scholarly journals EDP-convergence for nonlinear fast–slow reaction systems with detailed balance*

Nonlinearity ◽  
2021 ◽  
Vol 34 (8) ◽  
pp. 5762-5798
Author(s):  
Alexander Mielke ◽  
Mark A Peletier ◽  
Artur Stephan
Keyword(s):  
Detailed Balance ◽  
Reaction Systems ◽  
Slow Reaction

scholarly journals Coarse-graining via EDP-convergence for linear fast-slow reaction systems

2020 ◽  
Vol 30 (09) ◽  
pp. 1765-1807 ◽  
Author(s):  
Alexander Mielke ◽  
Artur Stephan
Keyword(s):  
Detailed Balance ◽  
Reaction System ◽  
Gradient Flow ◽  
Coarse Graining ◽  
Reaction Rates ◽  
Coarse Grained ◽  
Gradient Structure ◽  
Reaction Systems ◽  
Fast Reactions ◽  
Finite State

We consider linear reaction systems with slow and fast reactions, which can be interpreted as master equations or Kolmogorov forward equations for Markov processes on a finite state space. We investigate their limit behavior if the fast reaction rates tend to infinity, which leads to a coarse-grained model where the fast reactions create microscopically equilibrated clusters, while the exchange mass between the clusters occurs on the slow time scale. Assuming detailed balance the reaction system can be written as a gradient flow with respect to the relative entropy. Focusing on the physically relevant cosh-type gradient structure we show how an effective limit gradient structure can be rigorously derived and that the coarse-grained equation again has a cosh-type gradient structure. We obtain the strongest version of convergence in the sense of the Energy-Dissipation Principle (EDP), namely EDP-convergence with tilting.

scholarly journals Modeling of Chemical Reaction Systems with Detailed Balance Using Gradient Structures

2020 ◽  
Vol 181 (6) ◽  
pp. 2257-2303 ◽  
Author(s):  
Jan Maas ◽  
Alexander Mielke
Keyword(s):  
Chemical Reaction ◽  
Detailed Balance ◽  
Gradient Flow ◽  
Flow Equation ◽  
Gradient Flows ◽  
Reaction Systems ◽  
Chemical Reaction Systems ◽  
Detailed Balance Condition ◽  
Homogeneous Chemical ◽  
Reaction Rate Equation

AbstractWe consider various modeling levels for spatially homogeneous chemical reaction systems, namely the chemical master equation, the chemical Langevin dynamics, and the reaction-rate equation. Throughout we restrict our study to the case where the microscopic system satisfies the detailed-balance condition. The latter allows us to enrich the systems with a gradient structure, i.e. the evolution is given by a gradient-flow equation. We present the arising links between the associated gradient structures that are driven by the relative entropy of the detailed-balance steady state. The limit of large volumes is studied in the sense of evolutionary $$\Gamma $$ Γ -convergence of gradient flows. Moreover, we use the gradient structures to derive hybrid models for coupling different modeling levels.

Dielectronic Recombination in the Average-Atom Model

1988 ◽  
Vol 102 ◽  
pp. 215
Author(s):  
R.M. More ◽  
G.B. Zimmerman ◽  
Z. Zinamon
Keyword(s):  
Detailed Balance ◽  
Systematic Errors ◽  
Dielectronic Recombination ◽  
Rate Equations ◽  
Strong Correlations ◽  
Dense Plasmas ◽  
Atom Model ◽  
New Feature ◽  
Average Atom Model ◽  
Attachment Process

Autoionization and dielectronic attachment are usually omitted from rate equations for the non–LTE average–atom model, causing systematic errors in predicted ionization states and electronic populations for atoms in hot dense plasmas produced by laser irradiation of solid targets. We formulate a method by which dielectronic recombination can be included in average–atom calculations without conflict with the principle of detailed balance. The essential new feature in this extended average atom model is a treatment of strong correlations of electron populations induced by the dielectronic attachment process.

EXISTENCE AND INVARIANCE FOR CONVECTION, DISPERSION, REACTION SYSTEMS WITH PRESSURE

1992 ◽  
Vol 12 (4) ◽  
pp. 443-456 ◽  
Author(s):  
Chunhong Xie ◽  
Taiping He ◽  
Guohong Bai
Keyword(s):  
Reaction Systems

The Rotterdam Initiative in Reducing Waste Discharges at Source

1991 ◽  
Vol 24 (10) ◽  
pp. 171-177
Author(s):  
T. Vellinga ◽  
J. P. J. Nijssen
Keyword(s):  
Public Relations ◽  
Action Plan ◽  
Detailed Balance ◽  
River Rhine ◽  
The North ◽  
Complementary Role ◽  
Action Programme ◽  
The North Sea ◽  
Set Up

Much of the material dredged from the port of Rotterdam is contaminated to such a degree that it must be placed in specially constructed sites. The aim of Rotterdam is to ensure that the dredged material will once again be clean. This will entail the thorough cleansing of the sources of the contamination of the sediment in the harbours and in the River Rhine. The Rotterdam Rhine Research Project (RRP) is one of the means to achieve this based on: technical research, legal research, public relations and dialogues with dischargers. The programme for five selected heavy metals is almost complete. For many heavy metal discharge points between Rotterdam and Rheinfelden, a specially devised independent load assessment has been carried out four times. Balance studies were used to determine the relative contributions of the point discharges to the total. Currently the results are being used in an attempt to negotiate agreements with a selected number of the major dischargers. At present, more detailed balance studies are being set up and exploratory measurements carried out for organic micropollutants. It may be concluded that the research is progressing successfully and methods and techniques developed seem satisfactory and broadly applicable. The Rhine Action Programme encompasses an international effort to improve the quality of the Rhine water. Although the RRP plays a modest complementary role to the Rhine Action Plan, there is no doubt of the value of this Rotterdam initiative. The mode of work followed in the RRP contains elements that can be of use in combatting the contamination of the North Sea by rivers other than the Rhine.

Mechanism of the propagation step in the anionic polymerization of styrene

1980 ◽  
Vol 45 (4) ◽  
pp. 1047-1055 ◽  
Author(s):  
Miroslav Kašpar ◽  
Jiří Trekoval
Keyword(s):  
Experimental Data ◽  
Induction Period ◽  
Anionic Polymerization ◽  
Reactive Intermediate ◽  
Coordination Polymerization ◽  
Kinetic Dependence ◽  
Mathematical Solution ◽  
Slow Reaction

The paper is dealing with an investigation of the kinetic dependence of the propagation step in the anionic coordination polymerization of styrene in benzene at 303 K with "living" oligostyryllithium as initiator at the onset of the reaction. A short but distinct induction period was found, indicating a preceding slow reaction leading to the formation of a reactive intermediate, which behaves as the initiator of the reaction. Using results obtained in the first paper of this series, a new mechanism of propagation has been suggested, the mathematical solution of which is correlated with experimental data.

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