5,600 research outputs found
Edgeworth expansions for slow-fast systems with finite time scale separation
We derive Edgeworth expansions that describe corrections to the Gaussian limiting behaviour of slow-fast systems. The Edgeworth expansion is achieved using a semi-group formalism for the transfer operator, where a Duhamel-Dyson series is used to asymptotically determine the corrections at any desired order of the time scale parameter ε. The corrections involve integrals over higher-order auto-correlation functions. We develop a diagrammatic representation of the series to control the combinatorial wealth of the asymptotic expansion in ε and provide explicit expressions for the first two orders. At a formal level, the expressions derived are valid in the case when the fast dynamics is stochastic as well as when the fast dynamics is entirely deterministic. We corroborate our analytical results with numerical simulations and show that our method provides an improvement on the classical homogenization limit which is restricted to the limit of infinite time scale separation
Corrections to Einstein's relation for Brownian motion in a tilted periodic potential
In this paper we revisit the problem of Brownian motion in a tilted periodic
potential. We use homogenization theory to derive general formulas for the
effective velocity and the effective diffusion tensor that are valid for
arbitrary tilts. Furthermore, we obtain power series expansions for the
velocity and the diffusion coefficient as functions of the external forcing.
Thus, we provide systematic corrections to Einstein's formula and to linear
response theory. Our theoretical results are supported by extensive numerical
simulations. For our numerical experiments we use a novel spectral numerical
method that leads to a very efficient and accurate calculation of the effective
velocity and the effective diffusion tensor.Comment: 29 pages, 7 figures, submitted to the Journal of Statistical Physic
Kinetic decomposition for periodic homogenization problems
We develop an analytical tool which is adept for detecting shapes of
oscillatory functions, is useful in decomposing homogenization problems into
limit-problems for kinetic equations, and provides an efficient framework for
the validation of multi-scale asymptotic expansions. We apply it first to a
hyperbolic homogenization problem and transform it to a hyperbolic limit
problem for a kinetic equation. We establish conditions determining an
effective equation and counterexamples for the case that such conditions fail.
Second, when the kinetic decomposition is applied to the problem of enhanced
diffusion, it leads to a diffusive limit problem for a kinetic equation that in
turn yields the effective equation of enhanced diffusion
New homogenization approaches for stochastic transport through heterogeneous media
The diffusion of molecules in complex intracellular environments can be
strongly influenced by spatial heterogeneity and stochasticity. A key challenge
when modelling such processes using stochastic random walk frameworks is that
negative jump coefficients can arise when transport operators are discretized
on heterogeneous domains. Often this is dealt with through homogenization
approximations by replacing the heterogeneous medium with an
homogeneous medium. In this work, we present a new class
of homogenization approximations by considering a stochastic diffusive
transport model on a one-dimensional domain containing an arbitrary number of
layers with different jump rates. We derive closed form solutions for the th
moment of particle lifetime, carefully explaining how to deal with the internal
interfaces between layers. These general tools allow us to derive simple
formulae for the effective transport coefficients, leading to significant
generalisations of previous homogenization approaches. Here, we find that
different jump rates in the layers gives rise to a net bias, leading to a
non-zero advection, for the entire homogenized system. Example calculations
show that our generalized approach can lead to very different outcomes than
traditional approaches, thereby having the potential to significantly affect
simulation studies that use homogenization approximations.Comment: 9 pages, 2 figures, accepted version of paper published in The
Journal of Chemical Physic
Higher-order pathwise theory of fluctuations in stochastic homogenization
We consider linear elliptic equations in divergence form with stationary
random coefficients of integrable correlations. We characterize the
fluctuations of a macroscopic observable of a solution to relative order
, where is the spatial dimension; the fluctuations turn out to
be Gaussian. As for previous work on the leading order, this higher-order
characterization relies on a pathwise proximity of the macroscopic fluctuations
of a general solution to those of the (higher-order) correctors, via a
(higher-order) two-scale expansion injected into the homogenization commutator,
thus confirming the scope of this notion. This higher-order generalization
sheds a clearer light on the algebraic structure of the higher-order versions
of correctors, flux correctors, two-scale expansions, and homogenization
commutators. It reveals that in the same way as this algebra provides a
higher-order theory for microscopic spatial oscillations, it also provides a
higher-order theory for macroscopic random fluctuations, although both
phenomena are not directly related. We focus on the model framework of an
underlying Gaussian ensemble, which allows for an efficient use of
(second-order) Malliavin calculus for stochastic estimates. On the technical
side, we introduce annealed Calder\'on-Zygmund estimates for the elliptic
operator with random coefficients, which conveniently upgrade the known
quenched large-scale estimates.Comment: 57 page
Multiscale Surrogate Modeling and Uncertainty Quantification for Periodic Composite Structures
Computational modeling of the structural behavior of continuous fiber
composite materials often takes into account the periodicity of the underlying
micro-structure. A well established method dealing with the structural behavior
of periodic micro-structures is the so- called Asymptotic Expansion
Homogenization (AEH). By considering a periodic perturbation of the material
displacement, scale bridging functions, also referred to as elastic correctors,
can be derived in order to connect the strains at the level of the
macro-structure with micro- structural strains. For complicated inhomogeneous
micro-structures, the derivation of such functions is usually performed by the
numerical solution of a PDE problem - typically with the Finite Element Method.
Moreover, when dealing with uncertain micro-structural geometry and material
parameters, there is considerable uncertainty introduced in the actual stresses
experienced by the materials. Due to the high computational cost of computing
the elastic correctors, the choice of a pure Monte-Carlo approach for dealing
with the inevitable material and geometric uncertainties is clearly
computationally intractable. This problem is even more pronounced when the
effect of damage in the micro-scale is considered, where re-evaluation of the
micro-structural representative volume element is necessary for every occurring
damage. The novelty in this paper is that a non-intrusive surrogate modeling
approach is employed with the purpose of directly bridging the macro-scale
behavior of the structure with the material behavior in the micro-scale,
therefore reducing the number of costly evaluations of corrector functions,
allowing for future developments on the incorporation of fatigue or static
damage in the analysis of composite structural components.Comment: Appeared in UNCECOMP 201
Water waves over a rough bottom in the shallow water regime
This is a study of the Euler equations for free surface water waves in the
case of varying bathymetry, considering the problem in the shallow water
scaling regime. In the case of rapidly varying periodic bottom boundaries this
is a problem of homogenization theory. In this setting we derive a new model
system of equations, consisting of the classical shallow water equations
coupled with nonlocal evolution equations for a periodic corrector term. We
also exhibit a new resonance phenomenon between surface waves and a periodic
bottom. This resonance, which gives rise to secular growth of surface wave
patterns, can be viewed as a nonlinear generalization of the classical Bragg
resonance. We justify the derivation of our model with a rigorous mathematical
analysis of the scaling limit and the resulting error terms. The principal
issue is that the shallow water limit and the homogenization process must be
performed simultaneously. Our model equations and the error analysis are valid
for both the two- and the three-dimensional physical problems.Comment: Revised version, to appear in Annales de l'Institut Henri Poincar\'
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