Relaxation problem with quadratic noise

1986 ◽  
Vol 45 (1-2) ◽  
pp. 309-317 ◽  
Author(s):  
J. Luczka
Keyword(s):  
Relaxation Problem ◽  
Quadratic Noise

scholarly journals An asymptotic analysis for a generalized Cahn–Hilliard system with fractional operators

2021 ◽  
Author(s):  
Pierluigi Colli ◽  
Gianni Gilardi ◽  
Jürgen Sprekels
Keyword(s):  
Additional Term ◽  
Linear Operators ◽  
Phase Variable ◽  
Logarithmic Potentials ◽  
Relaxation Problem ◽  
System Equations ◽  
Indicator Functions ◽  
Regularity Results ◽  
Compact Resolvents ◽  
Densely Defined

AbstractIn the recent paper “Well-posedness and regularity for a generalized fractional Cahn–Hilliard system” (Colli et al. in Atti Accad Naz Lincei Rend Lincei Mat Appl 30:437–478, 2019), the same authors have studied viscous and nonviscous Cahn–Hilliard systems of two operator equations in which nonlinearities of double-well type, like regular or logarithmic potentials, as well as nonsmooth potentials with indicator functions, were admitted. The operators appearing in the system equations are fractional powers $$A^{2r}$$ A 2 r and $$B^{2\sigma }$$ B 2 σ (in the spectral sense) of general linear operators A and B, which are densely defined, unbounded, selfadjoint, and monotone in the Hilbert space $$L^2(\Omega )$$ L 2 ( Ω ) , for some bounded and smooth domain $$\Omega \subset {{\mathbb {R}}}^3$$ Ω ⊂ R 3 , and have compact resolvents. Existence, uniqueness, and regularity results have been proved in the quoted paper. Here, in the case of the viscous system, we analyze the asymptotic behavior of the solution as the parameter $$\sigma $$ σ appearing in the operator $$B^{2\sigma }$$ B 2 σ decreasingly tends to zero. We prove convergence to a phase relaxation problem at the limit, and we also investigate this limiting problem, in which an additional term containing the projection of the phase variable on the kernel of B appears.

Calculation of boltzmann equation collision integral and accurate solution of relaxation problem

1978 ◽  
Vol 12 (5) ◽  
pp. 749-757
Author(s):  
I. N. Kolyshkin ◽  
A. Ya. �nder ◽  
I. A. �nder
Keyword(s):  
Boltzmann Equation ◽  
Accurate Solution ◽  
Collision Integral ◽  
Relaxation Problem

scholarly journals Comparing Information Metrics for a Coupled Ornstein–Uhlenbeck Process

Entropy ◽  
2019 ◽  
Vol 21 (8) ◽  
pp. 775 ◽  
Author(s):  
James Heseltine ◽  
Eun-jin Kim
Keyword(s):  
Relative Entropy ◽  
Uhlenbeck Process ◽  
Relaxation Problem ◽  
Ornstein Uhlenbeck Process ◽  
Long Time ◽  
The Difference ◽  
Information Change ◽  
Information Metrics ◽  
Insight Into ◽  
Linear Geometry

It is often the case when studying complex dynamical systems that a statistical formulation can provide the greatest insight into the underlying dynamics. When discussing the behavior of such a system which is evolving in time, it is useful to have the notion of a metric between two given states. A popular measure of information change in a system under perturbation has been the relative entropy of the states, as this notion allows us to quantify the difference between states of a system at different times. In this paper, we investigate the relaxation problem given by a single and coupled Ornstein–Uhlenbeck (O-U) process and compare the information length with entropy-based metrics (relative entropy, Jensen divergence) as well as others. By measuring the total information length in the long time limit, we show that it is only the information length that preserves the linear geometry of the O-U process. In the coupled O-U process, the information length is shown to be capable of detecting changes in both components of the system even when other metrics would detect almost nothing in one of the components. We show in detail that the information length is sensitive to the evolution of subsystems.

Non-Markovian process driven by quadratic noise: Kramers-Moyal expansion and Fokker-Planck modeling

1995 ◽  
Vol 51 (4) ◽  
pp. 2933-2938 ◽  
Author(s):  
Jerzy Łuczka ◽  
Peter Hänggi ◽  
Adam Gadomski
Keyword(s):  
Markovian Process ◽  
Quadratic Noise ◽  
Fokker Planck

Towards proper parametrization in the exciton transfer and relaxation problem

1999 ◽  
Vol 83-84 ◽  
pp. 105-108 ◽  
Author(s):  
I. Barvı́k ◽  
V. Čápek ◽  
P. Heřman
Keyword(s):  
Relaxation Problem ◽  
Exciton Transfer

Two variants of Monte Carlo projection method for numerical solution of nonlinear Boltzmann equation

2019 ◽  
Vol 34 (3) ◽  
pp. 143-150
Author(s):  
Sergey V. Rogasinsky
Keyword(s):  
Boltzmann Equation ◽  
Projection Method ◽  
Orthonormal Basis ◽  
Statistical Modelling ◽  
Hermite Functions ◽  
Nonlinear Boltzmann Equation ◽  
Kinetic Boltzmann Equation ◽  
Relaxation Problem ◽  
Nonlinear Kinetic ◽  
Approximation Of A Function

Abstract The paper is focused on justification of a statistical modelling algorithm for solution of the nonlinear kinetic Boltzmann equation on the base of a projection method. Hermite functions are used as an orthonormal basis. The error of approximation of a function by a partial sum of Hermite functions series is estimated in the L2 norm. The estimates are compared for two variants of the projection method in the case of solutions to the homogeneous gas relaxation problem with a known solution.

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