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Marqués MI. Dynamics of a small particle in a fluctuating random light field. OPTICS LETTERS 2016; 41:796-799. [PMID: 26872191 DOI: 10.1364/ol.41.000796] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 06/05/2023]
Abstract
The dynamics of an electric dipole in a light field consisting of electromagnetic plane waves with polarizations randomly distributed and fluctuating phases is theoretically analyzed. The expression for the optical random-force fluctuations is derived and found to be proportional to the scattering cross section and to the square of the intensity divided by the frequency of the electromagnetic field. Under these fluctuations, and in the absence of damping, the dipole behaves like a super-diffusive particle with a kinetic energy growing linearly with time. The analytic predictions are tested against numerical simulations for the particular case of a resonant dipole.
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Abi Mansour A, Al Ghoul M. Scaling and crossover dynamics in the hyperbolic reaction-diffusion equations of initially separated components. PHYSICAL REVIEW. E, STATISTICAL, NONLINEAR, AND SOFT MATTER PHYSICS 2011; 84:026107. [PMID: 21929064 DOI: 10.1103/physreve.84.026107] [Citation(s) in RCA: 6] [Impact Index Per Article: 0.5] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 02/16/2011] [Revised: 04/17/2011] [Indexed: 05/31/2023]
Abstract
In this paper we investigate the dynamics of front propagation in the family of reactions (nA + mB (k)→ C) with initially segregated reactants in one dimension using hyperbolic reaction-diffusion equations with the mean-field approximation for the reaction rate. This leads to different dynamics than those predicted by their parabolic counterpart. Using perturbation techniques, we focus on the initial and intermediate temporal behavior of the center and width of the front and derive the different time scaling exponents. While the solution of the parabolic system yields a short time scaling as t(1/2) for the front center, width, and global reaction rate, the hyperbolic system exhibits linear scaling for those quantities. Moreover, those scaling laws are shown to be independent of the stoichiometric coefficients n and m. The perturbation results are compared with the full numerical solutions of the hyperbolic equations. The crossover time at which the hyperbolic regime crosses over to the parabolic regime is also studied. Conditions for static and moving fronts are also derived and numerically validated.
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Affiliation(s)
- Andrew Abi Mansour
- Program in Computational Science, American University of Beirut, Beirut, Lebanon
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Affiliation(s)
- Haim Taitelbaum
- a Department of Physics , Bar-Ilan University , Ramat-Gan , 52900 , Israel
| | - Zbigniew Koza
- a Department of Physics , Bar-Ilan University , Ramat-Gan , 52900 , Israel
- b Institute of Theoretical Physics, University of Wroclaw , 50204 , Wroclaw , Poland
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Yuste SB, Acedo L, Lindenberg K. Reaction front in an A+B-->C reaction-subdiffusion process. PHYSICAL REVIEW. E, STATISTICAL, NONLINEAR, AND SOFT MATTER PHYSICS 2004; 69:036126. [PMID: 15089380 DOI: 10.1103/physreve.69.036126] [Citation(s) in RCA: 39] [Impact Index Per Article: 2.0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 12/19/2003] [Indexed: 05/24/2023]
Abstract
We study the reaction front for the process A+B-->C in which the reagents move subdiffusively. Our theoretical description is based on a fractional reaction-subdiffusion equation in which both the motion and the reaction terms are affected by the subdiffusive character of the process. We design numerical simulations to check our theoretical results, describing the simulations in some detail because the rules necessarily differ in important respects from those used in diffusive processes. Comparisons between theory and simulations are on the whole favorable, with the most difficult quantities to capture being those that involve very small numbers of particles. In particular, we analyze the total number of product particles, the width of the depletion zone, the production profile of product and its width, as well as the reactant concentrations at the center of the reaction zone, all as a function of time. We also analyze the shape of the product profile as a function of time, in particular, its unusual behavior at the center of the reaction zone.
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Affiliation(s)
- S B Yuste
- Departamento de Física, Universidad de Extremadura, E-06071 Badajoz, Spain
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Sinder M, Pelleg J, Sokolovsky V, Meerovich V. Competing reactions with initially separated components in the asymptotic time region. PHYSICAL REVIEW. E, STATISTICAL, NONLINEAR, AND SOFT MATTER PHYSICS 2003; 68:022101. [PMID: 14525025 DOI: 10.1103/physreve.68.022101] [Citation(s) in RCA: 1] [Impact Index Per Article: 0.0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 01/13/2003] [Indexed: 05/24/2023]
Abstract
Two competing irreversible reactions with initially separated components and with essentially different reaction constants are theoretically studied in the asymptotic time region. The description of the two simultaneous reactions is reduced to the consideration of two reactions separated in space. It is shown that the reaction rate profile can have two maxima and their ratio is independent of time. The location and relative value of the maxima are functions of the reaction constants and initial concentrations.
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Affiliation(s)
- Misha Sinder
- Department of Materials Engineering, Ben Gurion University of the Negev, Beer-Sheva 84105, Israel
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Shipilevsky BM. Diffusion-controlled annihilation A+B-->0 with initially separated reactants: the death of an A particle island in the B particle sea. PHYSICAL REVIEW. E, STATISTICAL, NONLINEAR, AND SOFT MATTER PHYSICS 2003; 67:060101. [PMID: 16241185 DOI: 10.1103/physreve.67.060101] [Citation(s) in RCA: 7] [Impact Index Per Article: 0.3] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 02/21/2003] [Indexed: 05/04/2023]
Abstract
We consider the diffusion-controlled annihilation dynamics A+B-->0 with equal species diffusivities in the system where an island of particles A is surrounded by the uniform sea of particles B. We show that once the initial number of particles in the island is large enough, then at any system's dimensionality d the death of the majority of particles occurs in the universal scaling regime within which approximately 4/5 of the particles die at the island expansion stage and the remaining approximately 1/5 at the stage of its subsequent contraction. In the quasistatic approximation, the scaling of the reaction zone has been obtained for the cases of mean-field (d>or=d(c)) and fluctuation (d<d(c)) dynamics of the front.
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Park SH, Parus S, Kopelman R, Taitelbaum H. Gel-free experiments of reaction-diffusion front kinetics. PHYSICAL REVIEW. E, STATISTICAL, NONLINEAR, AND SOFT MATTER PHYSICS 2001; 64:055102. [PMID: 11735998 DOI: 10.1103/physreve.64.055102] [Citation(s) in RCA: 19] [Impact Index Per Article: 0.8] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 02/23/2001] [Indexed: 05/23/2023]
Abstract
We present a gel-free experimental system to study the kinetics of the reaction front in the A+B-->C reaction-diffusion system with initially-separated reactants. The experimental setup consists of a CCD camera monitoring the kinetics of the front formed in the reaction-diffusion process Cu(2+) + tetra [disodium ethyl bis(5-tetrazolylazo) acetate trihydrate] -->1:1 complex, in aqueous, gel-free solution, taking place inside a 150 microm gap between two flat microscope slides. The experimental results agree with the theoretical predictions for the time dependence of the front's width, height, and location, as well as the global reaction rate.
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Affiliation(s)
- S H Park
- Department of Chemistry, University of Michigan, Ann Arbor, Michigan 48109, USA
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Sinder M, Pelleg J. Asymptotic properties of a reversible A+B<-->C (static) reaction-diffusion process with initially separated reactants. PHYSICAL REVIEW. E, STATISTICAL PHYSICS, PLASMAS, FLUIDS, AND RELATED INTERDISCIPLINARY TOPICS 2000; 62:3340-8. [PMID: 11088833 DOI: 10.1103/physreve.62.3340] [Citation(s) in RCA: 19] [Impact Index Per Article: 0.8] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 04/10/2000] [Indexed: 04/15/2023]
Abstract
The asymptotic properties of the reaction front formed in a reversible reaction-diffusion process A+B<-->C (static) with initially separated reactants are investigated. The case of arbitrary nonzero values of the diffusion constants D(A) and D(B) and initial concentrations a(0) and b(0) of the reactants A and B is considered. The system is studied in the limit of t-->infinity and g-->0, where t and g are the time and the backward reaction rate constant, respectively. The dynamics of the reaction front is described as a crossover between the "irreversible" regime at times t<<g(-1) and the "reversible" regime at times t>>g(-1). The general properties of the crossover are studied with the help of an extended scaling approach formulated in this work. On the basis of the mean-field equations the analytical solutions in the reversible regime t>>g(-1) inside the reaction zone are discussed. It is shown that in the immobile reaction zone the reaction rate profile has two distinct maxima. This profile differs drastically from the usual single-maximum reaction rate profile inherent in the mobile reaction zone. The two-hump reaction zone profile is the result of the influence of C on the reaction rate in the reversible regime. Numerical computation of the mean-field kinetics equations supports the results of the asymptotic consideration.
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Affiliation(s)
- M Sinder
- Department of Materials Engineering, Ben Gurion University of the Negev, Beer Sheva 84105, Israel
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Sinder M, Pelleg J. Properties of the crossover from nonclassical to classical chemical kinetics in a reversible A+B<-->C reaction diffusion process. PHYSICAL REVIEW. E, STATISTICAL PHYSICS, PLASMAS, FLUIDS, AND RELATED INTERDISCIPLINARY TOPICS 1999; 60:R6259-62. [PMID: 11970609 DOI: 10.1103/physreve.60.r6259] [Citation(s) in RCA: 4] [Impact Index Per Article: 0.2] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 06/03/1999] [Indexed: 04/18/2023]
Abstract
We study the properties of the reaction front formed in a reversible reaction diffusion process A+B<-->C, with initially separated reactants. The case of the mobile C component is considered. In accordance with Chopard et al. [Phys. Rev. E 47, R40 (1993)] the dynamics of the front is described as a crossover between the "irreversible" regime at short times and the "reversible" regime at long times. A refined definition for the rate of C production is suggested, taking into account both the forward and the backward reaction rates. By this definition within the framework of the mean-field equations it is shown that the reversible regime is characterized by scaling of the local rate of C production as R(local) approximately t(-1) and by scaling of the global rate of C production as R(global) approximately t(-1/2). It is also established that in the considered special case of equal diffusion coefficients and equal initial concentrations, the macroscopic properties of the reaction front, such as the global rate of the C production R(global) and the concentration profiles of the components outside the front reaction, are unchanged through this crossover.
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Affiliation(s)
- M Sinder
- Department of Materials Engineering, Ben-Gurion University of the Negev, Beer Sheva 84105, Israel
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Sánchez AD, Bouzat S, Wio HS. Reaction kinetics of diffusing particles injected into a reactive substrate. PHYSICAL REVIEW. E, STATISTICAL PHYSICS, PLASMAS, FLUIDS, AND RELATED INTERDISCIPLINARY TOPICS 1999; 60:2677-83. [PMID: 11970069 DOI: 10.1103/physreve.60.2677] [Citation(s) in RCA: 5] [Impact Index Per Article: 0.2] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 02/02/1999] [Indexed: 11/07/2022]
Abstract
We analyze the kinetics of trapping (A+B-->B) and annihilation (A+B-->0) processes on a one-dimensional substrate with homogeneous distribution of immobile B particles while the A particles are supplied by a localized source. For the imperfect reaction case, we analyze both problems by means of a stochastic model and compare the results with numerical simulations. In addition, we present the exact analytical results of the stochastic model for the case of perfect trapping.
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Affiliation(s)
- A D Sánchez
- Centro Atómico Bariloche (CNEA) and Instituto Balseiro (CNEA and UNC), 8400-San Carlos de Bariloche, Argentina.
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Howard M, Cardy J. Fluctuation effects and multiscaling of the reaction-diffusion front for A+B to OE. ACTA ACUST UNITED AC 1999. [DOI: 10.1088/0305-4470/28/13/007] [Citation(s) in RCA: 48] [Impact Index Per Article: 1.9] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/12/2022]
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Yen A, Lin AL, Koo YEL, Vilensky B, Taitelbaum H, Kopelman R. Spatiotemporal Patterns and Nonclassical Kinetics of Competing Elementary Reactions: Chromium Complex Formation with Xylenol Orange in a Capillary. J Phys Chem A 1997. [DOI: 10.1021/jp962994e] [Citation(s) in RCA: 14] [Impact Index Per Article: 0.5] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/29/2022]
Affiliation(s)
- Andrew Yen
- Department of Chemistry, University of Michigan, Ann Arbor, Michigan 48109-1055, and Department of Physics, Bar-Ilan University, Ramat-Gan 52900, Israel
| | - Anna L. Lin
- Department of Chemistry, University of Michigan, Ann Arbor, Michigan 48109-1055, and Department of Physics, Bar-Ilan University, Ramat-Gan 52900, Israel
| | - Yong-Eun Lee Koo
- Department of Chemistry, University of Michigan, Ann Arbor, Michigan 48109-1055, and Department of Physics, Bar-Ilan University, Ramat-Gan 52900, Israel
| | - Baruch Vilensky
- Department of Chemistry, University of Michigan, Ann Arbor, Michigan 48109-1055, and Department of Physics, Bar-Ilan University, Ramat-Gan 52900, Israel
| | - Haim Taitelbaum
- Department of Chemistry, University of Michigan, Ann Arbor, Michigan 48109-1055, and Department of Physics, Bar-Ilan University, Ramat-Gan 52900, Israel
| | - Raoul Kopelman
- Department of Chemistry, University of Michigan, Ann Arbor, Michigan 48109-1055, and Department of Physics, Bar-Ilan University, Ramat-Gan 52900, Israel
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Taitelbaum H, Yen A, Kopelman R, Havlin S, Weiss GH. Effects of bias on the kinetics of A+B-->C with initially separated reactants. PHYSICAL REVIEW. E, STATISTICAL PHYSICS, PLASMAS, FLUIDS, AND RELATED INTERDISCIPLINARY TOPICS 1996; 54:5942-5947. [PMID: 9965810 DOI: 10.1103/physreve.54.5942] [Citation(s) in RCA: 7] [Impact Index Per Article: 0.3] [Reference Citation Analysis] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 05/22/2023]
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Yen A, Koo YE, Kopelman R. Experimental study of a crossover from nonclassical to classical chemical kinetics: An elementary and reversible A+B C reaction-diffusion process in a capillary. PHYSICAL REVIEW. E, STATISTICAL PHYSICS, PLASMAS, FLUIDS, AND RELATED INTERDISCIPLINARY TOPICS 1996; 54:2447-2450. [PMID: 9965351 DOI: 10.1103/physreve.54.2447] [Citation(s) in RCA: 12] [Impact Index Per Article: 0.4] [Reference Citation Analysis] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 05/22/2023]
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Taitelbaum H, Vilensky B, Lin A, Yen A, Koo YE, Kopelman R. Competing Reactions with Initially Separated Components. PHYSICAL REVIEW LETTERS 1996; 77:1640-1643. [PMID: 10063129 DOI: 10.1103/physrevlett.77.1640] [Citation(s) in RCA: 17] [Impact Index Per Article: 0.6] [Reference Citation Analysis] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 05/23/2023]
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Koza Z, Taitelbaum H. Motion of the reaction front in the A+B-->C reaction-diffusion system. PHYSICAL REVIEW. E, STATISTICAL PHYSICS, PLASMAS, FLUIDS, AND RELATED INTERDISCIPLINARY TOPICS 1996; 54:R1040-R1043. [PMID: 9965318 DOI: 10.1103/physreve.54.r1040] [Citation(s) in RCA: 42] [Impact Index Per Article: 1.5] [Reference Citation Analysis] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 11/07/2022]
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Molinàs-Mata P, Muñoz MA, Martínez DO, Barabási AL. Ballistic random walker. PHYSICAL REVIEW. E, STATISTICAL PHYSICS, PLASMAS, FLUIDS, AND RELATED INTERDISCIPLINARY TOPICS 1996; 54:968-971. [PMID: 9965149 DOI: 10.1103/physreve.54.968] [Citation(s) in RCA: 7] [Impact Index Per Article: 0.3] [Reference Citation Analysis] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 11/07/2022]
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Drory A. Theory of second- and higher-order stochastic processes. PHYSICAL REVIEW. E, STATISTICAL PHYSICS, PLASMAS, FLUIDS, AND RELATED INTERDISCIPLINARY TOPICS 1995; 51:5298-5314. [PMID: 9963262 DOI: 10.1103/physreve.51.5298] [Citation(s) in RCA: 1] [Impact Index Per Article: 0.0] [Reference Citation Analysis] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 05/22/2023]
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Cornell SJ. Refined simulations of the reaction front for diffusion-limited two-species annihilation in one dimension. PHYSICAL REVIEW. E, STATISTICAL PHYSICS, PLASMAS, FLUIDS, AND RELATED INTERDISCIPLINARY TOPICS 1995; 51:4055-4064. [PMID: 9963116 DOI: 10.1103/physreve.51.4055] [Citation(s) in RCA: 14] [Impact Index Per Article: 0.5] [Reference Citation Analysis] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 05/22/2023]
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Arora D, Bhatia DP, Prasad MA. Survival probability with random-force-dominated dynamics in the presence of traps. PHYSICAL REVIEW. E, STATISTICAL PHYSICS, PLASMAS, FLUIDS, AND RELATED INTERDISCIPLINARY TOPICS 1994; 49:5867-5870. [PMID: 9961918 DOI: 10.1103/physreve.49.5867] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 05/22/2023]
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Masoliver J, Weiss GH. Telegrapher's equations with variable propagation speeds. PHYSICAL REVIEW. E, STATISTICAL PHYSICS, PLASMAS, FLUIDS, AND RELATED INTERDISCIPLINARY TOPICS 1994; 49:3852-3854. [PMID: 9961672 DOI: 10.1103/physreve.49.3852] [Citation(s) in RCA: 8] [Impact Index Per Article: 0.3] [Reference Citation Analysis] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 05/22/2023]
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Masoliver J, Porr JM. Harmonic oscillators driven by colored noise: Crossovers, resonances, and spectra. PHYSICAL REVIEW. E, STATISTICAL PHYSICS, PLASMAS, FLUIDS, AND RELATED INTERDISCIPLINARY TOPICS 1993; 48:4309-4319. [PMID: 9961112 DOI: 10.1103/physreve.48.4309] [Citation(s) in RCA: 33] [Impact Index Per Article: 1.1] [Reference Citation Analysis] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 11/07/2022]
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Araujo M, Larralde H, Havlin S, Stanley HE. Scaling anomalies in reaction front dynamics of confined systems. PHYSICAL REVIEW LETTERS 1993; 71:3592-3595. [PMID: 10055016 DOI: 10.1103/physrevlett.71.3592] [Citation(s) in RCA: 20] [Impact Index Per Article: 0.6] [Reference Citation Analysis] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 05/23/2023]
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Heinrichs J. First-passage times and survival probabilities for particles moving in a field of random correlated forces. PHYSICAL REVIEW. E, STATISTICAL PHYSICS, PLASMAS, FLUIDS, AND RELATED INTERDISCIPLINARY TOPICS 1993; 48:2397-2401. [PMID: 9960871 DOI: 10.1103/physreve.48.2397] [Citation(s) in RCA: 5] [Impact Index Per Article: 0.2] [Reference Citation Analysis] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 04/12/2023]
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Masoliver J. Second-order dichotomous processes: Damped free motion, critical behavior, and anomalous superdiffusion. PHYSICAL REVIEW. E, STATISTICAL PHYSICS, PLASMAS, FLUIDS, AND RELATED INTERDISCIPLINARY TOPICS 1993; 48:121-135. [PMID: 9960574 DOI: 10.1103/physreve.48.121] [Citation(s) in RCA: 5] [Impact Index Per Article: 0.2] [Reference Citation Analysis] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 05/22/2023]
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Cornell S, Droz M. Steady-state reaction-diffusion front scaling for mA+nB-->. PHYSICAL REVIEW LETTERS 1993; 70:3824-3827. [PMID: 10053971 DOI: 10.1103/physrevlett.70.3824] [Citation(s) in RCA: 35] [Impact Index Per Article: 1.1] [Reference Citation Analysis] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 05/23/2023]
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Heinrichs J. Probability distributions for second-order processes driven by Gaussian noise. PHYSICAL REVIEW. E, STATISTICAL PHYSICS, PLASMAS, FLUIDS, AND RELATED INTERDISCIPLINARY TOPICS 1993; 47:3007-3012. [PMID: 9960353 DOI: 10.1103/physreve.47.3007] [Citation(s) in RCA: 3] [Impact Index Per Article: 0.1] [Reference Citation Analysis] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 05/22/2023]
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Chopard B, Droz M, Karapiperis T, Rácz Z. Properties of the reaction front in a reversible A+B. PHYSICAL REVIEW. E, STATISTICAL PHYSICS, PLASMAS, FLUIDS, AND RELATED INTERDISCIPLINARY TOPICS 1993; 47:R40-R43. [PMID: 9960074 DOI: 10.1103/physreve.47.r40] [Citation(s) in RCA: 17] [Impact Index Per Article: 0.5] [Reference Citation Analysis] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 05/22/2023]
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Larralde H, Araujo M, Havlin S, Stanley HE. Diffusion-reaction kinetics for A+B(static)-->C(inert) for one-dimensional systems with initially separated reactants. PHYSICAL REVIEW. A, ATOMIC, MOLECULAR, AND OPTICAL PHYSICS 1992; 46:R6121-R6123. [PMID: 9907992 DOI: 10.1103/physreva.46.r6121] [Citation(s) in RCA: 9] [Impact Index Per Article: 0.3] [Reference Citation Analysis] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 05/22/2023]
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Leyvraz F, Redner S. Spatial structure in diffusion-limited two-species annihilation. PHYSICAL REVIEW. A, ATOMIC, MOLECULAR, AND OPTICAL PHYSICS 1992; 46:3132-3147. [PMID: 9908482 DOI: 10.1103/physreva.46.3132] [Citation(s) in RCA: 21] [Impact Index Per Article: 0.7] [Reference Citation Analysis] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 05/22/2023]
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Taitelbaum H, Koo YE, Havlin S, Kopelman R, Weiss GH. Exotic behavior of the reaction front in the A+B-->C reaction-diffusion system. PHYSICAL REVIEW. A, ATOMIC, MOLECULAR, AND OPTICAL PHYSICS 1992; 46:2151-2154. [PMID: 9908355 DOI: 10.1103/physreva.46.2151] [Citation(s) in RCA: 34] [Impact Index Per Article: 1.1] [Reference Citation Analysis] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 05/22/2023]
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Larralde H, Araujo M, Havlin S, Stanley HE. Reaction front for A+B-->C diffusion-reaction systems with initially separated reactants. PHYSICAL REVIEW. A, ATOMIC, MOLECULAR, AND OPTICAL PHYSICS 1992; 46:855-859. [PMID: 9908186 DOI: 10.1103/physreva.46.855] [Citation(s) in RCA: 21] [Impact Index Per Article: 0.7] [Reference Citation Analysis] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 05/22/2023]
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