Deriving Effective Models for Multiscale Systems via Evolutionary $$\varGamma $$ Γ -Convergence

Mielke, Alexander

doi:10.1007/978-3-319-28028-8_12

Cited by 9 publications

(12 citation statements)

References 24 publications

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“…We now turn to the evolution. A gradient-flow structure is defined by a state space, a driving functional, and a dissipation metric [75,83]. The driving functional was identified above aŝ F; the large-deviation principle that we prove below will indicate that the state space for this gradient-flow structure is the metric space given by the set P 2 (R) of probability measures of finite second moment (i.e.…”

Section: Gradient Flowsmentioning

confidence: 90%

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Large Deviations and Gradient Flows for the Brownian One-Dimensional Hard-Rod System

2021

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We study a system of hard rods of finite size in one space dimension, which move by Brownian noise while avoiding overlap. We consider a scaling in which the number of particles tends to infinity while the volume fraction of the rods remains constant; in this limit the empirical measure of the rod positions converges almost surely to a deterministic limit evolution. We prove a large-deviation principle on path space for the empirical measure, by exploiting a one-to-one mapping between the hard-rod system and a system of non-interacting particles on a contracted domain. The large-deviation principle naturally identifies a gradient-flow structure for the limit evolution, with clear interpretations for both the driving functional (an ‘entropy’) and the dissipation, which in this case is the Wasserstein dissipation. This study is inspired by recent developments in the continuum modelling of multiple-species interacting particle systems with finite-size effects; for such systems many different modelling choices appear in the literature, raising the question how one can understand such choices in terms of more microscopic models. The results of this paper give a clear answer to this question, albeit for the simpler one-dimensional hard-rod system. For this specific system this result provides a clear understanding of the value and interpretation of different modelling choices, while giving hints for more general systems.

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Section: Gradient Flowsmentioning

confidence: 90%

“…75. The uniqueness of u is a direct consequence of the strict convexity of u 2 μ and the linear constraint (75).…”

Section: This Is Precisely (74)mentioning

confidence: 99%

Large Deviations and Gradient Flows for the Brownian One-Dimensional Hard-Rod System

2021

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“…The energy-dissipation principle formulation (2.6) of a gradient flow leads to a natural concept of gradient system convergence. A first version of this concept was formulated by Sandier and Serfaty [36] and generalizations have been used in a large number of proofs (see, e.g., [5,22,[24][25][26]29,39]). Definition 2.7 (Simple EDP convergence) A family of gradient systems (Q, E ε , R ε ) converges in the simple EDP sense to a gradient system (Q,…”

Section: Simple Edp Convergencementioning

confidence: 99%

Exploring families of energy-dissipation landscapes via tilting: three types of EDP convergence

Mielke

Montefusco

Peletier

2021

Continuum Mech. Thermodyn.

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We introduce two new concepts of convergence of gradient systems $$({\mathbf{Q}}, {{\mathcal {E}}}_\varepsilon ,{{\mathcal {R}}}_\varepsilon )$$ ( Q , E ε , R ε ) to a limiting gradient system $$({\mathbf{Q}},{{\mathcal {E}}}_0,{{\mathcal {R}}}_0)$$ ( Q , E 0 , R 0 ) . These new concepts are called ‘EDP convergence with tilting’ and ‘contact–EDP convergence with tilting.’ Both are based on the energy-dissipation-principle (EDP) formulation of solutions of gradient systems and can be seen as refinements of the Gamma-convergence for gradient flows first introduced by Sandier and Serfaty. The two new concepts are constructed in order to avoid the ‘unnatural’ limiting gradient structures that sometimes arise as limits in EDP convergence. EDP convergence with tilting is a strengthening of EDP convergence by requiring EDP convergence for a full family of ‘tilted’ copies of $$({\mathbf{Q}}, {{\mathcal {E}}}_\varepsilon ,{{\mathcal {R}}}_\varepsilon )$$ ( Q , E ε , R ε ) . It avoids unnatural limiting gradient structures, but many interesting systems are non-convergent according to this concept. Contact–EDP convergence with tilting is a relaxation of EDP convergence with tilting and still avoids unnatural limits but applies to a broader class of sequences $$({\mathbf{Q}}, {{\mathcal {E}}}_\varepsilon ,{{\mathcal {R}}}_\varepsilon )$$ ( Q , E ε , R ε ) . In this paper, we define these concepts, study their properties, and connect them with classical EDP convergence. We illustrate the different concepts on a number of test problems.

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“…The most frequently used method is to consider an ε-family of equations, where the occurring terms depend on the parameter ε, and then to pass to the limit as ε → ∞, where the limit equation corresponds to the unperturbed system. Another way to treat perturbed systems is to use an additional term in the equations like the term B t in (1.1) or even a combination of both as in [Mie16a], where the author considered the family of equations…”

Section: Introductionmentioning

confidence: 99%

“…to derive results on the so-called evolutionary Γ -convergence. Second, [Mie16a,p. 235] highlights with an example that in some cases it can be easier to treat a system with a nontrivial but exact gradient structure (X, E, Ψ ) perturbed gradient system (V, E, Ψ, B) with a simpler energy E and simpler dissipation potentials Ψ u .…”

Section: Introductionmentioning

confidence: 99%

An existence result and evolutionary $$\varGamma $$ Γ -convergence for perturbed gradient systems

2019

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The initial-value problem for the perturbed gradient flow B (t, u(t)are nonsmooth and supposed to be convex and nonconvex, respectively. The perturbationis assumed to be continuous and satisfies a growth condition. Under additional assumptions on the dissipation potential and the energy functional, existence of strong solutions is shown by proving convergence of a semi-implicit discretization scheme with a variational approximation technique. * The research was partially supported by DFG via SFB 910 via the subprojects A5 and A8

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Deriving Effective Models for Multiscale Systems via Evolutionary $$\varGamma $$ Γ -Convergence

Cited by 9 publications

References 24 publications

Large Deviations and Gradient Flows for the Brownian One-Dimensional Hard-Rod System

Large Deviations and Gradient Flows for the Brownian One-Dimensional Hard-Rod System

Exploring families of energy-dissipation landscapes via tilting: three types of EDP convergence

An existence result and evolutionary $$\varGamma $$ Γ -convergence for perturbed gradient systems

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