Mark Peletier

Mark Peletier
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Mathematics - Analysis of PDEs (30)
 
Mathematics - Mathematical Physics (16)
 
Mathematical Physics (16)
 
Mathematics - Probability (8)
 
Mathematics - Functional Analysis (7)
 
Mathematics - Dynamical Systems (4)
 
Nonlinear Sciences - Pattern Formation and Solitons (3)
 
Mathematics - Optimization and Control (3)
 
Physics - Soft Condensed Matter (2)
 
Mathematics - Numerical Analysis (1)
 
Mathematics - Classical Analysis and ODEs (1)
 
Mathematics - Differential Geometry (1)
 
Nonlinear Sciences - Adaptation and Self-Organizing Systems (1)
 
Physics - Statistical Mechanics (1)
 
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Quantitative Biology - Molecular Networks (1)
 
Physics - Atomic and Molecular Clusters (1)

Publications Authored By Mark Peletier

We study two specific measures of quality of chemical reaction networks, Precision and Sensitivity. The two measures arise in the study of sensory adaptation, in which the reaction network is viewed as an input-output system. Given a step change in input, Sensitivity is a measure of the magnitude of the response, while Precision is a measure of the degree to which the system returns to its original output for large time. Read More

Classical gradient systems have a linear relation between rates and driving forces. In generalized gradient systems we allow for arbitrary relations derived from general non-quadratic dissipation potentials. This paper describes two natural origins for these structures. Read More

In this paper we present a variational technique that handles coarse-graining and passing to a limit in a unified manner. The technique is based on a duality structure, which is present in many gradient flows and other variational evolutions, and which often arises from a large-deviations principle. It has three main features: (A) a natural interaction between the duality structure and the coarse-graining, (B) application to systems with non-dissipative effects, and (C) application to coarse-graining of approximate solutions which solve the equation only to some error. Read More

In this paper we analyse the behaviour of a pile-up of vertically periodic walls of edge dislocations at an obstacle, represented by a locked dislocation wall. Starting from a continuum non-local energy $E_\gamma$ modelling the interactions$-$at a typical length-scale of $1/\gamma$$-$of the walls subjected to a constant shear stress, we derive a first-order approximation of the energy $E_\gamma$ in powers of $1/\gamma$ by $\Gamma$-convergence, in the limit $\gamma\to\infty$. While the zero-order term in the expansion, the $\Gamma$-limit of $E_\gamma$, captures the `bulk' profile of the density of dislocation walls in the pile-up domain, the first-order term in the expansion is a `boundary-layer' energy that captures the profile of the density in the proximity of the lock. Read More

Within the framework of variational modelling we derive a one-phase moving boundary problem describing the motion of a semipermeable membrane enclosing a viscous liquid, driven by osmotic pressure and surface tension of the membrane. For this problem we prove the existence of classical solutions for a short time. Read More

We consider systems of $n$ parallel edge dislocations in a single slip system, represented by points in a two-dimensional domain; the elastic medium is modelled as a continuum. We formulate the energy of this system in terms of the empirical measure of the dislocations, and prove several convergence results in the limit $n\to\infty$. The main aim of the paper is to study the convergence of the evolution of the empirical measure as $n\to\infty$. Read More

We construct a stochastic model showing the relationship between noise, gradient flows and rate-independent systems. The model consists of a one-dimensional birth-death process on a lattice, with rates derived from Kramers' law as an approximation of a Brownian motion on a wiggly energy landscape. Taking various limits we show how to obtain a whole family of generalized gradient flows, ranging from quadratic to rate-independent ones, connected via '$L \log L$' gradient flows. Read More

We show how the mathematical structure of large-deviation principles matches well with the concept of coarse-graining. For those systems with a large-deviation principle, this may lead to a general approach to coarse-graining through the variational form of the large-deviation functional. Read More

We consider three one-dimensional continuous-time Markov processes on a lattice, each of which models the conduction of heat: the family of Brownian Energy Processes with parameter $m$, a Generalized Brownian Energy Process, and the Kipnis-Marchioro-Presutti process. The hydrodynamic limit of each of these three processes is a parabolic equation, the linear heat equation in the case of the BEP$(m)$ and the KMP, and a nonlinear heat equation for the GBEP($a$). We prove the hydrodynamic limit rigorously for the BEP$(m)$, and give a formal derivation for the GBEP($a$). Read More

We present an asymptotic analysis of a mesoscale energy for bilayer membranes that has been introduced and analyzed in two space dimensions by the second and third author (Arch. Ration. Mech. Read More

These are lecture notes for various Summer and Winter schools that I have given. The notes describe the methodology called Variational Modelling, and focus on the application to the modelling of gradient-flow systems. I describe the methodology itself in great detail, and explain why this is a rational modelling route. Read More

Motivated by the occurrence in rate functions of time-dependent large-deviation principles, we study a class of non-negative functions $\mathscr L$ that induce a flow, given by $\mathscr L(\rho_t,\dot\rho_t)=0$. We derive necessary and sufficient conditions for the unique existence of a generalized gradient structure for the induced flow, as well as explicit formulas for the corresponding driving entropy and dissipation functional. In particular, we show how these conditions can be given a probabilistic interpretation when $\mathscr L$ is associated to the large deviations of a microscopic particle system. Read More

We prove the equivalence between the notion of Wasserstein gradient flow for a one-dimensional nonlocal transport PDE with attractive/repulsive Newtonian potential on one side, and the notion of entropy solution of a Burgers-type scalar conservation law on the other. The solution of the former is obtained by spatially differentiating the solution of the latter. The proof uses an intermediate step, namely the $L^2$ gradient flow of the pseudo-inverse distribution function of the gradient flow solution. Read More

In this paper we discuss the connections between a Vlasov-Fokker-Planck equation and an underlying microscopic particle system, and we interpret those connections in the context of the GENERIC framework (\"Ottinger 2005). This interpretation provides (a) a variational formulation for GENERIC systems, (b) insight into the origin of this variational formulation, and (c) an explanation of the origins of the conditions that GENERIC places on its constitutive elements, notably the so-called degeneracy or non-interaction conditions. This work shows how the general connection between large-deviation principles on one hand and gradient-flow structures on the other hand extends to non-reversible particle systems. Read More

This paper unravels the problem of an idealised pile-up of n infinite, equi-spaced walls of edge dislocations at equilibrium. We define a dimensionless parameter that depends on the geometric, constitutive and loading parameters of the problem, and we identify five different scaling regimes corresponding to different values of that parameter for large n. For each of the cases we perform a rigorous micro-to-meso upscaling, and we obtain five expressions for the mesoscopic (continuum) internal stress. Read More

We propose three new discrete variational schemes that capture the conservative-dissipative structure of a generalized Kramers equation. The first two schemes are single-step minimization schemes while the third one combines a streaming and a minimization step. The cost functionals in the schemes are inspired by the rate functional in the Freidlin-Wentzell theory of large deviations for the underlying stochastic system. Read More

We consider a system of parallel straight edge dislocations and we analyse its asymptotic behaviour in the limit of many dislocations. The dislocations are represented by points in a plane, and they are arranged in vertical walls; each wall is free to move in the horizontal direction. The system is described by a discrete energy depending on the one-dimensional horizontal positions of the n walls; the energy contains contributions from repulsive pairwise interactions between all walls, a global shear stress forcing the walls to the left, and a pinned wall at x=0 that prevents the walls from leaving through the left boundary. Read More

This is a former PhD student's take on his teacher's scientific philosophy. I describe a set of 'principles' that I believe are conducive to good applied mathematics, and that I have learnt myself from observing Hans van Duijn in action. Read More

In recent work [1] we uncovered intriguing connections between Otto's characterisation of diffusion as entropic gradient flow [16] on one hand and large-deviation principles describing the microscopic picture (Brownian motion) on the other. In this paper, we sketch this connection, show how it generalises to a wider class of systems, and comment on consequences and implications. Specifically, we connect macroscopic gradient flows with large deviation principles, and point out the potential of a bigger picture emerging: we indicate that in some non- equilibrium situations, entropies and thermodynamic free energies can be derived via large deviation principles. Read More

We introduce a stochastic particle system that corresponds to the Fokker-Planck equation with decay in the many-particles limit, and study its large deviations. We show that the large-deviation rate functional corresponds to an energy-dissipation functional in a Gamma-convergence sense. Moreover, we prove that the resulting functional, which involves entropic terms and the Wasserstein metric, is again a variational formulation for the Fokker-Planck equation with decay. Read More

In the deformation of layered materials such as geological strata, or stacks of paper, mechanical properties compete with the geometry of layering. Smooth, rounded corners lead to voids between the layers, while close packing of the layers results in geometrically-induced curvature singularities. When voids are penalized by external pressure, the system is forced to trade off these competing effects, leading to sometimes striking periodic patterns. Read More

We study a singular-limit problem arising in the modelling of chemical reactions. At finite {\epsilon} > 0, the system is described by a Fokker-Planck convection-diffusion equation with a double-well convection potential. This potential is scaled by 1/{\epsilon}, and in the limit {\epsilon} -> 0, the solution concentrates onto the two wells, resulting into a limiting system that is a pair of ordinary differential equations for the density at the two wells. Read More

In this paper we derive an obstacle problem with a free boundary to describe the formation of voids at areas of intense geological folding. An elastic layer is forced by overburden pressure against a V-shaped rigid obstacle. Energy minimization leads to representation as a nonlinear fourth-order ordinary differential equation, for which we prove their exists a unique solution. Read More

We present the second of two articles on the small volume fraction limit of a nonlocal Cahn-Hilliard functional introduced to model microphase separation of diblock copolymers. After having established the results for the sharp-interface version of the functional (arXiv:0907.2224), we consider here the full diffuse-interface functional and address the limit in which epsilon and the volume fraction tend to zero but the number of minority phases (called particles) remains O(1). Read More

We study the connection between a system of many independent Brownian particles on one hand and the deterministic diffusion equation on the other. For a fixed time step $h>0$, a large-deviations rate functional $J_h$ characterizes the behaviour of the particle system at $t=h$ in terms of the initial distribution at $t=0$. For the diffusion equation, a single step in the time-discretized entropy-Wasserstein gradient flow is characterized by the minimization of a functional $K_h$. Read More

We shall prove new contraction properties of general transportation costs along nonnegative measure-valued solutions to Fokker-Planck equations in $R^d$, when the drift is a monotone (or $\lambda$-monotone) operator. A new duality approach to contraction estimates has been developed: it relies on the Kantorovich dual formulation of optimal transportation problems and on a variable-doubling technique. The latter is used to derive a new comparison property of solutions of the backward Kolmogorov (or dual) equation. Read More

We study the limit of high activation energy of a special Fokker-Planck equation, known as Kramers-Smoluchowski (K-S) equation. This equation governs the time evolution of the probability density of a particle performing a Brownian motion under the influence of a chemical potential H/e. We choose H having two wells corresponding to two chemical states A and B. Read More

We study a new formulation for the eikonal equation |grad u| =1 on a bounded subset of R^2. Considering a field P of orthogonal projections onto 1-dimensional subspaces, with divergence bounded in L^2, we prove existence and uniqueness for solutions of the equation P div P=0. We give a geometric description, comparable with the classical case, and we prove that such solutions exist only if the domain is a tubular neighbourhood of a regular closed curve. Read More

We study the H^{-1}-norm of the function 1 on tubular neighbourhoods of curves in R^2. We take the limit of small thickness epsilon, and we prove two different asymptotic results. The first is an asymptotic development for a fixed curve in the limit epsilon to 0, containing contributions from the length of the curve (at order epsilon^3), the ends (epsilon^4), and the curvature (epsilon^5). Read More

We consider a pattern-forming system in two space dimensions defined by an energy G_e. The functional G_e models strong phase separation in AB diblock copolymer melts, and patterns are represented by {0,1}-valued functions; the values 0 and 1 correspond to the A and B phases. The parameter e is the ratio between the intrinsic, material length scale and the scale of the domain. Read More

We develop a gradient-flow framework based on the Wasserstein metric for a parabolic moving-boundary problem that models crystal dissolution and precipitation. In doing so we derive a new weak formulation for this moving-boundary problem and we show that this formulation is well-posed. In addition, we develop a new uniqueness technique based on the framework of gradient flows with respect to the Wasserstein metric. Read More

We study a new formulation for the eikonal equation |grad u| =1 on a bounded subset of R^2. Instead of a vector field grad u, we consider a field P of orthogonal projections on 1-dimensional subspaces, with div P in L^2. We prove existence and uniqueness for solutions of the equation P div P=0. Read More

We study the stability of layered structures in a variational model for diblock copolymer-homopolymer blends. The main step consists of calculating the first and second derivative of a sharp-interface Ohta-Kawasaki energy for straight mono- and bilayers. By developing the interface perturbations in a Fourier series we fully characterise the stability of the structures in terms of the energy parameters. Read More

We study a variational model for a diblock-copolymer/homopolymer blend. The energy functional is a sharp-interface limit of a generalisation of the Ohta-Kawasaki energy. In one dimension, on the real line and on the torus, we prove existence of minimisers of this functional and we describe in complete detail the structure and energy of stationary points. Read More

We derive a new criterion for a real-valued function $u$ to be in the Sobolev space $W^{1,2}(\R^n)$. This criterion consists of comparing the value of a functional $\int f(u)$ with the values of the same functional applied to convolutions of $u$ with a Dirac sequence. The difference of these values converges to zero as the convolutions approach $u$, and we prove that the rate of convergence to zero is connected to regularity: $u\in W^{1,2}$ if and only if the convergence is sufficiently fast. Read More

We review and compare different computational variational methods applied to a system of fourth order equations that arises as a model of cylinder buckling. We describe both the discretization and implementation, in particular how to deal with a 1 dimensional null space. We show that we can construct many different solutions from a complex energy surface. Read More

Partial localization is the phenomenon of self-aggregation of mass into high-density structures that are thin in one direction and extended in the others. We give a detailed study of an energy functional that arises in a simplified model for lipid bilayer membranes. We demonstrate that this functional, defined on a class of two-dimensional spatial mass densities, exhibits partial localization and displays the `solid-like' behavior of cell membranes. Read More

We revisit the classical problem of the buckling of a long thin axially compressed cylindrical shell. By examining the energy landscape of the perfect cylinder we deduce an estimate of the sensitivity of the shell to imperfections. Key to obtaining this is the existence of a mountain pass point for the system. Read More

In this paper we consider a system of equations that describes a class of mass-conserving aggregation phenomena, including gravitational collapse and bacterial chemotaxis. In spatial dimensions strictly larger than two, and under the assumptions of radial symmetry, it is known that this system has at least two stable mechanisms of singularity formation (see e.g. Read More

A hallmark of a plethora of intracellular signaling pathways is the spatial separation of activation and deactivation processes that potentially results in precipitous gradients of activated proteins. The classical Metabolic Control Analysis (MCA), which quantifies the influence of an individual process on a system variable as the control coefficient, cannot be applied to spatially separated protein networks. The present paper unravels the principles that govern the control over the fluxes and intermediate concentrations in spatially heterogeneous reaction networks. Read More