A gradient structure for systems coupling reaction–diffusion effects in bulk and interfaces

Glitzky, Annegret; Mielke, Alexander

doi:10.1007/s00033-012-0207-y

Cited by 57 publications

(73 citation statements)

References 22 publications

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“…Detailed-balance, mass-action systems are a natural choice for 'non-dissipative' systems. They can be considered thermodynamically closed, and admit a free-energy functional F that drives the evolution in a gradient-flow structure [GM13,MM]. In this context, one can identify 'dissipation' with the instantaneous decrease of F , and the system is therefore non-dissipative in the sense that at all stationary points F is constant.…”

Section: Non-detailed-balance Chemical Reaction Networkmentioning

confidence: 99%

Precision and Sensitivity in Detailed-Balance Reaction Networks

Greef¹,

Masroor²,

Peletier³

et al. 2016

SIAM J. Appl. Math.

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Abstract. 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. High values of both are necessary for high-quality adaptation.We focus on reaction networks without dissipation, which we interpret as detailed-balance, massaction networks. We give various upper and lower bounds on the optimal values of Sensitivity and Precision, characterized in terms of the stoichiometry, by using a combination of ideas from matroid theory and differential-equation theory.Among other results, we show that this class of non-dissipative systems contains networks with arbitrarily high values of both Sensitivity and Precision. This good performance does come at a cost, however, since certain ratios of concentrations need to be large, the network has to be extensive, or the network should show strongly different time scales.

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Section: Non-detailed-balance Chemical Reaction Networkmentioning

confidence: 99%

Precision and Sensitivity in Detailed-Balance Reaction Networks

Greef¹,

Masroor²,

Peletier³

et al. 2016

SIAM J. Appl. Math.

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“…It is well known [7] that equation (1.1) in the density formulation (1.7) is the gradient flow of the Dirichlet form 17) in the weighted Hilbert space Z lin ε = L 2 ( ; γ ε ). In this approach the quadratic dissipation potential (which is also the squared metric velocity) of a curve u is…”

Section: Gradient Flows In a Metric Settingmentioning

confidence: 99%

“…• This fact is strongly related to the new variational formulation of reaction-diffusion systems, see [17,27]. In fact, it turns out that many reaction systems (even with nonlinear reactions of mass-action type) can be written as a gradient system with a well-chosen dissipation functional ψ * 0 .…”

Section: The Variational Approach: Basic Tools and Main Ideasmentioning

confidence: 99%

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Passing to the limit in a Wasserstein gradient flow: from diffusion to reaction

et al. 2011

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We study a singular-limit problem arising in the modelling of chemical reactions. At finite ε > 0, the system is described by a Fokker-Planck convection-diffusion equation with a double-well 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 et al. (SIAM J Math Anal, 42(4):1805-1825, 2010, using the linear structure of the equation. In this study we re-prove the result by using solely the Wasserstein gradient-flow structure of the system. In particular we make no use of the linearity, nor of the fact that it is a second-order 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 gradient-flow structure, we prove that the sequence of rescaled solutions is pre-compact in an appropriate topology. We then prove a Gamma-convergence 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 ε-problem converge to a solution of the limiting problem.

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“…[29][30][31][32][33], have the structure developed above. Except for recent work [4,13,34,35], the gradient structure was not displayed and exploited explicitly, only the Liapunov property of the free energy E was used for deriving a priori estimates.…”

Section: (B) Coupling Diffusion and Reactionmentioning

confidence: 99%

Gradient structures and geodesic convexity for reaction–diffusion systems

Liero

Mielke

2013

Phil. Trans. R. Soc. A.

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We consider systems of reaction–diffusion equations as gradient systems with respect to an entropy functional and a dissipation metric given in terms of a so-called Onsager operator, which is a sum of a diffusion part of Wasserstein type and a reaction part. We provide methods for establishing geodesic λ -convexity of the entropy functional by purely differential methods, thus circumventing arguments from mass transportation. Finally, several examples, including a drift–diffusion system, provide a survey on the applicability of the theory.

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A gradient structure for systems coupling reaction–diffusion effects in bulk and interfaces

Cited by 57 publications

References 22 publications

Precision and Sensitivity in Detailed-Balance Reaction Networks

Precision and Sensitivity in Detailed-Balance Reaction Networks

Passing to the limit in a Wasserstein gradient flow: from diffusion to reaction

Gradient structures and geodesic convexity for reaction–diffusion systems

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