Effective transport properties for flashing ratchets using homogenization theory

We study effective transport properties of Brownian motor models of molecular motors. The effective drift and diffusivity can be calculated by solving cell problems, given explicitly by homogenization theory. We briefly describe how this approach is equivalent to theWang-Peskin-Elston (WPE) [3] numerical algorithm for calculating effective transport properties of flashing ratchets. For an on-off flashing ratchet we examine the optimization of the Peclet number as a function of the free parameters of the system. We also present a numerical method for solving the cell equations for a flashing ratchet with Gaussian multiplicative noise. (© 2008 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Boundary value driven approach to computational homogenization

A novel approach to computational homogenization methods for the analysis of heterogeneous structures with large length scale differences between their micro- and macroscale is presented. The boundary value driven approach to homogenization (BVDH) is efficient and reliable compared to existing multi-scale frameworks and the limit cases of homogenization, respectively. Furthermore, the proposed approach is applicable to multi-physical problems such as thermomechanical analysis. (© 2014 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

A homogenization approach for the motion of motor proteins

We consider the asymptotic behavior of an evolving weakly coupled Fokker–Planck system of two equations set in a periodic environment. The magnitudes of the diffusion and the coupling are, respectively, proportional and inversely proportional to the size of the period. We prove that, as the period tends to zero, the solutions of the system either propagate (concentrate) with a fixed constant velocity (determined by the data) or do not move at all. The system arises in the modeling of motor proteins which can take two different states. Our result implies that, in the limit, the molecules either move along a filament with a fixed direction and constant speed or remain immobile.     

Probabilistic approach to homogenization of viscosity solutions of parabolic PDEs

In this paper, we develop the probabilistic approach to homogenization problems of viscosity solutions of systems of semilinear parabolic PDEs. Our main tool is the nonlinear Feynman-Kac formula. Received July 1998     

A Hilbert space approach to homogenization of linear ordinary differential equations including delay and memory terms

We present an abstract approach to homogenization in a Hilbert space setting. Related compactness results are obtained. Moreover, the homogenized equations may be computed explicitly, if periodicity is imposed. Examples for the applicability of our homogenization result for linear ordinary (integro‐)differential equations are given. Copyright © 2012 John Wiley & Sons, Ltd.     

Probabilistic approach to homogenization of a non-divergence form semilinear PDE with non-periodic coefficients

We consider a semilinear partial differential equation (PDE) of non-divergence form perturbed by a small parameter. We then study the asymptotic behavior of Sobolev solutions in the case where the coefficients admit limits in C?esaro sense. Neither periodicity nor ergodicity will be needed for the coefficients. In our situation, the limit (or averaged or effective) coefficients may have discontinuity. Our approach combines both probabilistic and PDEs arguments. The probabilistic one uses the weak convergence of solutions of backward stochastic differential equations (BSDE) in the Jakubowski S-topology, while the PDEs argument consists to built a solution, in a suitable Sobolev space, for the PDE limit. We finally show the existence and uniqueness for the associated averaged BSDE, then we deduce the uniqueness of the limit PDE from the uniqueness of the averaged BSDE.     

Asymmetric potentials and motor effect: a homogenization approach

We provide a mathematical analysis for the appearance of motor effects, i.e., the concentration (as Dirac masses) at one side of the domain, for the solution of a Fokker–Planck system with two components, one with an asymmetric potential and diffusion and one with pure diffusion. The system has been proposed as a model for motor proteins moving along molecular filaments. Its components describe the densities of different conformations of proteins.     

An approach to the homogenization of nonlinear elastomers via the theory of unbounded functionals

The homogenization process for some energies of integral type arising in the modelling of rubber-like elastomers is carried out. The main feature of the variational problems taken into account is the presence of pointwise oscillating constraints on the gradients of the admissible deformations. The classical homogenization result is established also in this framework, both for Dirichlet with affine boundary data, Neumann, and mixed problems, by proving that the limit energy is again of integral type, gradient constrained. An explicit computation for the homogenized integrand relative to energy density in a particular relevant case is derived.     

Variational approach for strain-driven,incremental homogenization of inelastic solids

This study is devoted to finite element modeling of phenomenological rate independent elastoplasticity coupled to damage. The new aspect concerns the treatment of both types of inelasticities. Consistent with physical observations these are interpreted as two pseudophases characterized by specific incremental quasihyperelastic potentials. On this basis, the governing macroscopic energetics is derived as a quasiconvex potential for macro-stresses and corresponding variational formulation is discretized and solved for two limit cases: pure plastic response and pure scalar damage response. (© 2005 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Intermittency and deterministic diffusion in chaotic ratchets

We consider the problem of deterministic transport of particles in an asymmetric periodic ratchet potential of the rocking type. When the inertial term is taken into account, the dynamics can be chaotic and modify the transport properties. We calculate the bifurcation diagram as a function of the amplitude of forcing and analyze in detail the crisis bifurcation that leads to current reversals. Near this bifurcation we obtain intermittency and anomalous deterministic diffusion.     

Ergodic theory and appication to nonconvex homogenization

We establish some ergodic theorems with the view to obtaining a convergence result of sequences of random Radon measures. We also give an application in stochastic homogenization of nonconvex integral functionals.     

A stochastic homogenization scheme for damaged composites

A stochastic homogenization scheme for the computation of effective elastic properties for damaged micro-heterogeneous materials is presented. The statistics is captured by uncertain material properties as well as stochastic interpolation between the upper and lower Hashin-Shtrikman bounds. The resulting multidimensional stochastic problem is solved via sophisticated Monte Carlo methods and applied to Ultra-High-Performance-Concrete. (© 2012 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

From homogenization to averaging in cellular flows

We consider an elliptic eigenvalue problem with a fast cellular flow of amplitude A  , in a two-dimensional domain with L²$L^2$ cells. For fixed A  , and L→∞$L\to \infty$, the problem homogenizes, and has been well studied. Also well studied is the limit when L   is fixed, and A→∞$A\to \infty$. In this case the solution equilibrates along stream lines.     

A discussion about the homogenization of moving interfaces

There has been considerable interest lately in the homogenization theory for first- and second-order partial differential equations in periodic/almost periodic and random, stationary, ergodic environments. Of special interest is the study of the averaged behavior of moving interfaces. In this note we revisit the last issue. We present several new results concerning interfaces moving by either oscillatory first-order or curvature dependent coupled with oscillatory forcing normal velocity in periodic environments and analyze in detail their behavior. Under sharp assumptions we show that such fronts may homogenize, get trapped or oscillate.     

Generalized Besicovitch spaces and applications to deterministic homogenization

The purpose of the present work is to introduce a framework which enables us to study nonlinear homogenization problems. The starting point is the theory of algebras with mean value. Very often in physics, from very simple experimental data, one gets complicated structure phenomena. These phenomena are represented by functions which are permanent in mean, but complicated in detail. In addition the functions are subject to the verification of a functional equation which in general is nonlinear. The problem is therefore to give an interpretation of these phenomena using functions having the following qualitative properties: they are functions that represent a phenomenon on a large scale, and which vary irregularly, undergoing nonperiodic oscillations on a fine scale. In this work we study the qualitative properties of spaces of such functions, which we call generalized Besicovitch spaces, and we prove general compactness results related to these spaces. We then apply these results in order to study some new homogenization problems. One important achievement of this work is the resolution of the generalized weakly almost periodic homogenization problem for a nonlinear pseudo-monotone parabolic-type operator. We also give the answer to the question raised by Frid and Silva in their paper [35] [H. Frid, J. Silva, Homogenization of nonlinear pde’s in the Fourier-Stieltjes algebras, SIAM J. Math. Anal, 41 (4) (2009) 1589-1620] as regards whether there exist or do not exist ergodic algebras that are not subalgebras of the Fourier-Stieltjes algebra.