Modeling of Chemical Reaction Systems with Detailed Balance Using Gradient Structures

Maas, Jan; Mielke, Alexander

doi:10.1007/s10955-020-02663-4

Cited by 25 publications

(38 citation statements)

References 58 publications

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“…The gradient‐flow classical structure for a driving functional U is

𝔾 false(n false) \overset{n}{true} = - scriptD U;

starting with Mielke, 28 provided there exists

𝕂 false(n false) = 𝔾^{- 1} false(n false)

, the following formulation was proposed and successfully used into, e.g., 29–36

\overset{n}{true} = - 𝕂 false(n false) false[scriptD U false],

where the Onsager structure

𝕂

can be splitted additively into different contributions. For instance, in our case, we may set:

𝕂 = 𝕂_{D} + 𝕂_{R}, 𝕂_{D} false[\cdot false] = - div S false(n false) \nabla false[\cdot false], 𝕂_{R} false[\cdot false] = H false[\cdot false] .

…”

Section: The Continuum Model For Isothermal and Rigid Scintillators: The Boundary Value Problemmentioning

confidence: 99%

“…In a series of recent papers, 32–36 dealing with the gradient structure of () it is shown that also () admits a gradient structure with

𝕂 = 𝕂_{R}

for detailed balanced recombination mechanisms and hence, the energy dissipation methods can be used with success to get asymptotic decay estimates.…”

Section: The Approximated Modelsmentioning

confidence: 99%

“…In detail, into Maas and Mielke, 36 the well‐posedness of () was obtained, in the sense that for all initial data

n_{o} \in scriptM

there exists an unique global solution

n : false[0, τ false) \to scriptM

. Moreover it was proved that since the recombination term satisfies the detailed balance conditions for the equilibrium state n ∞ = c , then () admits the gradient structure:

\overset{n}{true} = - H scriptD U false(n false),

with H given by ().…”

Section: The Approximated Modelsmentioning

confidence: 99%

See 2 more Smart Citations

A brief overview of existence results and decay time estimates for a mathematical modeling of scintillating crystals

Davı́

2021

Math Methods in App Sciences

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Inorganic scintillating crystals can be modelled as continua with microstructure.For rigid and isothermal crystals, the evolution of charge carriers becomes in this way described by a reaction-diffusion-drift equation coupled with the Poisson equation of electrostatic. Here, we give a survey of the available existence and asymptotic decays results for the resulting boundary value problem, the latter being a direct estimate of the scintillation decay time. We also show how to recover various approximated models which encompass also the two most used phenomenological models for scintillators, namely, the kinetic and diffusive ones. Also for these cases, we show, whenever it is possible, which existence and asymptotic decays estimate results are known to date. KEYWORDS entropy methods, existence of solutions of PDE, exponential rate of convergence, reaction-diffusion-drift equations, scintillators MSC CLASSIFICATION 35K57; 35B40; 35B45 INTRODUCTIONA scintillator crystal is a material which converts ionizing radiations into photons in the frequency range of visible light, hence its name. It acts as a true "wavelength shifter" and in such a role is used as radiation sensor into high-energy physics, in medical imaging and in security applications. 1 The physics of scintillation, which is a complex multi-scale phenomenon (see, e.g., Vasil'ev and Getkin 2 ) can be described within a continuum approach at three scales: at a microscopic scale, the incoming energy E generates a population of charged energy carriers which moves in straight directions for few nanometer 3 and whose density N = N(E) can be found by the means of approximated solutions of the Bethe-Bloch equation. 4-7* These energy carriers wander and migrate within a greater region either generating other energy carriers or recombining with emission of photons h𝜈. In the process, some energy is lost, and a scintillator is a material in which

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“…The gradient‐flow classical structure for a driving functional U is

𝔾 false(n false) \overset{n}{true} = - scriptD U;

starting with Mielke, 28 provided there exists

𝕂 false(n false) = 𝔾^{- 1} false(n false)

, the following formulation was proposed and successfully used into, e.g., 29–36

\overset{n}{true} = - 𝕂 false(n false) false[scriptD U false],

where the Onsager structure

𝕂

can be splitted additively into different contributions. For instance, in our case, we may set:

𝕂 = 𝕂_{D} + 𝕂_{R}, 𝕂_{D} false[\cdot false] = - div S false(n false) \nabla false[\cdot false], 𝕂_{R} false[\cdot false] = H false[\cdot false] .

…”

Section: The Continuum Model For Isothermal and Rigid Scintillators: The Boundary Value Problemmentioning

confidence: 99%

“…In a series of recent papers, 32–36 dealing with the gradient structure of () it is shown that also () admits a gradient structure with

𝕂 = 𝕂_{R}

for detailed balanced recombination mechanisms and hence, the energy dissipation methods can be used with success to get asymptotic decay estimates.…”

Section: The Approximated Modelsmentioning

confidence: 99%

“…In detail, into Maas and Mielke, 36 the well‐posedness of () was obtained, in the sense that for all initial data

n_{o} \in scriptM

there exists an unique global solution

n : false[0, τ false) \to scriptM

. Moreover it was proved that since the recombination term satisfies the detailed balance conditions for the equilibrium state n ∞ = c , then () admits the gradient structure:

\overset{n}{true} = - H scriptD U false(n false),

with H given by ().…”

Section: The Approximated Modelsmentioning

confidence: 99%

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A brief overview of existence results and decay time estimates for a mathematical modeling of scintillating crystals

Davı́

2021

Math Methods in App Sciences

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show abstract

“…Given compactness and local equilibrium, the hydrodynamic limit of interacting particle systems is usually proven at the level of the stochastic process [KL99]. In this work, we take a different route and, still not aiming at a full rigorous proof, apply a heuristic method by Maas and Mielke that uses gradient structures [MM20]. These are geometric structures that were introduced in both physical [Grm85,Mor86] and mathematical [DGMT80,CV90] contexts to model dissipative phenomena, in particular in the literature of nonequilibrium thermodynamics [Ons31, G Ö97].…”

Section: Introductionmentioning

confidence: 99%

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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“…Already the analysis of the arising equations for the pair distribution is not fully covered by the existing PDE theory, it is also open whether it is more suitable to study the formulation with the pair distribution or rather the one with the conditional distribution. Also the analysis of gradient structures or generalizations at the level of the pair distribution is an interesting topic for further investigations, related to the evolution of gradient structures like in [13] and possibly with other structures as in [42] for discrete models. An even more challenging question is the analysis of pair closure relations for the BBGKY-type hierarchy, for which one would possibly need similar quantitative estimates as for the mean-field limit in [47,54], which currently seem out of reach.…”

mentioning

confidence: 99%

Kinetic equations for processes on co-evolving networks

Burger¹

2022

KRM

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<p style='text-indent:20px;'>The aim of this paper is to derive macroscopic equations for processes on large co-evolving networks, examples being opinion polarization with the emergence of filter bubbles or other social processes such as norm development. This leads to processes on graphs (or networks), where both the states of particles in nodes as well as the weights between them are updated in time. In our derivation we follow the basic paradigm of statistical mechanics: We start from paradigmatic microscopic models and derive a Liouville-type equation in a high-dimensional space including not only the node states in the network (corresponding to positions in mechanics), but also the edge weights between them. We then derive a natural (finite) marginal hierarchy and pass to an infinite limit.</p><p style='text-indent:20px;'>We will discuss the closure problem for this hierarchy and see that a simple mean-field solution can only arise if the weight distributions between nodes of equal states are concentrated. In a more interesting general case we propose a suitable closure at the level of a two-particle distribution (including the weight between them) and discuss some properties of the arising kinetic equations. Moreover, we highlight some structure-preserving properties of this closure and discuss its analysis in a minimal model. We discuss the application of our theory to some agent-based models in literature and discuss some open mathematical issues.</p>

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Modeling of Chemical Reaction Systems with Detailed Balance Using Gradient Structures

Cited by 25 publications

References 58 publications

A brief overview of existence results and decay time estimates for a mathematical modeling of scintillating crystals

A brief overview of existence results and decay time estimates for a mathematical modeling of scintillating crystals

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

Kinetic equations for processes on co-evolving networks

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