Large deviations in stochastic heat-conduction processes provide a gradient-flow structure for heat conduction

Peletier, Mark A.; Redig, Frank; Vafayi, Kiamars

doi:10.1063/1.4894139

Cited by 27 publications

(24 citation statements)

References 39 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…[31] and our Sect. 7); as the limit of the simple symmetric exclusion process describing particles hopping on a lattice [3], with a gradient structure of a mixing entropy and a modified Wasserstein distance; and as the limit of oscillators that exchange energy ('heat') [35], with a gradient structure consisting of an alternative logarithmic entropy and again a modified Wasserstein distance. Rate-independent systems arise from taking further limits [7], and extensions to GENERIC have also been recognized [12].…”

Section: Gradient Flows and Large Deviations Of Markov Processesmentioning

confidence: 99%

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

Mielke

Montefusco

Peletier

2021

Continuum Mech. Thermodyn.

Self Cite

View full text Add to dashboard Cite

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.

show abstract

Section: Gradient Flows and Large Deviations Of Markov Processesmentioning

confidence: 99%

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

Mielke

Montefusco

Peletier

2021

Continuum Mech. Thermodyn.

Self Cite

View full text Add to dashboard Cite

show abstract

“…Different combinations of choices, however, can lead to the same equation (see e.g. [85] or [31,Eq. 2.1]).…”

Section: Variational Modellingmentioning

confidence: 99%

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

2021

View full text Add to dashboard Cite

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.

show abstract

“…We also refer to [PRV14,MPR14] for discussion of different gradient structures for the heat equation or for finite-state Markov processes. Thus, we emphasize that the gradient structure of a given ODE has additional physical information, e.g.…”

Section: The Energy-dissipation Principle For Gradient Systemmentioning

confidence: 99%

A gradient system with a wiggly energy and relaxed EDP-convergence

2019

View full text Add to dashboard Cite

For gradient systems depending on a microstructure, it is desirable to derive a macroscopic gradient structure describing the effective behavior of the microscopic scale on the macroscopic evolution. We introduce a notion of evolutionary Gamma-convergence that relates the microscopic energy and the microscopic dissipation potential with their macroscopic limits via Gamma-convergence. This new notion generalizes the concept of EDP-convergence, which was introduced in [26], and is now called relaxed EDP-convergence. Both notions are based on De Giorgi’s energy-dissipation principle (EDP), however the special structure of the dissipation functional in terms of the primal and dual dissipation potential is, in general, not preserved under Gamma-convergence. By using suitable tiltings we study the kinetic relation directly and, thus, are able to derive a unique macroscopic dissipation potential. The wiggly-energy model of Abeyaratne-Chu-James (1996) serves as a prototypical example where this nontrivial limit passage can be fully analyzed.

show abstract

Large deviations in stochastic heat-conduction processes provide a gradient-flow structure for heat conduction

Cited by 27 publications

References 39 publications

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

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

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

A gradient system with a wiggly energy and relaxed EDP-convergence

Contact Info

Product

Resources

About