arXiv subjects

Felix Otto

Publications and source records attributed to Felix Otto.

97 records · Page 6Linked to original sources

arXiv2012

An optimal error estimate in stochastic homogenization of discrete elliptic equations

This paper is the companion article to [Ann. Probab. 39 (2011) 779--856]. We consider a discrete elliptic equation on the $d$-dimensional lattice $\mathbb{Z}^d$ with random coefficients $A$ of the simplest type: They are identically distributed and independent from edge to edge. On scales large w.r.t. the lattice spacing (i.e., unity), the solution operator is known to behave like the solution operator of a (continuous) elliptic equation with constant deterministic coefficients. This symmetric "homogenized" matrix $A_{\mathrm{hom}}=a_{\mathrm{hom}}\mathrm{Id}$ is characterized by $ξ\cdot A_{\mathrm{hom}}ξ=<(ξ+\nablaϕ)\cdot A(ξ+\nablaϕ)>$ for any direction $ξ\in\mathbb{R}^d$, where the random field $ϕ$ (the "corrector") is the unique solution of $-\nabla^*\cdot A(ξ+\nablaϕ)=0$ in $\mathbb{Z}^d$ such that $ϕ(0)=0$, $\nablaϕ$ is stationary and $<\nablaϕ>=0$, $<\cdot>$ denoting the ensemble average (or expectation).

math.PR↗

arXiv2011

The Formation and Coarsening of the Concertina Pattern

The concertina is a magnetization pattern in elongated thin-film elements of a soft material. It is a ubiquitous domain pattern that occurs in the process of magnetization reversal in direction of the long axis of the small element. Van den Berg argued that this pattern grows out of the flux closure domains as the external field is reduced. Based on experimental observations and theory, we argue that in sufficiently elongated thin-film elements, the concertina pattern rather bifurcates from an oscillatory buckling mode. Using a reduced model derived by asymptotic analysis and investigated by numerical simulation, we quantitatively predict the average period of the concertina pattern and qualitatively predict its hysteresis. In particular, we argue that the experimentally observed coarsening of the concertina pattern is due to secondary bifurcations related to an Eckhaus instability. We also link the concertina pattern to the magnetization ripple and discuss the effect of a weak (crystalline or induced) anisotropy.

cond-mat.mtrl-sci↗

arXiv2011

An optimal variance estimate in stochastic homogenization of discrete elliptic equations

We consider a discrete elliptic equation on the $d$-dimensional lattice $\mathbb{Z}^d$ with random coefficients $A$ of the simplest type: they are identically distributed and independent from edge to edge. On scales large w.r.t. the lattice spacing (i.e., unity), the solution operator is known to behave like the solution operator of a (continuous) elliptic equation with constant deterministic coefficients. This symmetric ``homogenized'' matrix $A_{\mathrm {hom}}=a_{\mathrm {hom}}\operatorname {Id}$ is characterized by $ξ\cdot A_{\mathrm {hom}}ξ=\langle(ξ+\nablaϕ)\cdot A(ξ+\nablaϕ)\rangle$ for any direction $ξ\in\mathbb {R}^d$, where the random field $ϕ$ (the ``corrector'') is the unique solution of $-\nabla^*\cdot A(ξ+\nablaϕ)=0$ such that $ϕ(0)=0$, $\nablaϕ$ is stationary and $\langle\nablaϕ\rangle=0$, $\langle\cdot\rangle$ denoting the ensemble average (or expectation). It is known (``by ergodicity'') that the above ensemble average of the energy density $\mathcal {E}=(ξ+\nablaϕ)\cdot A(ξ+\nablaϕ)$, which is a stationary random field, can be recovered by a system average. We quantify this by proving that the variance of a spatial average of $\mathcal {E}$ on length scales $L$ satisfies the optimal estimate, that is, $\operatorname {var}[\sum \mathcal {E}η_L]\lesssim L^{-d}$, where the averaging function [i.e., $\sumη_L=1$, $\operatorname {supp}(η_L)\subset\{|x|\le L\}$] has to be smooth in the sense that $|\nablaη_L|\lesssim L^{-1-d}$. In two space dimensions (i.e., $d=2$), there is a logarithmic correction. This estimate is optimal since it shows that smooth averages of the energy density $\mathcal {E}$ decay in $L$ as if $\mathcal {E}$ would be independent from edge to edge (which it is not for $d>1$). This result is of practical significance, since it allows to estimate the dominant error when numerically computing $a_{\mathrm {hom}}$.

math.PR↗

arXiv2010

A compactness result for Landau state in thin-film micromagnetics

We deal with a nonconvex and nonlocal variational problem coming from thin-film micromagnetics. It consists in a free-energy functional depending on two small parameters $\eps$ and $η$ and defined over $S^2-$vector fields $m$ that are tangent at the boundary of a two-dimensional domain $Ω$. We are interested in the behavior of minimizers as $\eps, η\to 0$. The minimizers tend to be in-plane away from a region of length scale $\eps$ (generically, an interior vortex ball or two boundary vortex balls) and of vanishing divergence, so that $S^1-$transition layers of length scale $η$ (Néel walls) are enforced by the boundary condition. We first prove an upper bound for the minimal energy that corresponds to the cost of a vortex and the configuration of Néel walls associated to the viscosity solution, so-called Landau state. Our main result concerns the compactness of vector fields $m_{\eps, η}$ of energies close to the Landau state in the regime where a vortex is energetically more expensive than a Néel wall. Our method uses techniques developed for the Ginzburg-Landau type problems for the concentration of energy on vortex balls, together with an approximation argument of $S^2-$vector fields by $S^1-$vector fields away from the vortex balls.

math.AP↗

arXiv2007

Long-time convergence of an Adaptive Biasing Force method

We propose a proof of convergence of an adaptive method used in molecular dynamics to compute free energy profiles. Mathematically, it amounts to studying the long-time behavior of a stochastic process which satisfies a non-linear stochastic differential equation, where the drift depends on conditional expectations of some functionals of the process. We use entropy techniques to prove exponential convergence to the stationary state.

math.AP↗

arXiv2003

Cross-over in scaling laws: A simple example from micromagnetics

Scaling laws for characteristic length scales (in time or in the model parameters) are both experimentally robust and accessible for rigorous analysis. In multiscale situations cross--overs between different scaling laws are observed. We give a simple example from micromagnetics. In soft ferromagnetic films, the geometric character of a wall separating two magnetic domains depends on the film thickness. We identify this transition from a Néel wall to an Asymmetric Bloch wall by rigorously establishing a cross--over in the specific wall energy.

math-ph↗

arXiv2000

Two-Dimensional Modeling of Soft Ferromagnetic Films

We examine the response of a soft ferromagnetic film to an in-plane applied magnetic field. Our theory, based on asymptotic analysis of the micromagnetic energy in the thin-film limit, proceeds in two steps: first we determine the magnetic charge density by solving a convex variational problem; then we construct an associated magnetization field using a robust numerical method. Experimental results show good agreement with the theory. Our analysis is consistent with prior work by van den Berg and by Bryant and Suhl, but it goes much further; in particular it applies even for large fields which penetrate the sample.

cond-mat.mtrl-sci↗