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Abad E. On-lattice coalescence and annihilation of immobile reactants in loopless lattices and beyond. PHYSICAL REVIEW. E, STATISTICAL, NONLINEAR, AND SOFT MATTER PHYSICS 2004; 70:031110. [PMID: 15524509 DOI: 10.1103/physreve.70.031110] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 08/27/2003] [Revised: 06/03/2004] [Indexed: 05/24/2023]
Abstract
We study the behavior of the chemical reactions A+A-->A+S and A+A-->S+S (where the reactive species A and the inert species S are both assumed to be immobile) embedded on Bethe lattices of arbitrary coordination number z and on a two-dimensional (2D) square lattice. For the Bethe lattice case, exact solutions for the coverage in the A species in terms of the initial condition are obtained. In particular, our results hold for the important case of an infinite one-dimensional (1D) lattice (z=2). The method is based on an expansion in terms of conditional probabilities which exploits a Markovian property of these systems. Along the same lines, an approximate solution for the case of a 2D square lattice is developed. The effect of dilution in a random initial condition is discussed in detail, both for the lattice coverage and for the spatial distribution of reactants.
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Affiliation(s)
- E Abad
- Centre for Nonlinear Phenomena and Complex Systems, Université Libre de Bruxelles, CP 231, 1050 Bruxelles, Belgium.
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Mobilia M, Redner S. Majority versus minority dynamics: phase transition in an interacting two-state spin system. PHYSICAL REVIEW. E, STATISTICAL, NONLINEAR, AND SOFT MATTER PHYSICS 2003; 68:046106. [PMID: 14683001 DOI: 10.1103/physreve.68.046106] [Citation(s) in RCA: 12] [Impact Index Per Article: 0.6] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 06/04/2003] [Indexed: 05/23/2023]
Abstract
We introduce a simple model of opinion dynamics in which binary-state agents evolve due to the influence of agents in a local neighborhood. In a single update step, a fixed-size group is defined and all agents in the group adopt the state of the local majority with probability p or that of the local minority with probability 1-p. For group size G=3, there is a phase transition at p(c)=2/3 in all spatial dimensions. For p>p(c), the global majority quickly predominates, while for p<p(c), the system is driven to a mixed state in which the densities of agents in each state are equal. For p=p(c), the average magnetization (the difference in the density of agents in the two states) is conserved and the system obeys classical voter model dynamics. In one dimension and within a Kirkwood decoupling scheme, the final magnetization in a finite-length system has a nontrivial dependence on the initial magnetization for all p not equal p(c), in agreement with numerical results. At p(c), the exact two-spin correlation functions decay algebraically toward the value 1 and the system coarsens as in the classical voter model.
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Affiliation(s)
- M Mobilia
- Center for BioDynamics, Center for Polymer Studies & Department of Physics, Boston University, Boston, Massachusetts 02215, USA.
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Mai J, Sokolov IM, Blumen A. Directed particle diffusion under "burnt bridges" conditions. PHYSICAL REVIEW. E, STATISTICAL, NONLINEAR, AND SOFT MATTER PHYSICS 2001; 64:011102. [PMID: 11461220 DOI: 10.1103/physreve.64.011102] [Citation(s) in RCA: 27] [Impact Index Per Article: 1.2] [Reference Citation Analysis] [Abstract] [MESH Headings] [Track Full Text] [Subscribe] [Scholar Register] [Received: 10/27/2000] [Indexed: 05/23/2023]
Abstract
We study random walks on a one-dimensional lattice that contains weak connections, so-called "bridges." Each time the walker crosses the bridge from the left or attempts to cross it from the right, the bridge may be destroyed with probability p; this restricts the particle's motion and directs it. Our model, which incorporates asymmetric aspects in an otherwise symmetric hopping mechanism, is very akin to "Brownian ratchets" and to front propagation in autocatalytic A+B-->2A reactions. The analysis of the model and Monte Carlo simulations show that for large p the velocity of the directed motion is extremely sensitive to the distribution of bridges, whereas for small p the velocity can be understood based on a mean-field analysis. The single-particle model advanced by us here allows an almost quantitative understanding of the front's position in the A+B-->2A many-particle reaction.
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Affiliation(s)
- J Mai
- Theoretische Polymerphysik, Universität Freiburg, Hermann-Herder-Strasse 3, D-79104 Freiburg, Germany
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Abad E, Grosfils P, Nicolis G. Nonlinear reactive systems on a lattice viewed as Boolean dynamical systems. PHYSICAL REVIEW. E, STATISTICAL, NONLINEAR, AND SOFT MATTER PHYSICS 2001; 63:041102. [PMID: 11308814 DOI: 10.1103/physreve.63.041102] [Citation(s) in RCA: 3] [Impact Index Per Article: 0.1] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 08/22/2000] [Indexed: 05/23/2023]
Abstract
We present a stochastic, time-discrete Boolean model that mimics the mesoscopic dynamics of the desorption reactions A+A-->A+S and A+A-->S+S in a one-dimensional lattice. In the continuous-time limit, we derive a hierarchy of dynamical equations for the subset of moments involving contiguous lattice sites. The solution of the hierarchy allows to compute the exact dynamics of the mean coverage for both microscopic and coarse-grained initial conditions, which turn out to be different from the mean field predictions. The evolution equations for the mean coverage and the second-order moments are shown to be equivalent to those provided by a time-continuous master equation. The important role of higher-order fluctuations is brought out by the failure of a truncation scheme retaining only two-particle fluctuation correlations.
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Affiliation(s)
- E Abad
- Centre for Nonlinear Phenomena and Complex Systems, Université Libre de Bruxelles, Campus Plaine, Code Postal 231, B-1050 Brussels, Belgium.
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Mai J, Sokolov IM, Blumen A. Front propagation in one-dimensional autocatalytic reactions: the breakdown of the classical picture at small particle concentrations. PHYSICAL REVIEW. E, STATISTICAL PHYSICS, PLASMAS, FLUIDS, AND RELATED INTERDISCIPLINARY TOPICS 2000; 62:141-5. [PMID: 11088445 DOI: 10.1103/physreve.62.141] [Citation(s) in RCA: 12] [Impact Index Per Article: 0.5] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 01/26/2000] [Indexed: 04/15/2023]
Abstract
The autocatalytic scheme A+B-->2A in a discrete particle system is studied in one dimension via Monte Carlo simulations. We find considerable differences in the results for the front velocities and front forms compared to the classical, continuous picture, which is only valid in the limit of very small reaction probabilities p. Interestingly, we also obtain front propagation velocities fairly below the classical minimal velocity.
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Affiliation(s)
- J Mai
- Theoretische Polymerphysik, Universitat Freiburg, Hermann-Herder Strasse 3, D-79104 Freiburg im Breisgau, Germany
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Sokolov I, Blumen A. Bimolecular reactions in condensed matter: scales of mixing and homogenization. J Mol Liq 2000. [DOI: 10.1016/s0167-7322(99)00120-8] [Citation(s) in RCA: 1] [Impact Index Per Article: 0.0] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 10/17/2022]
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Bonnier B. Static A+B-->0 annihilation process in arbitrary dimension. PHYSICAL REVIEW. E, STATISTICAL PHYSICS, PLASMAS, FLUIDS, AND RELATED INTERDISCIPLINARY TOPICS 2000; 61:1270-4. [PMID: 11046404 DOI: 10.1103/physreve.61.1270] [Citation(s) in RCA: 3] [Impact Index Per Article: 0.1] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 04/07/1999] [Indexed: 11/07/2022]
Abstract
The pair annihilation A+B-->0 of static particles distributed at random in a d-dimensional space is studied in the large time regime for a tunneling law. The consequences of a superposition approximation used to close the hierarchy of equations describing the process are analytically explored. It is shown that the density and pair correlations of surviving particles have scaling expressions which display the ordering and clustering effects.
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Affiliation(s)
- B Bonnier
- Centre de Physique Theorique et de Modelisation, ESA 5468 du CNRS, Universite Bordeaux I, 351 Cours de la Liberation, 33405 Talence Cedex, France
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Mai J, Kuzovkov VN, Niessen WV. A Lotka-type model for oscillations in surface reactions. ACTA ACUST UNITED AC 1999. [DOI: 10.1088/0305-4470/30/12/009] [Citation(s) in RCA: 7] [Impact Index Per Article: 0.3] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/11/2022]
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Bonnier B, Brown R, Pommiers E. Self-organization in the static pair annihilation process. ACTA ACUST UNITED AC 1999. [DOI: 10.1088/0305-4470/28/18/008] [Citation(s) in RCA: 6] [Impact Index Per Article: 0.2] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/12/2022]
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Reigada R, Sagués F, Sokolov IM, Sancho JM, Blumen A. Spatial organization in the A+B→0 reaction under confined-scale mixing. J Chem Phys 1997. [DOI: 10.1063/1.474470] [Citation(s) in RCA: 14] [Impact Index Per Article: 0.5] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/15/2022] Open
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Baranovskii SD, Hensel F, Golub JE, Thomas P. Long-time behavior of the diffusion-controlled A+B→0 reaction with hopping energy relaxation. J Chem Phys 1997. [DOI: 10.1063/1.473057] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/14/2022] Open
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Reigada R, Sagués F, Sokolov IM, Sancho JM, Blumen A. Kinetics of the A+B→0 reaction under steady and turbulent flows. J Chem Phys 1996. [DOI: 10.1063/1.472896] [Citation(s) in RCA: 12] [Impact Index Per Article: 0.4] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/14/2022] Open
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Kuzovkov VN, Kotomin EA, von Niessen W. The kinetics of the bimolecular A+B→0 reaction in condensed matter: Effects of non‐equilibrium charge screening. J Chem Phys 1996. [DOI: 10.1063/1.472782] [Citation(s) in RCA: 8] [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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Mai J, Sokolov IM, Blumen A. Front Propagation and Local Ordering in One-Dimensional Irreversible Autocatalytic Reactions. PHYSICAL REVIEW LETTERS 1996; 77:4462-4465. [PMID: 10062544 DOI: 10.1103/physrevlett.77.4462] [Citation(s) in RCA: 11] [Impact Index Per Article: 0.4] [Reference Citation Analysis] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 05/23/2023]
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Oshanin G, Sokolov IM, Argyrakis P, Blumen A. Fluctuation‐dominated A+B→0 kinetics under short‐ranged interparticle interactions. J Chem Phys 1996. [DOI: 10.1063/1.472466] [Citation(s) in RCA: 10] [Impact Index Per Article: 0.4] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/14/2022] Open
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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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Hoyuelos M. Steady-state density in annihilation of immobile reactants with input of particles. PHYSICAL REVIEW. E, STATISTICAL PHYSICS, PLASMAS, FLUIDS, AND RELATED INTERDISCIPLINARY TOPICS 1994; 50:2597-2599. [PMID: 9962296 DOI: 10.1103/physreve.50.2597] [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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Sokolov IM, Blumen A. Kinetics in coagulation-annihilation processes. PHYSICAL REVIEW. E, STATISTICAL PHYSICS, PLASMAS, FLUIDS, AND RELATED INTERDISCIPLINARY TOPICS 1994; 50:2335-2338. [PMID: 9962242 DOI: 10.1103/physreve.50.2335] [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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Kuzovkov V, Kotomin E. Pair and triple correlations in the A+B-->B diffusion-controlled reaction. PHYSICAL REVIEW LETTERS 1994; 72:2105-2108. [PMID: 10055789 DOI: 10.1103/physrevlett.72.2105] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 05/23/2023]
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Mai J, Kuzovkov VN, von Niessen W. A theoretical stochastic model for the A+1/2B2→0 reaction. J Chem Phys 1993. [DOI: 10.1063/1.464434] [Citation(s) in RCA: 28] [Impact Index Per Article: 0.9] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/14/2022] Open
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Kuzovkov V, Kotomin E. Dynamic particle aggregation in the bimolecularA+B→0 reaction. J Chem Phys 1993. [DOI: 10.1063/1.464468] [Citation(s) in RCA: 17] [Impact Index Per Article: 0.5] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/14/2022] Open
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Privman V. Exact solution of a phase separation model with conserved order parameter dynamics. PHYSICAL REVIEW LETTERS 1992; 69:3686-3688. [PMID: 10046887 DOI: 10.1103/physrevlett.69.3686] [Citation(s) in RCA: 13] [Impact Index Per Article: 0.4] [Reference Citation Analysis] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 05/23/2023]
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Searle TM. Unstirred annihilation reactions and pair correlations. PHYSICAL REVIEW LETTERS 1992; 69:3256-3259. [PMID: 10046770 DOI: 10.1103/physrevlett.69.3256] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 05/23/2023]
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Zumofen G, Klafter J, Blumen A. Stochastic and deterministic analysis of reactions: The fractal case. PHYSICAL REVIEW. A, ATOMIC, MOLECULAR, AND OPTICAL PHYSICS 1991; 44:8390-8393. [PMID: 9905997 DOI: 10.1103/physreva.44.8390] [Citation(s) in RCA: 8] [Impact Index Per Article: 0.2] [Reference Citation Analysis] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 05/22/2023]
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Lin JC. Closure schemes for joint density functions in diffusion-limited reactions. PHYSICAL REVIEW A 1991; 44:6706-6722. [PMID: 9905799 DOI: 10.1103/physreva.44.6706] [Citation(s) in RCA: 21] [Impact Index Per Article: 0.6] [Reference Citation Analysis] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 11/07/2022]
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Brown R, Efremov NA. New Monte Carlo simulations of many-particle stochastic dynamics: Growth of correlations and local self-ordering during annihilation of like particles. Chem Phys 1991. [DOI: 10.1016/0301-0104(91)80112-u] [Citation(s) in RCA: 5] [Impact Index Per Article: 0.2] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/25/2022]
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Sokolov IM, Blumen A. Distribution of striation thicknesses in reacting lamellar systems. PHYSICAL REVIEW. A, ATOMIC, MOLECULAR, AND OPTICAL PHYSICS 1991; 43:6545-6549. [PMID: 9905004 DOI: 10.1103/physreva.43.6545] [Citation(s) in RCA: 11] [Impact Index Per Article: 0.3] [Reference Citation Analysis] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 05/22/2023]
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Lin JC. Symmetry between diffusion-limited coagulation and annihilation. PHYSICAL REVIEW. A, ATOMIC, MOLECULAR, AND OPTICAL PHYSICS 1991; 43:5714-5716. [PMID: 9904890 DOI: 10.1103/physreva.43.5714] [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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Sokolov IM, Blumen A. Mixing effects in the A+B-->0 reaction-diffusion scheme. PHYSICAL REVIEW LETTERS 1991; 66:1942-1945. [PMID: 10043349 DOI: 10.1103/physrevlett.66.1942] [Citation(s) in RCA: 12] [Impact Index Per Article: 0.4] [Reference Citation Analysis] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 05/23/2023]
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Muzzio FJ, Ottino JM. Diffusion and reaction in a lamellar system: Self-similarity with finite rates of reaction. PHYSICAL REVIEW. A, ATOMIC, MOLECULAR, AND OPTICAL PHYSICS 1990; 42:5873-5884. [PMID: 9903866 DOI: 10.1103/physreva.42.5873] [Citation(s) in RCA: 17] [Impact Index Per Article: 0.5] [Reference Citation Analysis] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 05/22/2023]
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Schnörer H, Kuzovkov V, Blumen A. Bimolecular annihilation reactions with immobile reactants: Unequal reactant concentrations. J Chem Phys 1990. [DOI: 10.1063/1.459438] [Citation(s) in RCA: 10] [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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Sheu WS, Lindenberg K. Exact solution of the one-dimensional immobile trapping problem with and without sources. PHYSICAL REVIEW. A, ATOMIC, MOLECULAR, AND OPTICAL PHYSICS 1990; 42:5025-5028. [PMID: 9904618 DOI: 10.1103/physreva.42.5025] [Citation(s) in RCA: 3] [Impact Index Per Article: 0.1] [Reference Citation Analysis] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 04/11/2023]
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Schnörer H, Kuzovkov V, Blumen A. Bimolecular annihilation reactions with immobile reactants. J Chem Phys 1990. [DOI: 10.1063/1.457972] [Citation(s) in RCA: 32] [Impact Index Per Article: 0.9] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/14/2022] Open
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38
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Muzzio FJ, Ottino JM. Dynamics of a lamellar system with diffusion and reaction: Scaling analysis and global kinetics. PHYSICAL REVIEW. A, GENERAL PHYSICS 1989; 40:7182-7192. [PMID: 9902133 DOI: 10.1103/physreva.40.7182] [Citation(s) in RCA: 22] [Impact Index Per Article: 0.6] [Reference Citation Analysis] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 05/22/2023]
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