Large deviations of a velocity jump process with a Hamilton-Jacobi approach

We study a random process on R n moving in straight lines and changing randomly its velocity at random exponential times. We focus more precisely on the Kolmogorov equation in the hyperbolic scale (t, x, v) $\to$ t $\epsilon$, x $\epsilon$, v, with $\epsilon$ \textgreater{} 0, before proceeding to a Hopf-Cole transform, which gives a kinetic equation on a potential. We show convergence as $\epsilon$ $\to$ 0 of the potential towards the viscosity solution of a Hamilton-Jacobi equation $\partial$t\"I + H ($\nabla$x\"I) = 0 where the hamiltonian may lack C 1 regularity, which is quite unseen in this type of studies. R{\'e}sum{\'e

Spontaneous polarization in an interfacial growth model for actin filament networks with a rigorous mechano-chemical coupling

Many processes in eukaryotic cells, including cell motility, rely on the growth of branched actin networks from surfaces. Despite its central role the mechano-chemical coupling mechanisms which guide the growth process are poorly understood, and a general continuum description combining growth and mechanics is lacking. We develop a theory that bridges the gap between mesoscale and continuum limit and propose a general framework providing the evolution law of actin networks growing under stress. This formulation opens an area for the systematic study of actin dynamics in arbitrary geometries. Our framework predicts a morphological instability of actin growth on a rigid sphere, leading to a spontaneous polarization of the network with a mode selection corresponding to a comet, as reported experimentally. We show that the mechanics of the contact between the network and the surface plays a crucial role, in that it determines directly the existence of the instability. We extract scaling laws relating growth dynamics and network properties offering basic perspectives for new experiments on growing actin networks.Comment: 7 pages, 4 figure

A constitutive law for cross-linked actin networks by homogenization techniques

Inspired by experiments on the actin driven propulsion of micrometer sized beads we develop and study a minimal mechanical model of a two-dimensional network of stiff elastic filaments grown from the surface of a cylinder. Starting out from a discrete model of the network structure and of its microscopic mechanical behavior we derive a macroscopic constitutive law by homogenization techniques. We calculate the axisymmetric equilibrium state and study its linear stability depending on the microscopic mechanical properties. We find that thin networks are linearly stable, whereas thick networks are unstable. The critical thickness for the change in stability depends on the ratio of the microscopic elastic constants. The instability is induced by the increase in the compressive load on the inner network layers as the thickness of the network increases. The here employed homogenization approach combined with more elaborate microscopic models can serve as a basis to study the evolution of polymerizing actin networks and the mechanism of actin driven motion.Comment: 19 pages, 7 figure

Behavior of a net of fibers linked by viscous interactions: theory and mechanical properties

International audienceThis paper presents an investigation of the macroscopic mechanical behavior of highly concentrated fiber suspensions for which the mechanical behavior is governed by local fiber-fiber interactions. The problem is approached by considering the case of a net of rigid fibers of uniform length, linked by viscous point interactions of power-law type. Those interactions may result in local forces and moments located at the contacting point between two fibers, and respectively power-law functions of the local linear and angular velocity at this point. Assuming the existence of an elementary representative volume which size is small compared to the size of the whole structure, the fiber net is regarded as a periodic assembly of identical cells. Macroscopic equilibrium and constitutive equations of the equivalent continuum are then obtained by the discrete and periodic media homogenization method, based on the use of asymptotic expansions. Depending on the order of magnitude of local translational viscosities and rotational viscosities, three types of the equivalent continua are proved to be possible. One of them leads to an effective Cosserat medium, the other ones being usual Cauchy media. Lastly, formulations that enable an effective computation of constitutive equations are detailed. They show that the equivalent continuum behaves like an anisotropic power-law fluid

FEMxDEM Multi-scale modelling applied to geomaterials

The multi-scale FEMxDEM approach is an innovative numerical method for geotechnical problems, using at the same time the Finite Element Method (FEM) at the engineering macro-scale and the Discrete Element Method (DEM) at the scale of the microstructure of the material. The link between scales is made via computational homogenization. In this way, the continuum numerical constitutive law and the corresponding tangent matrix are obtained directly from the discrete response of the microstructure [1,2,3]. In the proposed paper, a variety of operators, rather than the tangent consistent for the Newton-Raphson method, is tested in a challenging attempt to improve the poor convergence performance. The independence of the DEM computations is exploited to develop a parallelized code using an OpenMP paradigm. At the macro level, a second gradient constitutive relation is implemented in order to enrich the first gradient Cauchy relation bringing mesh-independency to the model. The second gradient regularization, together with the speedup provided by the parallelization allows by first time to the FEMxDEM model to be applied to real scale problems. Some results are given exhibiting the above findings with emphasis on aspects related to strain localisation

Some spaces are more equal than others

It has generally been thought that in perturbed Friedmann-Lemaitre-Robertson-Walker models of the Universe, global topology should not have any feedback effects on dynamics. However, a weak-field limit heuristical argument, assuming a finite particle horizon for the transmission of gravitational signals, shows that a residual acceleration effect can occur. The nature of this effect differs algebraically between different constant curvature 3-manifolds. This potentially provides a selection mechanism for the 3-manifold of comoving space.Comment: 4 pages, proceedings of the Grassmannian Conference in Fundamental Cosmology, Szczecin, 14-19 Sep 2009, to be refereed by Annalen der Physi

Actin based propulsion: Intriguing interplay between material properties and growth processes

Eukaryotic cells and intracellular pathogens such as bacteria or viruses utilize the actin polymerization machinery to propel themselves forward. Thereby, the onset of motion and choice of direction may be the result of a spontaneous symmetry-breaking or might be triggered by external signals and preexisting asymmetries, e.g. through a previous septation in bacteria. Although very complex, a key feature of cellular motility is the ability of actin to form dense polymeric networks, whose microstructure is tightly regulated by the cell. These polar actin networks produce the forces necessary for propulsion but may also be at the origin of a spontaneous symmetry-breaking. Understanding the exact role of actin dynamics in cell motility requires multiscale approaches which capture at the same time the polymer network structure and dynamics on the scale of a few nanometers and the macroscopic distribution of elastic stresses on the scale of the whole cell. In this chapter we review a selection of theories on how mechanical material properties and growth processes interact to induce the onset of actin based motion.Comment: 16 pages, 14 figures, chapter in book "Cell mechanics: from single scale-based models to multiscale modelling

The Neumann problem in thin domains with very highly oscillatory boundaries

In this paper we analyze the behavior of solutions of the Neumann problem posed in a thin domain of the type $R^\epsilon = \{(x_1,x_2) \in \R^2 \; | \; x_1 \in (0,1), \, - \, \epsilon \, b(x_1) < x_2 < \epsilon \, G(x_1, x_1/\epsilon^\alpha) \}$ with $\alpha>1$ and $\epsilon > 0$, defined by smooth functions $b(x)$ and $G(x,y)$, where the function $G$ is supposed to be $l(x)$-periodic in the second variable $y$. The condition $\alpha > 1$ implies that the upper boundary of this thin domain presents a very high oscillatory behavior. Indeed, we have that the order of its oscillations is larger than the order of the amplitude and height of $R^\epsilon$ given by the small parameter $\epsilon$. We also consider more general and complicated geometries for thin domains which are not given as the graph of certain smooth functions, but rather more comb-like domains.Comment: 20 pages, 4 figure