Validity of the Law of Mass Action in Three-Dimensional Coagulation Processes

Winkler, Anton A.; Frey, Erwin

doi:10.1103/physrevlett.108.108301

Cited by 23 publications

(50 citation statements)

References 25 publications

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“…For a cubic lattice with lattice spacing a = 1 and diffusion constant D = 1, by numerical integration we find that μ −1 = λ −1 + 0.252731009858(3). This value is corroborated by our numerical simulations, where we have considered the longtime decay of the density ρ ∼ μ −1 t −1 [17]. We now proceed to derive the flow equation to the renormalized reaction rate λ k for general reaction kernels λ(z), whose interaction may extend over several sites.…”

Section: A Derivation Of the Macroscopic Decay Ratementioning

confidence: 59%

“…In the following we study reaction kernels which allow for a particularly precise numerical solution, repeating for completeness the calculations given in the Supplemental Material of [17]. Consider the reaction kernels whose twodimensional versions are depicted in Fig.…”

Section: Appendix: the Macroscopic Decay Rate For Selected Reaction Kmentioning

confidence: 99%

“…(29) indeed cancels. We remark that the correction terms are also amenable to perturbative treatment [17,49]. We have run simulations for a range of different models (one-site objects with both finite and infinitely large reaction rates, and two examples of extended objects that react immediately on contact) for the three-dimensional system, where…”

Section: B Universal Correction To the Law Of Mass Actionmentioning

confidence: 99%

“…Finally, in Sec. IV we turn to the treatment of the process above the critical dimension in a broader and more detailed discussion than is given in our previous paper on this topic [17].…”

Section: Introductionmentioning

confidence: 99%

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Long-range and many-body effects in coagulation processes

Winkler

Frey

2013

Phys. Rev. E

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We study the problem of diffusing particles which coalesce upon contact. With the aid of a nonperturbative renormalization group, we first analyze the dynamics emerging below the critical dimension two, where strong fluctuations imply anomalously slow decay. Above two dimensions, the long-time, low-density behavior is known to conform with the law of mass action. For this case, we establish an exact mapping between the physics at the microscopic scale (lattice structure, particle shape and size) and the macroscopic decay rate in the law of mass action. In addition, we identify a term violating this classical law. It originates in long-range and many-particle fluctuations and is a simple, universal function of the macroscopic decay rate.

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Section: A Derivation Of the Macroscopic Decay Ratementioning

confidence: 59%

Section: Appendix: the Macroscopic Decay Rate For Selected Reaction Kmentioning

confidence: 99%

Section: B Universal Correction To the Law Of Mass Actionmentioning

confidence: 99%

“…Finally, in Sec. IV we turn to the treatment of the process above the critical dimension in a broader and more detailed discussion than is given in our previous paper on this topic [17].…”

Section: Introductionmentioning

confidence: 99%

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Long-range and many-body effects in coagulation processes

Winkler

Frey

2013

Phys. Rev. E

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“…Introductory texts are also provided by Cardy's (lecture) notes [91,293]. Roughly speaking, path integral representations of the chemical master equation (24) have been used to assess how a macroscopic law of mass action changes due to fluctuations, both below [96,130,132,[294][295][296][297][298][299][300][301][302][303][304][305] and above the (upper) critical dimension [295,306,307], using either perturbative [14, 70, 130-133, 294-301, 303-305, 308-316] or non-perturbative [306,307,[317][318][319] techniques. All of these articles focus on stochastic processes with spatial degrees of freedom for which alternative analytical approaches are scarce.…”

Section: History Of Stochastic Path Integralsmentioning

confidence: 99%

Master equations and the theory of stochastic path integrals

2017

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Abstract. This review provides a pedagogic and self-contained introduction to master equations and to their representation by path integrals. Since the 1930s, master equations have served as a fundamental tool to understand the role of fluctuations in complex biological, chemical, and physical systems. Despite their simple appearance, analyses of masters equations most often rely on lownoise approximations such as the Kramers-Moyal or the system size expansion, or require ad-hoc closure schemes for the derivation of low-order moment equations. We focus on numerical and analytical methods going beyond the low-noise limit and provide a unified framework for the study of master equations. After deriving the forward and backward master equations from the Chapman-Kolmogorov equation, we show how the two master equations can be cast into either of four linear partial differential equations (PDEs). Three of these PDEs are discussed in detail. The first PDE governs the time evolution of a generalized probability generating function whose basis depends on the stochastic process under consideration. Spectral methods, WKB approximations, and a variational approach have been proposed for the analysis of the PDE. The second PDE is novel and is obeyed by a distribution that is marginalized over an initial state. It proves useful for the computation of mean extinction times. The third PDE describes the time evolution of a "generating functional", which generalizes the so-called Poisson representation. Subsequently, the solutions of the PDEs are expressed in terms of two path integrals: a "forward" and a "backward" path integral. Combined with inverse transformations, one obtains two distinct path integral representations of the conditional probability distribution solving the master equations. We exemplify both path integrals in analysing elementary chemical reactions. Moreover, we show how a well-known path integral representation of averaged observables can be recovered from them. Upon expanding the forward and the backward path integrals around stationary paths, we then discuss and extend a recent method for the computation of rare event probabilities. Besides, we also derive path integral representations for processes with continuous state spaces whose forward and backward master equations admit Kramers-Moyal expansions. A truncation of the backward expansion at the level of a diffusion approximation recovers a classic path integral representation of the (backward) Fokker-Planck equation. One can rewrite this path integral in terms of an Onsager-Machlup function and, for purely diffusive Brownian motion, it simplifies to the path integral of Wiener. To make this review accessible to a broad community, we have used the language of probability theory rather than quantum (field) theory and do not assume any knowledge of the latter. The probabilistic structures underpinning various technical concepts, such as coherent states, the Doi-shift, and normal-ordered observables, are thereby made explicit. arXiv:1609.02849v2 [cond-mat...

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Universal Aspects of Neutron Halos in Light Exotic Nuclei

Frederico

2014

Few-Body Syst

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Validity of the Law of Mass Action in Three-Dimensional Coagulation Processes

Cited by 23 publications

References 25 publications

Long-range and many-body effects in coagulation processes

Long-range and many-body effects in coagulation processes

Master equations and the theory of stochastic path integrals

Universal Aspects of Neutron Halos in Light Exotic Nuclei

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