1
|
Shipilevsky BM. Diffusion-controlled formation and collapse of a d-dimensional A-particle island in the B-particle sea. Phys Rev E 2017; 95:062137. [PMID: 28709311 DOI: 10.1103/physreve.95.062137] [Citation(s) in RCA: 5] [Impact Index Per Article: 0.7] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 04/14/2017] [Indexed: 11/07/2022]
Abstract
We consider diffusion-controlled evolution of a d-dimensional A-particle island in the B-particle sea at propagation of the sharp reaction front A+B→0 at equal species diffusivities. The A-particle island is formed by a localized (point) A-source with a strength λ that acts for a finite time T. We reveal the conditions under which the island collapse time t_{c} becomes much longer than the injection period T (long-living island) and demonstrate that regardless of d the evolution of the long-living island radius r_{f}(t) is described by the universal law ζ_{f}=r_{f}/r_{f}^{M}=sqrt[eτ|lnτ|], where τ=t/t_{c} and r_{f}^{M} is the maximal island expansion radius at the front turning point t_{M}=t_{c}/e. We find that in the long-living island regime the ratio t_{c}/T changes with the increase of the injection period T by the law ∝(λ^{2}T^{2-d})^{1/d}, i.e., increases with the increase of T in the one-dimensional (1D) case, does not change with the increase of T in the 2D case and decreases with the increase of T in the 3D case. We derive the scaling laws for particles death in the long-living island and determine the limits of their applicability. We demonstrate also that these laws describe asymptotically the evolution of the d-dimensional spherical island with a uniform initial particle distribution generalizing the results obtained earlier for the quasi-one-dimensional geometry. As striking results, we present a systematic analysis of the front relative width evolution for fluctuation, logarithmically modified, and mean-field regimes, and we demonstrate that in a wide range of parameters the front remains sharp up to a narrow vicinity of the collapse point.
Collapse
Affiliation(s)
- Boris M Shipilevsky
- Institute of Solid State Physics, Chernogolovka, Moscow district, 142432, Russia
| |
Collapse
|
2
|
Weber MF, Frey E. Master equations and the theory of stochastic path integrals. REPORTS ON PROGRESS IN PHYSICS. PHYSICAL SOCIETY (GREAT BRITAIN) 2017; 80:046601. [PMID: 28306551 DOI: 10.1088/1361-6633/aa5ae2] [Citation(s) in RCA: 31] [Impact Index Per Article: 4.4] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 06/06/2023]
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 master equations most often rely on low-noise 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.
Collapse
Affiliation(s)
- Markus F Weber
- Arnold Sommerfeld Center for Theoretical Physics and Center for NanoScience, Department of Physics, Ludwig-Maximilians-Universität München, Theresienstraße 37, 80333 München, Germany
| | - Erwin Frey
- Arnold Sommerfeld Center for Theoretical Physics and Center for NanoScience, Department of Physics, Ludwig-Maximilians-Universität München, Theresienstraße 37, 80333 München, Germany
| |
Collapse
|
3
|
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.
Collapse
Affiliation(s)
- Andrew Abi Mansour
- Program in Computational Science, American University of Beirut, Beirut, Lebanon
| | | |
Collapse
|
4
|
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
| |
Collapse
|
5
|
Shipilevsky BM. Death of an A -particle island in the B -particle sea: propagation and evolution of the reaction front A+B<-->C. Phys Rev E 2009; 79:021117. [PMID: 19391716 DOI: 10.1103/physreve.79.021117] [Citation(s) in RCA: 3] [Impact Index Per Article: 0.2] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 08/29/2008] [Indexed: 11/07/2022]
Abstract
We present a systematic theory of propagation and evolution of the reaction front A+B<-->C in the reaction-diffusion system where an island of particles A is surrounded by the uniform sea of particles B . In the first part of the work we give a systematic analysis of the crossover from the irreversible to reversible regime of front propagation in terms of the quasistatic approximation (QSA) and derive the key condition for the island death in the quasiequilibrium front regime. We show that the same as in the case of pure annihilation A+B-->0 the QSA enables the description of the quasiequilibrium front propagation only to a critical point t{c} on approaching to which the QSA is violated. In the second part of the work under the assumption of a sufficiently large forward reaction constant k we derive the perturbative expansion in powers of 1k which gives the asymptotically exact description of the quasiequilibrium front evolution up to t-->infinity. We demonstrate that below some critical value of the reduced backward reaction constant g<g{c} there appear two turning points on the front trajectory, the first of which arises at the sharp localized front stage and is due to the finite number of island particles whereas the second is a consequence of radical transformation of the front structure at the passage through the critical point (delocalization of the front). We find a remarkable property of self-similarity of the passage through the critical point, we derive scaling laws for such passage and show that in the limit g-->0 these laws lead to a striking phenomenon of an abrupt delocalization of the front.
Collapse
Affiliation(s)
- Boris M Shipilevsky
- Institute of Solid State Physics, Chernogolovka, Moscow district 142432, Russia
| |
Collapse
|
6
|
Täuber UC, Howard M, Vollmayr-Lee BP. Applications of field-theoretic renormalization group methods to reaction–diffusion problems. ACTA ACUST UNITED AC 2005. [DOI: 10.1088/0305-4470/38/17/r01] [Citation(s) in RCA: 217] [Impact Index Per Article: 11.4] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/11/2022]
|
7
|
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.
Collapse
Affiliation(s)
- Misha Sinder
- Department of Materials Engineering, Ben Gurion University of the Negev, Beer-Sheva 84105, Israel
| | | | | | | |
Collapse
|
8
|
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.
Collapse
|
9
|
Sinder M. Theory for competing reactions with initially separated components. PHYSICAL REVIEW. E, STATISTICAL, NONLINEAR, AND SOFT MATTER PHYSICS 2002; 65:037104. [PMID: 11909315 DOI: 10.1103/physreve.65.037104] [Citation(s) in RCA: 4] [Impact Index Per Article: 0.2] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 09/09/2001] [Indexed: 05/23/2023]
Abstract
The asymptotic long-time properties of a system with initially separated components and two competing irreversible reactions A1+B-->C1 and A2+B-->C2 are studied. It is shown that the system is characterized by a single reaction zone, with width growing like t(1/6), in which both reactions occur. Numerical computations of the mean-field kinetic equations confirm these asymptotic results.
Collapse
Affiliation(s)
- Misha Sinder
- Department of Materials Engineering, Ben Gurion University of the Negev, Beer-Sheva 84105, Israel
| |
Collapse
|
10
|
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.
Collapse
Affiliation(s)
- S H Park
- Department of Chemistry, University of Michigan, Ann Arbor, Michigan 48109, USA
| | | | | | | |
Collapse
|
11
|
O'Donoghue SJ, Bray AJ. Persistence in the one-dimensional A+B--> Ø reaction-diffusion model. PHYSICAL REVIEW. E, STATISTICAL, NONLINEAR, AND SOFT MATTER PHYSICS 2001; 64:041105. [PMID: 11690008 DOI: 10.1103/physreve.64.041105] [Citation(s) in RCA: 3] [Impact Index Per Article: 0.1] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 05/08/2001] [Revised: 06/19/2001] [Indexed: 05/23/2023]
Abstract
The persistence properties of a set of random walkers obeying the A+B--> Ø reaction, with equal initial density of particles and homogeneous initial conditions, is studied using two definitions of persistence. The probability P(t) that an annihilation process has not occurred at a given site has the asymptotic form P(t) approximately const+t(-straight theta), where straight theta is the persistence exponent (type I persistence). We argue that, for a density of particles rho>>1, this nontrivial exponent is identical to that governing the persistence properties of the one-dimensional diffusion equation, partial differential(t)straight phi= partial differential(xx)straight phi, where straight theta approximately 0.1207 [S. N. Majumdar, C. Sire, A. J. Bray, and S. J. Cornell, Phys. Rev. Lett. 77, 2867 (1996)]. In the case of an initial low density, rho(0)<<1, we find straight theta approximately 1/4 asymptotically. The probability that a site remains unvisited by any random walker (type II persistence) is also investigated and found to decay with a stretched exponential form, P(t) approximately exp(-constxrho(1/2)(0)t(1/4)), provided rho(0)<<1. A heuristic argument for this behavior, based on an exactly solvable toy model, is presented.
Collapse
Affiliation(s)
- S J O'Donoghue
- Department of Physics and Astronomy, University of Manchester, Manchester M13 9PL, United Kingdom
| | | |
Collapse
|
12
|
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.
Collapse
Affiliation(s)
- M Sinder
- Department of Materials Engineering, Ben Gurion University of the Negev, Beer Sheva 84105, Israel
| | | |
Collapse
|
13
|
Sinder M, Pelleg J. Crossover from nonclassical to classical chemical kinetics in an initially separated A + B<-->C reaction-diffusion system with arbitrary diffusion constants. PHYSICAL REVIEW. E, STATISTICAL PHYSICS, PLASMAS, FLUIDS, AND RELATED INTERDISCIPLINARY TOPICS 2000; 61:4935-4942. [PMID: 11031536 DOI: 10.1103/physreve.61.4935] [Citation(s) in RCA: 5] [Impact Index Per Article: 0.2] [Reference Citation Analysis] [Abstract] [MESH Headings] [Track Full Text] [Subscribe] [Scholar Register] [Received: 12/06/1999] [Indexed: 05/23/2023]
Abstract
The asymptotic long-time properties of the reaction front formed in a reversible reaction-diffusion process A + B<-->C with initially separated reactants are investigated. The case of arbitrary nonzero values of the diffusion constants DA, DB, DC of the components A, B, C and the initial concentrations a0 and b0 of A and B is considered. The system is studied in the limit of g-->0, where g is the backward reaction rate constant. In accordance with previous work, 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. It is shown that through this crossover the macroscopic properties of the reaction front, such as the global rate of C production, the motion of the reaction zone center, and the concentration profiles of the components outside the reaction front, are unchanged. The concentration profiles of the components inside the reaction zone are described by quasistatic equations. The results of the theoretical consideration are confirmed by computing the mean-field kinetics equations.
Collapse
Affiliation(s)
- M Sinder
- Department of Materials Engineering, Ben-Gurion University of the Negev, Beer Sheva, Israel
| | | |
Collapse
|
14
|
O'Shaughnessy B, Vavylonis D. Interfacial reactions: mixed order kinetics and segregation effects. PHYSICAL REVIEW LETTERS 2000; 84:3193-3196. [PMID: 11019045 DOI: 10.1103/physrevlett.84.3193] [Citation(s) in RCA: 9] [Impact Index Per Article: 0.4] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 04/09/1998] [Revised: 11/09/1999] [Indexed: 05/23/2023]
Abstract
We study A-B reaction kinetics at a fixed interface separating A and B bulks. Initially, the number of reactions R(t) approximately tn(infinity)(A)n(infinity)(B) is second order in the far-field densities n(infinity)(A), n(infinity)(B). First order kinetics, governed by diffusion from the dilute bulk, onset at long times: R(t) approximately x(t)n(infinity)(A), where x(t) approximately t(1/z) is the rms molecular displacement. Below a critical dimension, d<d(c) = z-1, mean-field theory is invalid: a new regime appears, R(t) approximately x(d+1)(t)n(infinity)(A)n(infinity)(B), and long time A-B segregation (similar to bulk A+B-->0) leads to anomalous decay of interfacial densities. Numerical simulations for z = 2 support the theory.
Collapse
Affiliation(s)
- B O'Shaughnessy
- Department of Chemical Engineering, Columbia University, New York, New York 10027, USA
| | | |
Collapse
|
15
|
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.
Collapse
Affiliation(s)
- M Sinder
- Department of Materials Engineering, Ben-Gurion University of the Negev, Beer Sheva 84105, Israel
| | | |
Collapse
|
16
|
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.
Collapse
Affiliation(s)
- A D Sánchez
- Centro Atómico Bariloche (CNEA) and Instituto Balseiro (CNEA and UNC), 8400-San Carlos de Bariloche, Argentina.
| | | | | |
Collapse
|
17
|
Goldschmidt YY, Hinrichsen H, Howard M, Täuber UC. Nonequilibrium critical behavior in unidirectionally coupled stochastic processes. PHYSICAL REVIEW. E, STATISTICAL PHYSICS, PLASMAS, FLUIDS, AND RELATED INTERDISCIPLINARY TOPICS 1999; 59:6381-408. [PMID: 11969623 DOI: 10.1103/physreve.59.6381] [Citation(s) in RCA: 9] [Impact Index Per Article: 0.4] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 09/10/1998] [Indexed: 04/18/2023]
Abstract
Phase transitions from an active into an absorbing, inactive state are generically described by the critical exponents of directed percolation (DP), with upper critical dimension d(c)=4. In the framework of single-species reaction-diffusion systems, this universality class is realized by the combined processes A-->A+A, A+A-->A, and A-->0. We study a hierarchy of such DP processes for particle species A,B,..., unidirectionally coupled via the reactions A-->B, ...(with rates mu(AB),...). When the DP critical points at all levels coincide, multicritical behavior emerges, with density exponents beta(i) which are markedly reduced at each hierarchy level i> or =2. This scenario can be understood on the basis of the mean-field rate equations, which yield beta(i)=1/2(i-1) at the multicritical point. Using field-theoretic renormalization-group techniques in d=4-epsilon dimensions, we identify a new crossover exponent phi, and compute phi=1+O(epsilon(2)) in the multicritical regime (for small mu(AB)) of the second hierarchy level. In the active phase, we calculate the fluctuation correction to the density exponent on the second hierarchy level, beta(2)=1/2-epsilon/8+O(epsilon(2)). Outside the multicritical region, we discuss the crossover to ordinary DP behavior, with the density exponent beta(1)=1-epsilon/6+O(epsilon(2)). Monte Carlo simulations are then employed to confirm the crossover scenario, and to determine the values for the new scaling exponents in dimensions d< or =3, including the critical initial slip exponent. Our theory is connected to specific classes of growth processes and to certain cellular automata, and the above ideas are also applied to unidirectionally coupled pair annihilation processes. We also discuss some technical as well as conceptual problems of the loop expansion, and suggest some possible interpretations of these difficulties.
Collapse
Affiliation(s)
- Y Y Goldschmidt
- Department of Physics and Astronomy, University of Pittsburgh, Pittsburgh, PA 15260, USA
| | | | | | | |
Collapse
|
18
|
Drouffe JM, Godrèche C. Phase ordering and persistence in a class of stochastic processes interpolating between the Ising and voter models. ACTA ACUST UNITED AC 1999. [DOI: 10.1088/0305-4470/32/2/003] [Citation(s) in RCA: 45] [Impact Index Per Article: 1.8] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/11/2022]
|
19
|
|
20
|
|
21
|
Howard M, Godrèche C. Persistence in the Voter model: continuum reaction-diffusion approach. ACTA ACUST UNITED AC 1999. [DOI: 10.1088/0305-4470/31/11/001] [Citation(s) in RCA: 25] [Impact Index Per Article: 1.0] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/12/2022]
|
22
|
Rey PA, Droz M. A renormalization group study of a class of reaction-diffusion models, with particles input. ACTA ACUST UNITED AC 1999. [DOI: 10.1088/0305-4470/30/4/013] [Citation(s) in RCA: 25] [Impact Index Per Article: 1.0] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/11/2022]
|
23
|
Rey PA, Cardy J. Asymptotic form of the approach to equilibrium in reversible recombination reactions. ACTA ACUST UNITED AC 1999. [DOI: 10.1088/0305-4470/32/9/008] [Citation(s) in RCA: 28] [Impact Index Per Article: 1.1] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/12/2022]
|
24
|
|
25
|
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]
|
26
|
Howard MJ, Barkema GT. Shear flows and segregation in the reaction A+B-->. PHYSICAL REVIEW. E, STATISTICAL PHYSICS, PLASMAS, FLUIDS, AND RELATED INTERDISCIPLINARY TOPICS 1996; 53:5949-5956. [PMID: 9964953 DOI: 10.1103/physreve.53.5949] [Citation(s) in RCA: 4] [Impact Index Per Article: 0.1] [Reference Citation Analysis] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 05/22/2023]
|
27
|
Barkema GT, Howard MJ, Cardy JL. Reaction-diffusion front for A+B-->. PHYSICAL REVIEW. E, STATISTICAL PHYSICS, PLASMAS, FLUIDS, AND RELATED INTERDISCIPLINARY TOPICS 1996; 53:R2017-R2020. [PMID: 9964596 DOI: 10.1103/physreve.53.r2017] [Citation(s) in RCA: 17] [Impact Index Per Article: 0.6] [Reference Citation Analysis] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 05/22/2023]
|
28
|
Cornell S, Koza Z, Droz M. Dynamic multiscaling of the reaction-diffusion front for mA+nB-->0. PHYSICAL REVIEW. E, STATISTICAL PHYSICS, PLASMAS, FLUIDS, AND RELATED INTERDISCIPLINARY TOPICS 1995; 52:3500-3505. [PMID: 9963827 DOI: 10.1103/physreve.52.3500] [Citation(s) in RCA: 40] [Impact Index Per Article: 1.4] [Reference Citation Analysis] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 11/07/2022]
|