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Hofmann J. Corrections to reaction-diffusion dynamics above the upper critical dimension. Phys Rev E 2022; 105:024127. [PMID: 35291187 DOI: 10.1103/physreve.105.024127] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 06/09/2021] [Accepted: 02/03/2022] [Indexed: 06/14/2023]
Abstract
Reaction-diffusion models are common in many areas of statistical physics, where they describe the late-time dynamics of chemical reactions. Using a Bose gas representation, which maps the real-time dynamics of the reactants to the imaginary-time evolution of an interacting Bose gas, we consider corrections to the late-time scaling of k-particle annihilation processes kA→∅ above the upper critical dimension, where mean-field theory sets the leading order. We establish that the leading corrections are not given by a small renormalization of the reaction rate due to k-particle memory effects, but instead set by higher-order correlation functions that capture memory effects of subclusters of reactants. Drawing on methods developed for ultracold quantum gases and nuclear physics, we compute these corrections exactly for various annihilation processes with k>2.
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Affiliation(s)
- Johannes Hofmann
- Department of Physics, Gothenburg University, 41296 Gothenburg, Sweden
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2
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Park SC. One-dimensional annihilating random walk with long-range interaction. Phys Rev E 2020; 102:042112. [PMID: 33212649 DOI: 10.1103/physreve.102.042112] [Citation(s) in RCA: 1] [Impact Index Per Article: 0.3] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 07/17/2020] [Accepted: 09/01/2020] [Indexed: 11/07/2022]
Abstract
We study the annihilating random walk with long-range interaction in one dimension. Each particle performs random walks on a one-dimensional ring in such a way that the probability of hopping toward the nearest particle is W=[1-ɛ(x+μ)^{-σ}]/2 (the probability of moving away from its nearest particle is 1-W), where x is the distance from the hopping particle to its nearest particle and ɛ, μ, and σ are parameters. For positive (negative) ɛ, a particle is effectively repulsed (attracted) by its nearest particle and each hopping is generally biased. On encounter, two particles are immediately removed from the system. We first study the survival probability and the mean spreading behaves in the long-time limit if there are only two particles in the beginning. Then we study how the density decays to zero if all sites are occupied at the outset. We find that the asymptotic behaviors are classified by seven categories: (i) σ>1 or ɛ=0, (ii) σ=1 and 2ɛ>1, (iii) σ=1 and 2ɛ=1, (iv) σ=1 and 2ɛ<1, (v) σ<1 and ɛ>0, (vi) σ=0 and ɛ<0, and (vii) 0<σ<1 and ɛ<0. The asymptotic behaviors in each category are universal in the sense that μ (and sometimes ɛ) cannot affect the asymptotic behaviors.
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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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3
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Patel M, Kumar R, Kishor K, Mlsna T, Pittman CU, Mohan D. Pharmaceuticals of Emerging Concern in Aquatic Systems: Chemistry, Occurrence, Effects, and Removal Methods. Chem Rev 2019; 119:3510-3673. [DOI: 10.1021/acs.chemrev.8b00299] [Citation(s) in RCA: 827] [Impact Index Per Article: 165.4] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 01/31/2023]
Affiliation(s)
- Manvendra Patel
- School of Environmental Sciences, Jawaharlal Nehru University, New Delhi 110067, India
| | - Rahul Kumar
- School of Environmental Sciences, Jawaharlal Nehru University, New Delhi 110067, India
| | - Kamal Kishor
- School of Environmental Sciences, Jawaharlal Nehru University, New Delhi 110067, India
| | - Todd Mlsna
- Department of Chemistry, Mississippi State University, Mississippi State, Mississippi 39762, United States
| | - Charles U. Pittman
- Department of Chemistry, Mississippi State University, Mississippi State, Mississippi 39762, United States
| | - Dinesh Mohan
- School of Environmental Sciences, Jawaharlal Nehru University, New Delhi 110067, India
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4
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Hoze N, Holcman D. Stochastic coagulation-fragmentation processes with a finite number of particles and applications. ANN APPL PROBAB 2018. [DOI: 10.1214/17-aap1334] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/19/2022]
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5
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Weinstein BT, Lavrentovich MO, Möbius W, Murray AW, Nelson DR. Genetic drift and selection in many-allele range expansions. PLoS Comput Biol 2017; 13:e1005866. [PMID: 29194439 PMCID: PMC5728587 DOI: 10.1371/journal.pcbi.1005866] [Citation(s) in RCA: 20] [Impact Index Per Article: 2.9] [Reference Citation Analysis] [Abstract] [MESH Headings] [Grants] [Track Full Text] [Download PDF] [Figures] [Journal Information] [Subscribe] [Scholar Register] [Received: 06/08/2017] [Revised: 12/13/2017] [Accepted: 11/01/2017] [Indexed: 12/15/2022] Open
Abstract
We experimentally and numerically investigate the evolutionary dynamics of four competing strains of E. coli with differing expansion velocities in radially expanding colonies. We compare experimental measurements of the average fraction, correlation functions between strains, and the relative rates of genetic domain wall annihilations and coalescences to simulations modeling the population as a one-dimensional ring of annihilating and coalescing random walkers with deterministic biases due to selection. The simulations reveal that the evolutionary dynamics can be collapsed onto master curves governed by three essential parameters: (1) an expansion length beyond which selection dominates over genetic drift; (2) a characteristic angular correlation describing the size of genetic domains; and (3) a dimensionless constant quantifying the interplay between a colony’s curvature at the frontier and its selection length scale. We measure these parameters with a new technique that precisely measures small selective differences between spatially competing strains and show that our simulations accurately predict the dynamics without additional fitting. Our results suggest that the random walk model can act as a useful predictive tool for describing the evolutionary dynamics of range expansions composed of an arbitrary number of genotypes with different fitnesses. Population expansions occur naturally during the spread of invasive species and have played a role in our evolutionary history when humans migrated out of Africa. We use a colony of non-motile bacteria expanding into unoccupied, nutrient-rich territory on an agar plate as a model system to explore how an expanding population’s spatial structure impacts its evolutionary dynamics. Spatial structure is present in expanding microbial colonies because daughter cells migrate only a small distance away from their mothers each generation. Generally, the constituents of expansions occurring in nature and in the lab have different genetic compositions (genotypes, or alleles if a single gene differs), each instilling different fitnesses, which compete to proliferate at the frontier. Here, we show that a random-walk model can accurately predict the dynamics of four expanding strains of E. coli with different fitnesses; each strain represents a competing allele. Our results can be extended to describe any number of competing genotypes with different fitnesses in a naturally occurring expansion as long as the underlying motility of the organisms does not cause our model to break down. Our model can also be used to precisely measure small selective differences between spatially competing genotypes in controlled laboratory settings.
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Affiliation(s)
- Bryan T. Weinstein
- School of Engineering and Applied Sciences, Harvard University, Cambridge, Massachusetts, United States of America
- * E-mail:
| | - Maxim O. Lavrentovich
- Department of Physics and Astronomy, University of Pennsylvania, Philadelphia, Pennsylvania, United States of America
| | - Wolfram Möbius
- Living Systems Institute, University of Exeter, Exeter, United Kingdom
- Physics and Astronomy, College of Engineering, Mathematics and Physical Sciences, University of Exeter, Exeter, United Kingdom
- Department of Physics, Harvard University, Cambridge, Massachusetts, United States of America
| | - Andrew W. Murray
- FAS Center for Systems Biology, Harvard University, Cambridge, Massachusetts, United States of America
- Department of Molecular and Cellular Biology, Harvard University, Cambridge, Massachusetts, United States of America
| | - David R. Nelson
- School of Engineering and Applied Sciences, Harvard University, Cambridge, Massachusetts, United States of America
- Department of Physics, Harvard University, Cambridge, Massachusetts, United States of America
- FAS Center for Systems Biology, Harvard University, Cambridge, Massachusetts, United States of America
- Department of Molecular and Cellular Biology, Harvard University, Cambridge, Massachusetts, United States of America
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6
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Crampe N, Ragoucy E, Rittenberg V, Vanicat M. Integrable dissipative exclusion process: Correlation functions and physical properties. Phys Rev E 2016; 94:032102. [PMID: 27739772 DOI: 10.1103/physreve.94.032102] [Citation(s) in RCA: 13] [Impact Index Per Article: 1.6] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 03/25/2016] [Indexed: 11/07/2022]
Abstract
We study a one-parameter generalization of the symmetric simple exclusion process on a one-dimensional lattice. In addition to the usual dynamics (where particles can hop with equal rates to the left or to the right with an exclusion constraint), annihilation and creation of pairs can occur. The system is driven out of equilibrium by two reservoirs at the boundaries. In this setting the model is still integrable: it is related to the open XXZ spin chain through a gauge transformation. This allows us to compute the full spectrum of the Markov matrix using Bethe equations. We also show that the stationary state can be expressed in a matrix product form permitting to compute the multipoints correlation functions as well as the mean value of the lattice and the creation-annihilation currents. Finally, the variance of the lattice current is computed for a finite-size system. In the thermodynamic limit, it matches the value obtained from the associated macroscopic fluctuation theory.
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Affiliation(s)
- N Crampe
- Laboratoire Charles Coulomb (L2C), UMR 5221 CNRS-Université de Montpellier, Montpellier, F-France
| | - E Ragoucy
- Laboratoire de Physique Théorique LAPTh, CNRS and Université de Savoie, 9 chemin de Bellevue, BP 110, F-74941 Annecy-le-Vieux Cedex, France
| | - V Rittenberg
- Physikalisches Institut der Universitaet Bonn, Nussallee 12 D-53115 Bonn, Federal Republic of Germany
| | - M Vanicat
- Laboratoire de Physique Théorique LAPTh, CNRS and Université de Savoie, 9 chemin de Bellevue, BP 110, F-74941 Annecy-le-Vieux Cedex, France
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7
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Ostvar S, Wood BD. A non-scale-invariant form for coarse-grained diffusion-reaction equations. J Chem Phys 2016. [DOI: 10.1063/1.4962421] [Citation(s) in RCA: 5] [Impact Index Per Article: 0.6] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/14/2022] Open
Affiliation(s)
- Sassan Ostvar
- School of Chemical, Biological and Environmental Engineering, Oregon State University, Corvallis, Oregon 97331, USA
| | - Brian D. Wood
- School of Chemical, Biological and Environmental Engineering, Oregon State University, Corvallis, Oregon 97331, USA
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8
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Biswas S, Larralde H, Leyvraz F. Ballistic annihilation with superimposed diffusion in one dimension. Phys Rev E 2016; 93:022136. [PMID: 26986317 DOI: 10.1103/physreve.93.022136] [Citation(s) in RCA: 3] [Impact Index Per Article: 0.4] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 10/29/2015] [Indexed: 11/07/2022]
Abstract
We consider a one-dimensional system with particles having either positive or negative velocity, and these particles annihilate on contact. Diffusion is superimposed on the ballistic motion of the particle. The annihilation may represent a reaction in which the two particles yield an inert species. This model has been the subject of previous work, in which it was shown that the particle concentration decays faster than either the purely ballistic or the purely diffusive case. We report on previously unnoticed behavior for large times when only one of the two species remains, and we also unravel the underlying fractal structure present in the system. We also consider in detail the case in which the initial concentration of right-going particles is 1/2+ɛ, with ɛ≠0. It is shown that remarkably rich behavior arises, in which two crossover times are observed as ɛ→0.
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Affiliation(s)
- Soham Biswas
- Instituto de Ciencias Físicas, Universidad Nacional Autónoma de México, Cuernavaca 62210, Mexico
| | - Hernán Larralde
- Instituto de Ciencias Físicas, Universidad Nacional Autónoma de México, Cuernavaca 62210, Mexico
| | - Francois Leyvraz
- Instituto de Ciencias Físicas, Universidad Nacional Autónoma de México, Cuernavaca 62210, Mexico
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9
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Göppel T, Palyulin VV, Gerland U. The efficiency of driving chemical reactions by a physical non-equilibrium is kinetically controlled. Phys Chem Chem Phys 2016; 18:20135-43. [DOI: 10.1039/c6cp01034b] [Citation(s) in RCA: 4] [Impact Index Per Article: 0.5] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/21/2022]
Abstract
A model system illustrates how the coupling efficiency of a physical non-equilibrium to a chemical reaction is affected by the relative timescales of the respective kinetics.
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Affiliation(s)
- Tobias Göppel
- Physics of Complex Biosystems
- Physics Department
- Technical University of Munich
- D-85748 Garching
- Germany
| | - Vladimir V. Palyulin
- Physics of Complex Biosystems
- Physics Department
- Technical University of Munich
- D-85748 Garching
- Germany
| | - Ulrich Gerland
- Physics of Complex Biosystems
- Physics Department
- Technical University of Munich
- D-85748 Garching
- Germany
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10
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Lazaridis F, Savara A, Argyrakis P. Reaction efficiency effects on binary chemical reactions. J Chem Phys 2014; 141:104103. [PMID: 25217900 DOI: 10.1063/1.4894791] [Citation(s) in RCA: 2] [Impact Index Per Article: 0.2] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/15/2022] Open
Abstract
We study the effect of the variation of reaction efficiency in binary reactions. We use the well-known A + B → 0 model, which has been extensively studied in the past. We perform simulations on this model where we vary the efficiency of reaction, i.e., when two particles meet they do not instantly react, as has been assumed in previous studies, but they react with a probability γ, where γ is in the range 0 < γ < 1. Our results show that at small γ values the system is reaction limited, but as γ increases it crosses over to a diffusion limited behavior. At early times, for small γ values, the particle density falls slower than for larger γ values. This fall-off goes over a crossover point, around the value of γ = 0.50 for high initial densities. Under a variety of conditions simulated, we find that the crossover point was dependent on the initial concentration but not on the lattice size. For intermediate and long times simulations, all γ values (in the depleted reciprocal density versus time plot) converge to the same behavior. These theoretical results are useful in models of epidemic reactions and epidemic spreading, where a contagion from one neighbor to the next is not always successful but proceeds with a certain probability, an analogous effect with the reaction probability examined in the current work.
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Affiliation(s)
- Filippos Lazaridis
- Department of Physics, University of Thessaloniki, Thessaloniki 54124, Greece
| | - Aditya Savara
- Chemical Sciences Division, Oak Ridge National Laboratory, Oak Ridge, Tennessee 37830, USA
| | - Panos Argyrakis
- Department of Physics, University of Thessaloniki, Thessaloniki 54124, Greece
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11
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Spendier K, Kenkre VM. Analytic solutions for some reaction-diffusion scenarios. J Phys Chem B 2013; 117:15639-50. [PMID: 23883422 DOI: 10.1021/jp406322t] [Citation(s) in RCA: 14] [Impact Index Per Article: 1.3] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/28/2022]
Abstract
Motivated currently by the problem of coalescence of receptor clusters in mast cells in the general subject of immune reactions, and formerly by the investigation of exciton trapping and sensitized luminescence in molecular systems and aggregates, we present analytic expressions for survival probabilities of moving entities undergoing diffusion and reaction on encounter. Results we provide cover several novel situations in simple 1-d systems as well as higher-dimensional counterparts along with a useful compendium of such expressions in chemical physics and allied fields. We also emphasize the importance of the relationship of discrete sink term analysis to continuum boundary condition studies.
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Affiliation(s)
- Kathrin Spendier
- Consortium of the Americas for Interdisciplinary Science, University of New Mexico , Albuquerque, New Mexico 87131, United States
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13
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Allam J, Sajjad MT, Sutton R, Litvinenko K, Wang Z, Siddique S, Yang QH, Loh WH, Brown T. Measurement of a reaction-diffusion crossover in exciton-exciton recombination inside carbon nanotubes using femtosecond optical absorption. PHYSICAL REVIEW LETTERS 2013; 111:197401. [PMID: 24266488 DOI: 10.1103/physrevlett.111.197401] [Citation(s) in RCA: 2] [Impact Index Per Article: 0.2] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 01/02/2013] [Revised: 10/02/2013] [Indexed: 06/02/2023]
Abstract
Exciton-exciton recombination in isolated semiconducting single-walled carbon nanotubes was studied using femtosecond transient absorption. Under sufficient excitation to saturate the optical absorption, we observed an abrupt transition between reaction- and diffusion-limited kinetics, arising from reactions between incoherent localized excitons with a finite probability of ~0.2 per encounter. This represents the first experimental observation of a crossover between classical and critical kinetics in a 1D coalescing random walk, which is a paradigm for the study of nonequilibrium systems.
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Affiliation(s)
- J Allam
- Advanced Technology Institute and Department of Physics, University of Surrey, Guildford GU2 7XH, United Kingdom
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14
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Zeraati S, Jafarpour FH, Hinrichsen H. Phase transition in an exactly solvable reaction-diffusion process. PHYSICAL REVIEW. E, STATISTICAL, NONLINEAR, AND SOFT MATTER PHYSICS 2013; 87:062120. [PMID: 23848640 DOI: 10.1103/physreve.87.062120] [Citation(s) in RCA: 1] [Impact Index Per Article: 0.1] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 03/18/2013] [Indexed: 06/02/2023]
Abstract
We study a nonconserved one-dimensional stochastic process which involves two species of particles A and B. The particles diffuse asymmetrically and react in pairs as A∅↔AA↔BA↔A∅ and B∅↔BB↔AB↔B∅. We show that the stationary state of the model can be calculated exactly by using matrix product techniques. The model exhibits a phase transition at a particular point in the phase diagram which can be related to a condensation transition in a particular zero-range process. We determine the corresponding critical exponents and provide a heuristic explanation for the unusually strong corrections to scaling seen in the vicinity of the critical point.
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Affiliation(s)
- Somayeh Zeraati
- Bu-Ali Sina University, Physics Department, 65174-4161 Hamedan, Iran.
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15
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Winkler AA, Frey E. Long-range and many-body effects in coagulation processes. PHYSICAL REVIEW. E, STATISTICAL, NONLINEAR, AND SOFT MATTER PHYSICS 2013; 87:022136. [PMID: 23496488 DOI: 10.1103/physreve.87.022136] [Citation(s) in RCA: 2] [Impact Index Per Article: 0.2] [Reference Citation Analysis] [Abstract] [MESH Headings] [Track Full Text] [Subscribe] [Scholar Register] [Received: 10/29/2012] [Indexed: 06/01/2023]
Abstract
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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Affiliation(s)
- Anton A Winkler
- Arnold Sommerfeld Center for Theoretical Physics and Center for NanoScience, Department of Physics, Ludwig-Maximilians-Universität München, Theresienstrasse 37, D-80333 München, Germany
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16
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Siitonen AJ, Bachilo SM, Tsyboulski DA, Weisman RB. Evidence for long-lived, optically generated quenchers of excitons in single-walled carbon nanotubes. NANO LETTERS 2012; 12:33-38. [PMID: 22142025 DOI: 10.1021/nl2028238] [Citation(s) in RCA: 3] [Impact Index Per Article: 0.3] [Reference Citation Analysis] [Abstract] [MESH Headings] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 05/29/2023]
Abstract
The nonlinear dependence of near-infrared photoluminescence (PL) emission on excitation intensity has been measured for individual nanotubes representing six different (n,m) species. Significant deviations from linearity are observed for intensities as low as ~100 W/cm(2), and an approximate inverse correlation is found between nonlinearity and PL action cross section (brightness). A model in which all PL nonlinearity arises from exciton-exciton annihilation is insufficient to account for the experimental data using realistic parameters. It is proposed that additional nonlinear quenching arises from photoinduced quenching states or species with longer lifetimes than emissive excitons. Evidence is also found for metastable photogenerated PL quenchers with lifetimes up to 20 s.
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Affiliation(s)
- Anni J Siitonen
- Department of Chemistry and Richard E. Smalley Institute for Nanoscale Science and Technology, Rice University, 6100 Main Street, Houston, Texas 77005, USA
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Abstract
ABSTRACTIt has been well established by theory and simulations that the reaction rates of diffusion-limited reactions can be affected by the spatial dimension in which they occur. The types of reactions A + B → C, A + A → A, and A + C→, C have been shown, theoretically and/or by simulation, to exhibit non-classical reaction kinetics in low and fractal dimensions. We present here experimental results from several 1D and fractal systems.An A+B → C type reaction was experimentally investigated in a long, thin capillary tube in which the reactants, A and B, are initially segregated. This initial segregation of reactants means that the net diffusion is along the length of the capillary only, making the system effectively 1D and allowing some of the properties of the resulting reaction front to be studied. The reaction rate of excitonic fusion, A + A →A, as well as trapping, A + C→ C, reactions were observed via phosphorescence(P) and delayed fluorescence(F) of naphthalene within the channels of Vycor glass, in isotopically mixed naphthalene crystals and in the isolated chains of dilute polymer blends. In these experiments, the non-classical kinetics are measured in terms of the heterogeneity exponent, h, from the equation: Rate ∼ F =; kt-hpn,, which gives the time dependence of the rate coefficient. Classically h =; 0, while h =; 1/2 in 1D, as well as in the fractal dimensions discussed here, for A + A → A as well as A + C →C type reactions.
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18
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Korolev KS, Avlund M, Hallatschek O, Nelson DR. Genetic demixing and evolution in linear stepping stone models. REVIEWS OF MODERN PHYSICS 2010; 82:1691-1718. [PMID: 21072144 PMCID: PMC2975585 DOI: 10.1103/revmodphys.82.1691] [Citation(s) in RCA: 130] [Impact Index Per Article: 9.3] [Reference Citation Analysis] [Abstract] [Grants] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 05/17/2023]
Abstract
Results for mutation, selection, genetic drift, and migration in a one-dimensional continuous population are reviewed and extended. The population is described by a continuous limit of the stepping stone model, which leads to the stochastic Fisher-Kolmogorov-Petrovsky-Piscounov equation with additional terms describing mutations. Although the stepping stone model was first proposed for population genetics, it is closely related to "voter models" of interest in nonequilibrium statistical mechanics. The stepping stone model can also be regarded as an approximation to the dynamics of a thin layer of actively growing pioneers at the frontier of a colony of micro-organisms undergoing a range expansion on a Petri dish. The population tends to segregate into monoallelic domains. This segregation slows down genetic drift and selection because these two evolutionary forces can only act at the boundaries between the domains; the effects of mutation, however, are not significantly affected by the segregation. Although fixation in the neutral well-mixed (or "zero-dimensional") model occurs exponentially in time, it occurs only algebraically fast in the one-dimensional model. An unusual sublinear increase is also found in the variance of the spatially averaged allele frequency with time. If selection is weak, selective sweeps occur exponentially fast in both well-mixed and one-dimensional populations, but the time constants are different. The relatively unexplored problem of evolutionary dynamics at the edge of an expanding circular colony is studied as well. Also reviewed are how the observed patterns of genetic diversity can be used for statistical inference and the differences are highlighted between the well-mixed and one-dimensional models. Although the focus is on two alleles or variants, q-allele Potts-like models of gene segregation are considered as well. Most of the analytical results are checked with simulations and could be tested against recent spatial experiments on range expansions of inoculations of Escherichia coli and Saccharomyces cerevisiae.
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Affiliation(s)
- K S Korolev
- Department of Physics and FAS Center for Systems Biology, Harvard University, Cambridge, Massachusetts 02138, USA
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19
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Romero AH, Lacasta AM, Sancho JM, Lindenberg K. Numerical study of A+A→0 and A+B→0 reactions with inertia. J Chem Phys 2007; 127:174506. [DOI: 10.1063/1.2779327] [Citation(s) in RCA: 10] [Impact Index Per Article: 0.6] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/15/2022] Open
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20
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Morelli LG, Cerdeira HA. Aggregation process on complex networks. PHYSICAL REVIEW. E, STATISTICAL, NONLINEAR, AND SOFT MATTER PHYSICS 2004; 69:051107. [PMID: 15244808 DOI: 10.1103/physreve.69.051107] [Citation(s) in RCA: 1] [Impact Index Per Article: 0.1] [Reference Citation Analysis] [Abstract] [MESH Headings] [Track Full Text] [Subscribe] [Scholar Register] [Received: 11/21/2003] [Indexed: 05/24/2023]
Abstract
We study the dynamics of the aggregation of particles and the evolution of the mass distribution, on a complex network which is built following the Watts-Strogatz model. The particles perform random walks following the links on the network, and aggregate when they meet other particles. On disordered networks the density of particles decays as t(-1), while on regular networks it decays as t(-1/2). For intermediate levels of network disorder the dynamics follows that of regular networks at intermediate density, and for low density the disorder of the network becomes relevant and the density decays as t(-1). The crossover time between these two regimes scales with network disorder as t approximately p(-2). We study also an annealed model for the aggregation process, in which the quenched disorder of the network is replaced by stochastic long range jumps in the particle dynamics. The annealed model is found to obey a different scaling with network disorder, with a crossover time t approximately p(-1).
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Affiliation(s)
- Luis G Morelli
- Abdus Salam International Center for Theoretical Physics, P O Box 586, 34100 Trieste, Italy.
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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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Oshanin G, Burlatsky SF. Adsorption of reactive particles on a random catalytic chain: an exact solution. PHYSICAL REVIEW. E, STATISTICAL, NONLINEAR, AND SOFT MATTER PHYSICS 2003; 67:016115. [PMID: 12636572 DOI: 10.1103/physreve.67.016115] [Citation(s) in RCA: 3] [Impact Index Per Article: 0.1] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 09/19/2002] [Indexed: 05/24/2023]
Abstract
We study equilibrium properties of a catalytically activated annihilation A+A-->0 reaction taking place on a one-dimensional chain of length N (N--> infinity ) in which some segments (placed at random, with mean concentration p) possess special, catalytic properties. Annihilation reaction takes place as soon as any two A particles land onto two vacant sites at the extremities of the catalytic segment, or when any A particle lands onto a vacant site on a catalytic segment while the site at the other extremity of this segment is already occupied by another A particle. Noncatalytic segments are inert with respect to reaction and here two adsorbed A particles harmlessly coexist. For both "annealed" and "quenched" disorder in placement of the catalytic segments, we calculate exactly the disorder-averaged pressure per site. Explicit asymptotic formulas for the particle mean density and the compressibility are also presented.
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Affiliation(s)
- G Oshanin
- Laboratoire de Physique Théorique des Liquides, Université Paris 6, 4 Place Jussieu, 75252 Paris, France
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Karimipour V. General reaction-diffusion processes with separable equations for correlation functions. PHYSICAL REVIEW. E, STATISTICAL, NONLINEAR, AND SOFT MATTER PHYSICS 2002; 66:041114. [PMID: 12443184 DOI: 10.1103/physreve.66.041114] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 04/21/2002] [Revised: 07/25/2002] [Indexed: 05/24/2023]
Abstract
We consider general multispecies models of reaction diffusion processes and obtain a set of constraints on the rates which give rise to closed systems of equations for correlation functions. Our results are valid in any dimension and on any type of lattice. We also show that under these conditions the evolution equations for two point functions at different times are also closed. As an example we introduce a class of two species models that may be useful for the description of voting processes or the spreading of epidemics.
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Affiliation(s)
- V Karimipour
- Department of Physics, Sharif University of Technology, P.O. Box 11365-9161, Tehran, Iran.
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Mobilia M, Bares PA. Generalized empty-interval method applied to a class of one-dimensional stochastic models. PHYSICAL REVIEW. E, STATISTICAL, NONLINEAR, AND SOFT MATTER PHYSICS 2001; 64:066123. [PMID: 11736252 DOI: 10.1103/physreve.64.066123] [Citation(s) in RCA: 1] [Impact Index Per Article: 0.0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 07/09/2001] [Indexed: 05/23/2023]
Abstract
In this work we study, on a finite and periodic lattice, a class of one-dimensional (bimolecular and single-species) reaction-diffusion models that cannot be mapped onto free-fermion models. We extend the conventional empty-interval method, also called interparticle distribution function (IPDF) method, by introducing a string function, which is simply related to relevant physical quantities. As an illustration, we specifically consider a model that cannot be solved directly by the conventional IPDF method and that can be viewed as a generalization of the voter model and/or as an epidemic model. We also consider the reversible diffusion-coagulation model with input of particles and determine other reaction-diffusion models that can be mapped onto the latter via suitable similarity transformations. Finally we study the problem of the propagation of a wave front from an inhomogeneous initial configuration and note that the mean-field scenario predicted by Fisher's equation is not valid for the one-dimensional (microscopic) models under consideration.
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Affiliation(s)
- M Mobilia
- Institute of Theoretical Physics, Swiss Federal Institute of Technology of Lausanne, CH-1015 Lausanne EPFL, Switzerland
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Mobilia M, Bares PA. Solution of a one-dimensional stochastic model with branching and coagulation reactions. PHYSICAL REVIEW. E, STATISTICAL, NONLINEAR, AND SOFT MATTER PHYSICS 2001; 64:045101. [PMID: 11690073 DOI: 10.1103/physreve.64.045101] [Citation(s) in RCA: 1] [Impact Index Per Article: 0.0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 04/17/2001] [Indexed: 05/23/2023]
Abstract
We solve a one-dimensional stochastic model of interacting particles on a chain. Particles can have branching and coagulation reactions; they can also appear on an empty site and disappear spontaneously. This model, which can be viewed as an epidemic model and/or as a generalization of the voter model, is treated analytically beyond the conventional solvable situations. With help of a suitably chosen string function, which is simply related to the density and the noninstantaneous two-point correlation functions of the particles, exact expressions of the density and of the noninstantaneous two-point correlation functions, as well as the relaxation spectrum are obtained on a finite and periodic lattice.
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Affiliation(s)
- M Mobilia
- Institute of Theoretical Physics, Swiss Federal Institute of Technology of Lausanne, CH-1015 Lausanne, Switzerland
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Yuste SB, Lindenberg K. Subdiffusion-limited A+A reactions. PHYSICAL REVIEW LETTERS 2001; 87:118301. [PMID: 11531549 DOI: 10.1103/physrevlett.87.118301] [Citation(s) in RCA: 20] [Impact Index Per Article: 0.9] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 04/02/2001] [Indexed: 05/23/2023]
Abstract
We consider the coagulation dynamics A+A-->A and A+A <==> A and the annihilation dynamics A+A-->0 for particles moving subdiffusively in one dimension. This scenario combines the "anomalous kinetics" and "anomalous diffusion" problems, each of which leads to interesting dynamics separately and to even more interesting dynamics in combination. Our analysis is based on the fractional diffusion equation.
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Affiliation(s)
- S B Yuste
- Department of Chemistry and Biochemistry 0340, University of California San Diego, La Jolla, California 92093-0340, USA
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Fontes LR, Isopi M, Newman CM, Stein DL. Aging in 1D discrete spin models and equivalent systems. PHYSICAL REVIEW LETTERS 2001; 87:110201. [PMID: 11531509 DOI: 10.1103/physrevlett.87.110201] [Citation(s) in RCA: 7] [Impact Index Per Article: 0.3] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 03/07/2001] [Indexed: 05/23/2023]
Abstract
We derive exact expressions for a number of aging functions that are scaling limits of nonequilibrium correlations, R(t(w),t(w)+t) as t(w)-->infinity, t/t(w)-->theta, in the 1D homogenous q-state Potts model for all q with T = 0 dynamics following a quench from T = infinity. One such quantity is <sigma-->(0)(t(w));sigma-->(n)(t(w)+t)> when n/square root of ([t(w))-->z. Exact, closed-form expressions are also obtained when an interlude of T = infinity dynamics occurs. Our derivations express the scaling limit via coalescing Brownian paths and a "Brownian space-time spanning tree," which also yields other aging functions, such as the persistence probability of no spin flip at 0 between t(w) and t(w)+t.
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Affiliation(s)
- L R Fontes
- Instituto de Matemática e Estatística, Universidade de São Paulo, 05315-970 São Paulo, Brasil
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Exactly Solvable Models for Many-Body Systems Far from Equilibrium. PHASE TRANSITIONS AND CRITICAL PHENOMENA 2001. [DOI: 10.1016/s1062-7901(01)80015-x] [Citation(s) in RCA: 155] [Impact Index Per Article: 6.7] [Reference Citation Analysis] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 12/24/2022]
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Kim H, Shin KJ. Single species diffusion-influenced reaction A+A-->alphaA: validity of the smoluchowski approach. PHYSICAL REVIEW. E, STATISTICAL PHYSICS, PLASMAS, FLUIDS, AND RELATED INTERDISCIPLINARY TOPICS 2000; 61:3426-34. [PMID: 11088119 DOI: 10.1103/physreve.61.3426] [Citation(s) in RCA: 10] [Impact Index Per Article: 0.4] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 06/30/1999] [Revised: 11/05/1999] [Indexed: 11/07/2022]
Abstract
We investigate the single species diffusion-influenced reaction, A+A-->alphaA with a finite reactivity in all dimensions. The reaction model includes a pure coagulation (alpha=1) or a pure annihilation (alpha=0) model. We apply the hierarchical Smoluchowski approach to study the dimensional aspects of the fluctuation, reactivity, particle size, and alpha(0</=alpha</=1). The theoretical results are compared with those of the Monte Carlo simulations in one, two, and three regular dimensions. The simulation results reveal that the classical Smoluchowski approach is exact in the short time limit in all dimensions and in the long time limit in three dimensions. The hierarchical Smoluchowski approach is found to be numerically exact at all times in two and three dimensions. A numerical method to obtain the exact result of the annihilation for a finite reactivity in one dimension is presented. We also propose a quite accurate analytic solution for an arbitrary alpha for the infinite reactivity in one dimension.
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Affiliation(s)
- H Kim
- Department of Chemistry and Center for Molecular Catalysis, Seoul National University, Seoul 151-742, Korea
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Hinrichsen H, Sandow S, Peschel I. On matrix product ground states for reaction - diffusion models. ACTA ACUST UNITED AC 1999. [DOI: 10.1088/0305-4470/29/11/005] [Citation(s) in RCA: 61] [Impact Index Per Article: 2.4] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/12/2022]
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Redner S, ben-Avraham D. Nearest-neighbour distances of diffusing particles from a single trap. ACTA ACUST UNITED AC 1999. [DOI: 10.1088/0305-4470/23/22/003] [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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Braunstein L, Martin HO, Grynberg MD, Roman HE. Effects of probability of reaction on annihilation reactions in one dimension. ACTA ACUST UNITED AC 1999. [DOI: 10.1088/0305-4470/25/5/009] [Citation(s) in RCA: 33] [Impact Index Per Article: 1.3] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/11/2022]
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Derrida B. Exponents appearing in the zero-temperature dynamics of the 1D Potts model. ACTA ACUST UNITED AC 1999. [DOI: 10.1088/0305-4470/28/6/006] [Citation(s) in RCA: 54] [Impact Index Per Article: 2.2] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/12/2022]
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Simon H. Concentration for one and two-species one-dimensional reaction-diffusion systems. ACTA ACUST UNITED AC 1999. [DOI: 10.1088/0305-4470/28/23/013] [Citation(s) in RCA: 52] [Impact Index Per Article: 2.1] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/12/2022]
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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]
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Lin AL, Kopelman R, Argyrakis P. Diffusion-Controlled Elementary Reactions in Tubular Confinement: Extreme Nonclassicality, Segregation, and Anomalous Scaling Laws for Dimensional Crossovers. J Phys Chem A 1997. [DOI: 10.1021/jp9628904] [Citation(s) in RCA: 17] [Impact Index Per Article: 0.6] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/30/2022]
Affiliation(s)
- Anna L. Lin
- Department of Chemistry, University of Michigan, Ann Arbor, Michigan 48109-1055
| | - Raoul Kopelman
- Department of Chemistry, University of Michigan, Ann Arbor, Michigan 48109-1055
| | - Panos Argyrakis
- Department of Physics, University of Thessaloniki, 54006 Thessaloniki, Greece
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Sibona GJ, Budde CE, Condat CA. Diffusion- and drift-controlled reactions in two and three dimensions. PHYSICAL REVIEW. E, STATISTICAL PHYSICS, PLASMAS, FLUIDS, AND RELATED INTERDISCIPLINARY TOPICS 1996; 54:6232-6241. [PMID: 9965843 DOI: 10.1103/physreve.54.6232] [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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Lin AL, Kopelman R, Argyrakis P. Dimensional crossovers and anomalous scaling of single and reacting random walkers in baguettelike lattices: Monte Carlo simulations of the number of distinct sites visited and of bimolecular A+A and A+B reactions. PHYSICAL REVIEW. E, STATISTICAL PHYSICS, PLASMAS, FLUIDS, AND RELATED INTERDISCIPLINARY TOPICS 1996; 54:R5893-R5896. [PMID: 9965939 DOI: 10.1103/physreve.54.r5893] [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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Monthus C. Exponents appearing in heterogeneous reaction-diffusion models in one dimension. PHYSICAL REVIEW. E, STATISTICAL PHYSICS, PLASMAS, FLUIDS, AND RELATED INTERDISCIPLINARY TOPICS 1996; 54:4844-4859. [PMID: 9965666 DOI: 10.1103/physreve.54.4844] [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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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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Molski A. Fluctuation theory of the coupling between diffusion and reaction in 1D: coalescence with point reactivities. Chem Phys Lett 1996. [DOI: 10.1016/s0009-2614(96)00166-2] [Citation(s) in RCA: 3] [Impact Index Per Article: 0.1] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 12/01/2022]
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Martinez HL. Kinetics of nonstationary, single species, bimolecular, diffusion‐influenced irreversible reactions. J Chem Phys 1996. [DOI: 10.1063/1.471661] [Citation(s) in RCA: 7] [Impact Index Per Article: 0.3] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/14/2022] Open
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Lin A, Kopelman R, Argyrakis P. Nonclassical kinetics in three dimensions: Simulations of elementary A+B and A+A reactions. PHYSICAL REVIEW. E, STATISTICAL PHYSICS, PLASMAS, FLUIDS, AND RELATED INTERDISCIPLINARY TOPICS 1996; 53:1502-1509. [PMID: 9964413 DOI: 10.1103/physreve.53.1502] [Citation(s) in RCA: 34] [Impact Index Per Article: 1.2] [Reference Citation Analysis] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 11/07/2022]
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Lindenberg K, Argyrakis P, Kopelman R. The Hierarchies of Nonclassical Regimes for Diffusion-Limited Binary Reactions. ACTA ACUST UNITED AC 1996. [DOI: 10.1007/978-1-4612-3992-5_12] [Citation(s) in RCA: 1] [Impact Index Per Article: 0.0] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 03/07/2023]
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48
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Bonnier B, Pommiers E. Segregation in the static pair annihilation process: Exact results. PHYSICAL REVIEW. E, STATISTICAL PHYSICS, PLASMAS, FLUIDS, AND RELATED INTERDISCIPLINARY TOPICS 1995; 52:5873-5876. [PMID: 9964102 DOI: 10.1103/physreve.52.5873] [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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Balboni D, Rey PA, Droz M. Universality of a class of annihilation-coagulation models. PHYSICAL REVIEW. E, STATISTICAL PHYSICS, PLASMAS, FLUIDS, AND RELATED INTERDISCIPLINARY TOPICS 1995; 52:6220-6226. [PMID: 9964138 DOI: 10.1103/physreve.52.6220] [Citation(s) in RCA: 6] [Impact Index Per Article: 0.2] [Reference Citation Analysis] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 05/22/2023]
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Grynberg MD, Stinchcombe RB. Dynamics of adsorption-desorption processes as a soluble problem of many fermions. PHYSICAL REVIEW. E, STATISTICAL PHYSICS, PLASMAS, FLUIDS, AND RELATED INTERDISCIPLINARY TOPICS 1995; 52:6013-6024. [PMID: 9964117 DOI: 10.1103/physreve.52.6013] [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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