An inequality connecting entropy distance, Fisher Information and large deviations

Hilder, Bastian; Peletier, Mark A.; Sharma, Upanshu; Tse, Oliver

doi:10.1016/j.spa.2019.07.012

Cited by 13 publications

(22 citation statements)

References 31 publications

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“…While this is a hard question to answer in general, considerable progress has been made in the case of some specific systems in two seemingly independent directions. One direction that is tailored to allow for non-dissipative effects is the study of so-called FIR inequalities, first introduced for the many-particle limit of Vlasov-type nonlinear diffusions [DLPS17], independent particles on a graph [HPST20] and chemical reactions [RZ21,Sec. 5].…”

Section: Introductionmentioning

confidence: 99%

“…Inspired by FIR-inequalities and MFT, we set up an abstract framework whose central outcome will be a series of decompositions of the integrand L into distinct dissipative and non-dissipative components. These decompositions generalise: (1) the connection between large deviations and dissipative systems from [MPR14] to include non-dissipative effects, (2) the known cases of FIR inequalities [HPST20] to a general setting, and (3) MFT to non-quadratic action functions.…”

Section: Introductionmentioning

confidence: 99%

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Variational structures beyond gradient flows: a macroscopic fluctuation-theory perspective

Patterson,

Renger,

Sharma

2021

Preprint

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Macroscopic equations arising out of stochastic particle systems in detailed balance (called dissipative systems or gradient flows) have a natural variational structure, which can be derived from the large-deviation rate functional for the density of the particle system. While large deviations can be studied in considerable generality, these variational structures are often restricted to systems in detailed balance. Using insights from macroscopic fluctuation theory, in this work we aim to generalise this variational connection beyond dissipative systems by augmenting densities with fluxes, which encode non-dissipative effects. Our main contribution is an abstract framework, which for a given flux-density cost and a quasipotential, provides a decomposition into dissipative and non-dissipative components and a generalised orthogonality relation between them. We then apply this abstract theory to various stochastic particle systems -independent copies of jump processes, zero-range processes, chemical-reaction networks in complex balance and lattice-gas models.

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Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Variational structures beyond gradient flows: a macroscopic fluctuation-theory perspective

Patterson,

Renger,

Sharma

2021

Preprint

Self Cite

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“…(2) We have seen that the coarse-graining method of Maas and Mielke, in the hydrodynamic limit of reaction-diffusion systems, has to be supplied with a limit procedure. Coarse-graining and limit, indeed, are two ingredients of the same recipe and fit well together in a variational framework [DLPS17,HPST20]. In a future work, we will demonstrate how the variational framework of these two papers can be applied to the hydrodynamic limit of reaction-diffusion systems.…”

Section: The Limit N → ∞mentioning

confidence: 74%

A route to the hydrodynamic limit of a reaction-diffusion master equation using gradient structures

Montefusco¹,

Schütte²,

Winkelmann³

2022

Preprint

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The reaction-diffusion master equation (RDME) is a lattice-based stochastic model for spatially resolved cellular processes. It is often interpreted as an approximation to spatially continuous reaction-diffusion models, which, in the limit of an infinitely large population, may be described by means of reaction-diffusion partial differential equations (RDPDEs). Analyzing and understanding the relation between different mathematical models for reaction-diffusion dynamics is a research topic of steady interest. In this work, we explore a route to the hydrodynamic limit of the RDME which uses gradient structures. Specifically, we elaborate on a method introduced in [J. Maas, A. Mielke.: Modeling of chemical reactions systems with detailed balance using gradient structures. J. Stat. Phys. (181), 2257-2303] in the context of well-mixed reaction networks by showing that, once it is complemented with an appropriate limit procedure, it can be applied to spatially extended systems with diffusion. Under the assumption of detailed balance, we write down a gradient structure for the RDME and use the method to produce a gradient structure for its hydrodynamic limit, namely, for the corresponding RDPDE.

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“…4 and 7, there is a nontrivial interaction of the two terms, as a result of which the individual liminf estimates do not hold. [11,20,38]) A related line of evolutionary convergence in variational systems centers around convergence of the functional q [11,20], the authors use a duality formulation for J ε to combine a coarse-graining map and the limit ε → 0 into a single method.…”

Section: Remark 220 (Comparison Tomentioning

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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An inequality connecting entropy distance, Fisher Information and large deviations

Cited by 13 publications

References 31 publications

Variational structures beyond gradient flows: a macroscopic fluctuation-theory perspective

Variational structures beyond gradient flows: a macroscopic fluctuation-theory perspective

A route to the hydrodynamic limit of a reaction-diffusion master equation using gradient structures

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

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