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Ancona M, Michieletto D, Marenduzzo D. Competition between local erasure and long-range spreading of a single biochemical mark leads to epigenetic bistability. Phys Rev E 2021; 101:042408. [PMID: 32422714 DOI: 10.1103/physreve.101.042408] [Citation(s) in RCA: 7] [Impact Index Per Article: 2.3] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 11/28/2019] [Accepted: 02/19/2020] [Indexed: 11/07/2022]
Abstract
The mechanism through which cells determine their fate is intimately related to the spreading of certain biochemical (so-called epigenetic) marks along their genome. The mechanisms behind mark spreading and maintenance are not yet fully understood, and current models often assume a long-range infectionlike process for the dynamics of marks, due to the polymeric nature of the chromatin fiber which allows looping between distant sites. While these existing models typically consider antagonizing marks, here we propose a qualitatively different scenario which analyses the spreading of a single mark. We define a one-dimensional stochastic model in which mark spreading/infection occurs as a long-range process whereas mark erasure/recovery is a local process, with an enhanced rate at boundaries of infected domains. In the limiting case where our model exhibits absorbing states, we find a first-order-like transition separating the marked/infected phase from the unmarked/recovered phase. This suggests that our model, in this limit, belongs to the long-range compact directed percolation universality class. The abrupt nature of the transition is retained in a more biophysically realistic situation when a basal infection/recovery rate is introduced (thereby removing absorbing states). Close to the transition there is a range of bistability where both the marked/infected and unmarked/recovered states are metastable and long lived, which provides a possible avenue for controlling fate decisions in cells. Increasing the basal infection/recovery rate, we find a second transition between a coherent (marked or unmarked) phase, and a mixed, or random, one.
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Affiliation(s)
- Marco Ancona
- SUPA, School of Physics and Astronomy, University of Edinburgh, Edinburgh EH9 3FD, United Kingdom
| | - Davide Michieletto
- SUPA, School of Physics and Astronomy, University of Edinburgh, Edinburgh EH9 3FD, United Kingdom.,MRC Human Genetics Unit, Institute of Genetics and Molecular Medicine, University of Edinburgh, Edinburgh EH4 2XU, United Kingdom.,Centre for Mathematical Biology, and Department of Mathematical Sciences, University of Bath, North Road, Bath BA2 7AY, United Kingdom
| | - Davide Marenduzzo
- SUPA, School of Physics and Astronomy, University of Edinburgh, Edinburgh EH9 3FD, United Kingdom
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2
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Anjos FCD, Lyra ML, Gléria I, Argolo C, de Souza AJF. Emerging extreme value and Fermi-Dirac distributions in the Lévy branching and annihilating process. Phys Rev E 2020; 101:052136. [PMID: 32575329 DOI: 10.1103/physreve.101.052136] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 01/13/2020] [Accepted: 05/06/2020] [Indexed: 11/07/2022]
Abstract
We study the dynamics of the branching and annihilating process with long-range interactions. Static particles generate an offspring and annihilate upon contact. The branching distance is supposed to follow a Lévy-like power-law distribution with P(r)∝1/r^{α}. We analyze the long term behavior of the mean particles number and its fluctuations as a function of the parameter α that controls the range of the branching process. We show that the dynamic exponent associated with the particle number fluctuations varies continuously for α<4 while the particle number exponent only changes for α<3. A crossover from extreme value Frechet (at α=3) and Gumbell (for 2<α<3) distributions is developed, similar to the one reported in recent experiments with cw-pumped random fiber lasers presenting underlying gain and Lévy processes. We report the dependence of the relevant dynamical power-law exponents on α showing that explosive growth takes place for α≤2. Further, the average occupation number distribution is shown to evolve from the standard Fermi-Dirac form to the generalized one within the context of nonextensive statistics.
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Affiliation(s)
- F C Dos Anjos
- Instituto de Física, Universidade Federal de Alagoas, 57072-970 Maceió-AL, Brazil
| | - M L Lyra
- Instituto de Física, Universidade Federal de Alagoas, 57072-970 Maceió-AL, Brazil
| | - Iram Gléria
- Instituto de Física, Universidade Federal de Alagoas, 57072-970 Maceió-AL, Brazil
| | - C Argolo
- Instituto Federal de Ciência e Tecnologia do Estado de Alagoas, 57020-510 Maceió-AL, Brazil and Núcleo de Ciências Exatas - NCEx, Universidade Federal de Alagoas, 57309-005 Arapiraca-AL, Brazil
| | - Adauto J F de Souza
- Departamento de Física, Universidade Federal Rural de Pernambuco, 52171-900 Recife-PE, Brazil
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3
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Park SC. Crossover behaviors in branching annihilating attracting walk. Phys Rev E 2020; 101:052103. [PMID: 32575215 DOI: 10.1103/physreve.101.052103] [Citation(s) in RCA: 3] [Impact Index Per Article: 0.8] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 02/09/2020] [Accepted: 04/13/2020] [Indexed: 11/07/2022]
Abstract
We introduce branching annihilating attracting walk (BAAW) in one dimension. The attracting walk is implemented by a biased hopping in such a way that a particle prefers hopping to a nearest neighbor located on the side where the nearest particle is found within the range of attraction. We study the BAAW with four offspring by extensive Monte Carlo simulation. At first, we find the critical exponents of the BAAW with infinite range of attraction, which are different from those of the directed Ising (DI) universality class. Our results are consistent with the recent observation [Daga and Ray, Phys. Rev. E 99, 032104 (2019)2470-004510.1103/PhysRevE.99.032104]. Then, by studying crossover behaviors, we show that as far as the range of attraction is finite the BAAW belongs to the DI class. We conclude that the origin of non-DI critical behavior of the BAAW with infinite range of attraction is the long-range nature of the attraction.
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Affiliation(s)
- Su-Chan Park
- The Catholic University of Korea, Bucheon 14662, Republic of Korea
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4
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Park SC. Branching annihilating random walks with long-range attraction in one dimension. Phys Rev E 2020; 101:052125. [PMID: 32575194 DOI: 10.1103/physreve.101.052125] [Citation(s) in RCA: 2] [Impact Index Per Article: 0.5] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 03/09/2020] [Accepted: 04/30/2020] [Indexed: 11/07/2022]
Abstract
We introduce and numerically study the branching annihilating random walks with long-range attraction (BAWL). The long-range attraction makes hopping biased in such a manner that particle's hopping along the direction to the nearest particle has larger transition rate than hopping against the direction. Still, unlike the Lévy flight, a particle only hops to one of its nearest-neighbor sites. The strength of bias takes the form x^{-σ} with non-negative σ, where x is the distance to the nearest particle from a particle to hop. By extensive Monte Carlo simulations, we show that the critical decay exponent δ varies continuously with σ up to σ=1 and δ is the same as the critical decay exponent of the directed Ising (DI) universality class for σ≥1. Investigating the behavior of the density in the absorbing phase, we argue that σ=1 is indeed the threshold that separates the DI and non-DI critical behavior. We also show by Monte Carlo simulations that branching bias with symmetric hopping exhibits the same critical behavior as the BAWL.
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Affiliation(s)
- Su-Chan Park
- Department of Physics, The Catholic University of Korea, Bucheon 14662, Republic of Korea
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5
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Birnšteinová Š, Hnatič M, Lučivjanský T. Two-Species Reaction-Diffusion System: the Effect of Long-Range Spreading. EPJ WEB OF CONFERENCES 2020. [DOI: 10.1051/epjconf/202022602005] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/14/2022] Open
Abstract
We study fluctuation effects in the two-species reaction-diffusion system A + B → Ø and A + A → (Ø, A). In contrast to the usually assumed ordinary short-range diffusion spreading of the reactants we consider anomalous diffusion due to microscopic long-range hops. In order to describe the latter, we employ the Lévy stochastic ensemble. The probability distribution for the Lévy flights decays in d dimensions with the distance r according to a power-law r−d−σ. For anomalous diffusion (including Lévy flights) the critical dimension dc = σ depends on the control parameter σ, 0 < σ ≤ 2. The model is studied in terms of the field theoretic approach based on the Feynman diagrammatic technique and perturbative renormalization group method. We demonstrate the ideas behind the B particle density calculation.
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Daga B, Ray P. Universality classes of absorbing phase transitions in generic branching-annihilating particle systems with nearest-neighbor bias. Phys Rev E 2019; 99:032104. [PMID: 30999391 DOI: 10.1103/physreve.99.032104] [Citation(s) in RCA: 9] [Impact Index Per Article: 1.8] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 01/16/2019] [Indexed: 11/07/2022]
Abstract
We study absorbing phase transitions in systems of branching annihilating random walkers and pair contact process with diffusion on a one-dimensional ring, where the walkers hop to their nearest neighbor with a bias ε. For ε=0, three universality classes-directed percolation (DP), parity-conserving (PC), and pair contact process with diffusion (PCPD)-are typically observed in such systems. We find that the introduction of ε does not change the DP universality class but alters the other two universality classes. For nonzero ε, the PCPD class crosses over to DP, and the PC class changes to a new universality class.
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Affiliation(s)
- Bijoy Daga
- The Institute of Mathematical Sciences, C.I.T Campus, Taramani, Chennai-600113, India
| | - Purusattam Ray
- The Institute of Mathematical Sciences, C.I.T Campus, Taramani, Chennai-600113, India.,Homi Bhabha National Institute, Training School Complex, Anushakti Nagar, Mumbai-400094, India
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7
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Homrighausen I, Winkler AA, Frey E. Fluctuation effects in the pair-annihilation process with Lévy dynamics. PHYSICAL REVIEW. E, STATISTICAL, NONLINEAR, AND SOFT MATTER PHYSICS 2013; 88:012111. [PMID: 23944418 DOI: 10.1103/physreve.88.012111] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 10/31/2012] [Revised: 06/16/2013] [Indexed: 06/02/2023]
Abstract
We investigate the density decay in the pair-annihilation process A+A→∅ in the case when the particles perform anomalous diffusion on a cubic lattice. The anomalous diffusion is realized via Lévy flights, which are characterized by long-range jumps and lead to superdiffusive behavior. As a consequence, the critical dimension depends continuously on the control parameter of the Lévy flight distribution. This instance is used to study the system close to the critical dimension by means of the nonperturbative renormalization group theory. Close to the critical dimension, the assumption of well-stirred reactants is violated by anticorrelations between the particles, and the law of mass action breaks down. The breakdown of the law of mass action is known to be caused by long-range fluctuations. We identify three interrelated consequences of these fluctuations. First, despite being a nonuniversal quantity and thus depending on the microscopic details, the renormalized reaction rate λ(0) can be approximated by a universal law close to the critical dimension. The emergence of universality relies on the fact that long-range fluctuations suppress the influence of the underlying microscopic details. Second, as criticality is approached, the macroscopic reaction rate decreases such that the law of mass action loses its significance. And third, additional nonanalytic power law corrections complement the analytic law of mass action term. An increasing number of those corrections accumulate and give an essential contribution as the critical dimension is approached. We test our findings for two implementations of Lévy flights that differ in the way they cross over to the normal diffusion in the limit σ→2.
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Affiliation(s)
- Ingo Homrighausen
- Arnold Sommerfeld Center for Theoretical Physics and Center for NanoScience, Department of Physics, Ludwig-Maximilians-Universität München, Theresienstrasse 37, 80333 München, Germany
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8
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Benitez F, Wschebor N. Branching and annihilating random walks: exact results at low branching rate. PHYSICAL REVIEW. E, STATISTICAL, NONLINEAR, AND SOFT MATTER PHYSICS 2013; 87:052132. [PMID: 23767512 DOI: 10.1103/physreve.87.052132] [Citation(s) in RCA: 5] [Impact Index Per Article: 0.5] [Reference Citation Analysis] [Abstract] [MESH Headings] [Track Full Text] [Subscribe] [Scholar Register] [Received: 11/15/2012] [Indexed: 06/02/2023]
Abstract
We present some exact results on the behavior of branching and annihilating random walks, both in the directed percolation and parity conserving universality classes. Contrary to usual perturbation theory, we perform an expansion in the branching rate around the nontrivial pure annihilation (PA) model, whose correlation and response function we compute exactly. With this, the nonuniversal threshold value for having a phase transition in the simplest system belonging to the directed percolation universality class is found to coincide with previous nonperturbative renormalization group (RG) approximate results. We also show that the parity conserving universality class has an unexpected RG fixed point structure, with a PA fixed point which is unstable in all dimensions of physical interest.
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Affiliation(s)
- Federico Benitez
- LPTMC, CNRS-UMR 7600, Université Pierre et Marie Curie, 75252 Paris, France
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9
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Fiore CE, de Oliveira MJ. Robustness of first-order phase transitions in one-dimensional long-range contact processes. PHYSICAL REVIEW. E, STATISTICAL, NONLINEAR, AND SOFT MATTER PHYSICS 2013; 87:042101. [PMID: 23679367 DOI: 10.1103/physreve.87.042101] [Citation(s) in RCA: 1] [Impact Index Per Article: 0.1] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 11/24/2012] [Indexed: 06/02/2023]
Abstract
It has been proposed [Ginelli et al., Phys. Rev. E 71, 026121 (2005)] that, unlike the short-range contact process, the long-range counterpart may lead to the existence of a discontinuous phase transition in one dimension. Aiming to explore such a link, here we investigate thoroughly a family of long-range contact processes. They are introduced through the transition rate 1+aℓ(-σ), where ℓ is the length of inactive islands surrounding particles. In the former approach we reconsider the original model (called the σ-contact process) by considering distinct mechanisms of weakening the long-range interaction toward the short-range limit. In addition, we study the effect of different rules, including creation and annihilation by clusters of particles and distinct versions with infinitely many absorbing states. Our results show that for all examples presenting a single absorbing state, a discontinuous transition is possible for small σ. On the other hand, the presence of infinite absorbing states leads to a distinct scenario depending on the interactions at the perimeter of inactive sites.
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Affiliation(s)
- Carlos E Fiore
- Departamento de Física, Universidade Federal do Paraná, Caixa Postal 19044, 81531-000 Curitiba, Paraná, Brazil
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Benitez F, Wschebor N. Branching-rate expansion around annihilating random walks. PHYSICAL REVIEW. E, STATISTICAL, NONLINEAR, AND SOFT MATTER PHYSICS 2012; 86:010104. [PMID: 23005353 DOI: 10.1103/physreve.86.010104] [Citation(s) in RCA: 2] [Impact Index Per Article: 0.2] [Reference Citation Analysis] [Abstract] [MESH Headings] [Track Full Text] [Subscribe] [Scholar Register] [Received: 03/01/2012] [Indexed: 06/01/2023]
Abstract
We present some exact results for branching and annihilating random walks. We compute the nonuniversal threshold value of the annihilation rate for having a phase transition in the simplest reaction-diffusion system belonging to the directed percolation universality class. Also, we show that the accepted scenario for the appearance of a phase transition in the parity conserving universality class must be improved. In order to obtain these results we perform an expansion in the branching rate around pure annihilation, a theory without branching. This expansion is possible because we manage to solve pure annihilation exactly in any dimension.
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Affiliation(s)
- Federico Benitez
- LPTMC, CNRS-UMR 7600, Université Pierre et Marie Curie, 75252 Paris, France
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11
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Alcaraz FC, Rittenberg V. Pair annihilation reaction D+D-->0 in disordered media and conformal invariance. PHYSICAL REVIEW. E, STATISTICAL, NONLINEAR, AND SOFT MATTER PHYSICS 2007; 75:051110. [PMID: 17677025 DOI: 10.1103/physreve.75.051110] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 12/11/2006] [Revised: 03/27/2007] [Indexed: 05/16/2023]
Abstract
The raise and peel model is a stochastic model of a fluctuating interface separating a substrate covered with clusters of matter of different sizes and a rarefied gas of tiles. The stationary state is obtained when adsorption compensates the desorption of tiles. This model is generalized to an interface with defects (D) . The defects are either adjacent or separated by a cluster. If a tile hits the end of a cluster with a defect nearby, the defect hops at the other end of the cluster, changing its shape. If a tile hits two adjacent defects, the defects annihilate and are replaced by a small cluster. There are no defects in the stationary state. This model can be seen as describing the reaction D+D-->0 , in which the particles (defects) D hop at long distances, changing the medium, and annihilate. Between the hops the medium also changes (tiles hit clusters, changing their shapes). Several properties of this model are presented and some exact results are obtained using the connection of our model with a conformally invariant quantum chain.
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Affiliation(s)
- F C Alcaraz
- Instituto de Física de São Carlos, Universidade de São Paulo, Caixa Postal 369, São Carlos, São Paulo, Brazil.
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12
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Jiménez-Dalmaroni A. Directed percolation with incubation times. PHYSICAL REVIEW. E, STATISTICAL, NONLINEAR, AND SOFT MATTER PHYSICS 2006; 74:011123. [PMID: 16907076 DOI: 10.1103/physreve.74.011123] [Citation(s) in RCA: 1] [Impact Index Per Article: 0.1] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 03/07/2006] [Indexed: 05/11/2023]
Abstract
We introduce a model for directed percolation with a long-range temporal diffusion, while the spatial diffusion is kept short ranged. In an interpretation of directed percolation as an epidemic process, this non-Markovian modification can be understood as incubation times, which are distributed accordingly to a Lévy distribution. We argue that the best approach to find the effective action for this problem is through a generalization of the Cardy-Sugar method, adding the non-Markovian features into the geometrical properties of the lattice. We formulate a field theory for this problem and renormalize it up to one loop in a perturbative expansion. We solve the various technical difficulties that the integrations possess by means of an asymptotic analysis of the divergences. We show the absence of field renormalization at one-loop order, and we argue that this would be the case to all orders in perturbation theory. Consequently, in addition to the characteristic scaling relations of directed percolation, we find a scaling relation valid for the critical exponents of this theory. In this universality class, the critical exponents vary continuously with the Lévy parameter.
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Affiliation(s)
- Andrea Jiménez-Dalmaroni
- Rudolf Peierls Centre for Theoretical Physics, University of Oxford, 1 Keble Road, Oxford OX1 3NP, UK.
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13
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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]
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Vernon DC. Long range hops and the pair annihilation reaction A+A-->0: renormalization group and simulation. PHYSICAL REVIEW. E, STATISTICAL, NONLINEAR, AND SOFT MATTER PHYSICS 2003; 68:041103. [PMID: 14682919 DOI: 10.1103/physreve.68.041103] [Citation(s) in RCA: 6] [Impact Index Per Article: 0.3] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 04/17/2003] [Indexed: 05/24/2023]
Abstract
A simple example of a nonequilibrium system for which fluctuations are important is a system of particles which diffuse and may annihilate in pairs on contact. The renormalization group can be used to calculate the time dependence of the density of particles, and provides both an exact value for the exponent governing the decay of particles and an epsilon expansion for the amplitude of this power law. When the diffusion is anomalous, as when the particles perform Lévy flights, the critical dimension depends continuously on the control parameter for the Lévy distribution. The epsilon expansion can then become an expansion in a small parameter. We present the renormalization group calculation and compare these results with those of a simulation.
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Affiliation(s)
- Daniel C Vernon
- Department of Physics, Simon Fraser University, Burnaby, British Columbia, Canada V5A 1S6.
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Chen L, Deem MW. Reaction, Lévy flights, and quenched disorder. PHYSICAL REVIEW. E, STATISTICAL, NONLINEAR, AND SOFT MATTER PHYSICS 2002; 65:011109. [PMID: 11800679 DOI: 10.1103/physreve.65.011109] [Citation(s) in RCA: 4] [Impact Index Per Article: 0.2] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 07/09/2001] [Revised: 08/08/2001] [Indexed: 05/23/2023]
Abstract
We consider the A+A-->Ø reaction, where the transport of the particles is given by Lévy flights in a quenched random potential. With a common literature model of the disorder, the random potential can only increase the rate of reaction. With a model of the disorder that obeys detailed balance, however, the rate of reaction initially increases and then decreases as a function of the disorder strength. The physical behavior obtained with this second model is in accord with that for reactive turbulent flow, indicating that Lévy flight statistics can model aspects of turbulent fluid transport.
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Affiliation(s)
- Ligang Chen
- Department of Chemical Engineering, University of California, Los Angeles, California 90095-1592, USA
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