabilistic characterization through the resolution of a PDF equation can be obtained through two levels of approximations: first, mixed ensemble moments have to be approximated to lead to a closed system of equations and, second, the resulting nonlocal equations should be at least partially localized to ensure computational efficiency. We propose a new semi-local formulation based on a modified large-eddy diffusivity (LED) approach; the formulation retains most of the accuracy of a fully nonlocal approach while presenting the same order of algorithmic complexity as the standard LED approach. The accuracy of the approach is successfully tested against Monte Carlo simulations.
The influence of colored noise on the transient response of nonlinear dynamical systems is investigated by the generalized cell mapping (GCM) based on the short-time Gaussian approximation (STGA) scheme. The block matrix procedure is introduced into the GCM/STGA method to solve the storage problem caused by the dimensionality of the system. In addition, a parallel calculation strategy can be implemented due to the independence of the storage and the computation of the block matrixes. Taking the well-known Mathieu-Duffing oscillator as an example, the deterministic global properties are first computed with the digraph cell mapping method, in which the coexistence of multiple attractors is observed. Then, the evolutionary processes of transient probability density functions (PDFs) of the response under colored noise from different initial distributions are revealed. Due to the attractive characteristics of the attractor and disturbance generated by colored noise, it can be seen that it takes longer for transient responses from the attractor to achieve the steady-state than the case of the initial distribution which locates around the saddle. The evolutionary processes of transient responses are quite disparate from different initial distributions with the influence of the colored noise, although they both concentrate around the attractor alter enough time, which is consistent with the global structure without noise. Furthermore, the effects of the intensity and the correlation time of the colored noise on the mulistability are discussed. The stochastic P-bifurcation occurs with the increase of these two parameters, respectively. The evolutionary directions of the colored-noise-induced bifurcations are opposite in the two cases. Monte Carlo (MC) simulations are in good agreement with the results obtained by the GCM/STGA method based on the block matrix procedure. Copyright (C) EPLA, 2019
The need for accurate and efficient simulation of the noise background arises in statistical significance tests for periodic signals buried in colored noise. This paper discusses techniques for generating colored-noise sequences which simulate processes with a given spectral density. Matlab routines based on the fast fractional difference algorithm are presented. These routines can create various stochastic models (e.g., first-order autoregressive (AR (1)), power law (PL), autoregressive fractionally integrated moving average ((1, d, 0)), and generalized Gauss Markov (GGM)) that serve as possible candidate null hypotheses to test against in various scenarios. Allan variance and power spectral density (PSD) show that our algorithms are accurate and efficient, and can be easily implemented for stationary noise models and non-stationary PLs with spectral indices up to 2. Our algorithms can also be extended to produce a non-stationary PL with a range-limited steeper PSD (spectral indices up to about 4) using a GGM approximation with a proper break frequency. The red-noise leakage effect on the periodogram is further discussed. The result shows that our GGM approximation has a potential for alleviating the red-noise leakage in the PSD estimates. The Timmer & Koenig procedure is also included for comparison.