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22
Passing to the limit in a wasserstein gradient flow: From diffusion to reaction
 Calc. Var. Partial Differential Equations
"... Abstract. We study a singularlimit problem arising in the modelling of chemical reactions. At finite ε> 0, the system is described by a FokkerPlanck convectiondiffusion equation with a doublewell convection potential. This potential is scaled by 1/ε, and in the limit ε → 0, the solution conce ..."
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Cited by 17 (5 self)
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Abstract. We study a singularlimit problem arising in the modelling of chemical reactions. At finite ε> 0, the system is described by a FokkerPlanck convectiondiffusion equation with a doublewell convection potential. This potential is scaled by 1/ε, and in the limit ε → 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. This convergence has been proved in Peletier, Savaré, and Veneroni, SIAM Journal on Mathematical Analysis, 42(4):1805–1825, 2010, using the linear structure of the equation. In this paper we reprove the result by using solely the Wasserstein gradientflow structure of the system. In particular we make no use of the linearity, nor of the fact that it is a secondorder system. The first key step in this approach is a reformulation of the equation as the minimization of an action functional that captures the property of being a curve of maximal slope in an integrated form. The second important step is a rescaling of space. Using only the Wasserstein gradientflow structure, we prove that the sequence of rescaled solutions is precompact in an appropriate topology. We then prove a Gammaconvergence result for the functional in this topology, and we identify the limiting functional and the differential equation that it represents. A consequence of these results is that solutions of the εproblem converge to a solution of the limiting problem.
Variational formulation of the FokkerPlanck equation with decay: A particle approach. Arxiv preprint arXiv:1108.3181
, 2011
"... We introduce a stochastic particle system that corresponds to the Fokker–Planck equation with decay in the manyparticle limit, and study its large deviations. We show that the largedeviation rate functional corresponds to an energydissipation functional in a Moscoconvergence sense. Moreover, we ..."
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Cited by 12 (6 self)
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We introduce a stochastic particle system that corresponds to the Fokker–Planck equation with decay in the manyparticle limit, and study its large deviations. We show that the largedeviation rate functional corresponds to an energydissipation functional in a Moscoconvergence 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.
FROM DIFFUSION TO REACTION VIA ΓCONVERGENCE
, 2010
"... We study the limit of high activation energy of a special Fokker–Planck equation known as the Kramers–Smoluchowski equation (KS). 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/ε. We choose ..."
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Cited by 10 (2 self)
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We study the limit of high activation energy of a special Fokker–Planck equation known as the Kramers–Smoluchowski equation (KS). 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/ε. We choose H having two wells corresponding to two chemical states A and B. We prove that after a suitable rescaling the solution to KS converges, in the limit of high activation energy (ε → 0), to the solution of a simple system modeling the diffusion of A and B, and the reaction A ⇋ B. The aim of this paper is to give a rigorous proof of Kramers’s formal derivation and to embed chemical reactions and diffusion processes in a common variational framework which allows one to derive the former as a singular limit of the latter, thus establishing a connection between two worlds often regarded as separate. The singular limit is analyzed by means of Γconvergence in the space of finite Borel measures endowed with the weak∗ topology.
Chemical Reactions as ΓLimit of Diffusion
, 2012
"... We study the limit of high activation energy of a special Fokker–Planck equation known as the Kramers–Smoluchowski equation (KS). 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/ε. We choose ..."
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Cited by 3 (0 self)
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We study the limit of high activation energy of a special Fokker–Planck equation known as the Kramers–Smoluchowski equation (KS). 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/ε. We choose H having two wells corresponding to two chemical states A and B. We prove that after a suitable rescaling the solution to the KS converges, in the limit of high activation energy (ε → 0), to the solution of a simpler system modeling the spatial diffusion of A and B combined with the reaction A ⇋ B. With this result we give a rigorous proof of Kramers’s formal derivation, and we show how chemical reactions and diffusion processes can be embedded in a common framework. This allows one to derive a chemical reaction as a singular limit of a diffusion process, thus establishing a connection between two worlds often regarded as separate. The proof rests on two main ingredients. One is the formulation of the two disparate equations as evolution equations for measures. The second is a variational formulation of both equations that allows us to use the tools of variational calculus and, specifically, Γconvergence.
Lyapunov functionals for boundarydriven nonlinear driftdiffusions
, 2013
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Calculus of Variations Passing to the limit in a Wasserstein gradient flow: from diffusion to reaction
, 2012
"... ..."
Sufficient Conditions for Global Minimality of Metastable States in a Class of Nonconvex Functionals: A Simple Approach Via Quadratic
"... Abstract We consider massconstrained minimizers for a class of nonconvex energy functionals involving a doublewell potential. Based upon global quadratic lower bounds to the energy, we introduce a simple strategy to find sufficient conditions on a given critical point (metastable state) to be a g ..."
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Abstract We consider massconstrained minimizers for a class of nonconvex energy functionals involving a doublewell potential. Based upon global quadratic lower bounds to the energy, we introduce a simple strategy to find sufficient conditions on a given critical point (metastable state) to be a global minimizer. We show that this strategy works well for the one exact and known metastable state: the constant state. In doing so, we numerically derive an almost optimal lower bound for both the order–disorder transition curve of the Ohta–Kawasaki energy and the liquid–solid interface of the phasefield crystal energy. We discuss how this strategy extends to nonconstant computed metastable states, and the resulting symmetry issues that one must overcome. We give a preliminary analysis of these symmetry issues by addressing the global optimality of a computed lamellar structure for the Ohta–Kawasaki energy in one (1D) and two (2D) space dimensions. We also consider global optimality of a nonconstant state for a spatially inhomogenous perturbation of the 2D Ohta–Kawasaki energy. Finally we use one of our simple quadratic lower bounds to rigorously prove that for certain values of the Ohta–Kawasaki parameter and aspect ratio of an asymmetric torus, any global minimizer v(x) for the 1D problem is automatically a global minimizer for the 2D problem on the asymmetric torus. Communicated by Robert V. Kohn.
Energies, gradient flows, and large deviations: a modelling point of view
, 2012
"... Modelling is the art of taking a realworld situation and constructing some mathematics to describe it. It’s an art rather than a science, because it involves choices that can not be rationally justified. Personal preferences are important, and ‘taste ’ plays a major role. ..."
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Modelling is the art of taking a realworld situation and constructing some mathematics to describe it. It’s an art rather than a science, because it involves choices that can not be rationally justified. Personal preferences are important, and ‘taste ’ plays a major role.