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Junge M, Lyu H. The phase structure of asymmetric ballistic annihilation. ANN APPL PROBAB 2022. [DOI: 10.1214/21-aap1773] [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]
Affiliation(s)
| | - Hanbaek Lyu
- Department of Mathematics, University of Wisonsin–Madison
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Damron M, Gravner J, Junge M, Lyu H, Sivakoff D. Parking on transitive unimodular graphs. ANN APPL PROBAB 2019. [DOI: 10.1214/18-aap1443] [Citation(s) in RCA: 7] [Impact Index Per Article: 1.4] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/19/2022]
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García de Soria MI, Maynar P, Trizac E. Brownian motion under annihilation dynamics. PHYSICAL REVIEW. E, STATISTICAL, NONLINEAR, AND SOFT MATTER PHYSICS 2008; 78:061110. [PMID: 19256805 DOI: 10.1103/physreve.78.061110] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 09/03/2008] [Indexed: 05/27/2023]
Abstract
The behavior of a heavy tagged intruder immersed in a bath of particles evolving under ballistic annihilation dynamics is investigated. The Fokker-Planck equation for this system is derived and the peculiarities of the corresponding diffusive behavior are worked out. In the long time limit, the intruder velocity distribution function approaches a Gaussian form, but with a different temperature from its bath counterpart. As a consequence of the continuous decay of particles in the bath, the mean-squared displacement increases exponentially in the collision per particle time scale. Analytical results are finally successfully tested against Monte Carlo numerical simulations.
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Sheu WS, Wang SC. Effects of velocity relaxation on the anomalous kinetics of a one-dimensional A+A-->Ø reaction. PHYSICAL REVIEW. E, STATISTICAL, NONLINEAR, AND SOFT MATTER PHYSICS 2008; 78:046101. [PMID: 18999487 DOI: 10.1103/physreve.78.046101] [Citation(s) in RCA: 1] [Impact Index Per Article: 0.1] [Reference Citation Analysis] [Abstract] [MESH Headings] [Track Full Text] [Subscribe] [Scholar Register] [Received: 06/30/2008] [Revised: 08/07/2008] [Indexed: 05/27/2023]
Abstract
A one-dimensional A+A-->slashed circle reaction with alternating dichotomic velocities is investigated to study the effects of velocity relaxation on the anomalous kinetics of this reaction. While we keep the magnitudes constant, the particle velocities are allowed to change from one moving direction to the other with a relaxation time, beta. The kinetics of the reaction is studied for various relaxation times. Although the anomalous slower reaction rate is anticipated for the reaction in one-dimension, compared to the classical rate law, the rate is found to speed up at intermediate times for intermediate values of beta . The time evolution for the spatial distribution of particles is also discussed to elucidate the effects of the velocity relaxation times on the kinetics of the reaction.
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Affiliation(s)
- Wen-Shyan Sheu
- Department of Chemistry, Fu-Jen Catholic University, Hsin-Chuang, Taipei 242, Taiwan, Republic of China
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Maynar P, García de Soria MI, Schehr G, Barrat A, Trizac E. Dynamics of annihilation. II. Fluctuations of global quantities. PHYSICAL REVIEW. E, STATISTICAL, NONLINEAR, AND SOFT MATTER PHYSICS 2008; 77:051128. [PMID: 18643047 DOI: 10.1103/physreve.77.051128] [Citation(s) in RCA: 1] [Impact Index Per Article: 0.1] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 01/17/2008] [Indexed: 05/26/2023]
Abstract
We develop a theory for fluctuations and correlations in a gas evolving under ballistic annihilation dynamics. Starting from the hierarchy of equations governing the evolution of microscopic densities in phase space, we subsequently restrict our attention to a regime of spatial homogeneity, and obtain explicit predictions for the fluctuations and time correlation of the total number of particles, total linear momentum, and total kinetic energy. Cross correlations between these quantities are worked out as well. These predictions are successfully tested against molecular dynamics and Monte Carlo simulations. This provides strong support for the theoretical approach developed, including the hydrodynamic treatment of the spectrum of the linearized Boltzmann operator. This paper makes use of the spectral analysis reported in the preceding paper [Phys. Rev. E 77, 051127 (2008)].
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Affiliation(s)
- Pablo Maynar
- Laboratoire de Physique Théorique (CNRS UMR 8627), Bâtiment 210, Université Paris-Sud, 91405 Orsay Cedex, France
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García de Soria MI, Maynar P, Schehr G, Barrat A, Trizac E. Dynamics of annihilation. I. Linearized Boltzmann equation and hydrodynamics. PHYSICAL REVIEW. E, STATISTICAL, NONLINEAR, AND SOFT MATTER PHYSICS 2008; 77:051127. [PMID: 18643046 DOI: 10.1103/physreve.77.051127] [Citation(s) in RCA: 1] [Impact Index Per Article: 0.1] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 01/17/2008] [Indexed: 05/26/2023]
Abstract
We study the nonequilibrium statistical mechanics of a system of freely moving particles, in which binary encounters lead either to an elastic collision or to the disappearance of the pair. Such a system of ballistic annihilation therefore constantly loses particles. The dynamics of perturbations around the free decay regime is investigated using the spectral properties of the linearized Boltzmann operator, which characterize linear excitations on all time scales. The linearized Boltzmann equation is solved in the hydrodynamic limit by a projection technique, which yields the evolution equations for the relevant coarse-grained fields and expressions for the transport coefficients. We finally present the results of molecular dynamics simulations that validate the theoretical predictions.
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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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Gillespie DT, Lampoudi S, Petzold LR. Effect of reactant size on discrete stochastic chemical kinetics. J Chem Phys 2007; 126:034302. [PMID: 17249866 DOI: 10.1063/1.2424461] [Citation(s) in RCA: 15] [Impact Index Per Article: 0.9] [Reference Citation Analysis] [Abstract] [MESH Headings] [Grants] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/14/2022] Open
Abstract
This paper is aimed at understanding what happens to the propensity functions (rates) of bimolecular chemical reactions when the volume occupied by the reactant molecules is not negligible compared to the containing volume of the system. For simplicity our analysis focuses on a one-dimensional gas of N hard-rod molecules, each of length l. Assuming these molecules are distributed randomly and uniformly inside the real interval [0,L] in a nonoverlapping way, and that they have Maxwellian distributed velocities, the authors derive an expression for the probability that two rods will collide in the next infinitesimal time dt. This probability controls the rate of any chemical reaction whose occurrence is initiated by such a collision. The result turns out to be a simple generalization of the well-known result for the point molecule case l=0: the system volume L in the formula for the propensity function in the point molecule case gets replaced by the "free volume" L-Nl. They confirm the result in a series of one-dimensional molecular dynamics simulations. Some possible wider implications of this result are discussed.
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Lipowski A, Lipowska D, Ferreira AL. Molecular dynamics simulations of ballistic annihilation. PHYSICAL REVIEW. E, STATISTICAL, NONLINEAR, AND SOFT MATTER PHYSICS 2006; 73:032102. [PMID: 16605578 DOI: 10.1103/physreve.73.032102] [Citation(s) in RCA: 3] [Impact Index Per Article: 0.2] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 11/22/2005] [Revised: 01/17/2006] [Indexed: 05/08/2023]
Abstract
Using event-driven molecular dynamics we study one- and two-dimensional ballistic annihilation. We estimate exponents xi and gamma, which describe the long-time decay of the number of particles [n(t) approximately t-xi] and of their typical velocity [v(t) approximately t-gamma]. To a good accuracy our results confirm the scaling relation xi+gamma=1. In the two-dimensional case our results are in good agreement with those obtained from Boltzmann kinetic theory.
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Affiliation(s)
- Adam Lipowski
- Faculty of Physics, Adam Mickiewicz University, 61-614 Poznań, Poland
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Ben-Naim E, Machta B, Machta J. Power-law velocity distributions in granular gases. PHYSICAL REVIEW. E, STATISTICAL, NONLINEAR, AND SOFT MATTER PHYSICS 2005; 72:021302. [PMID: 16196551 DOI: 10.1103/physreve.72.021302] [Citation(s) in RCA: 3] [Impact Index Per Article: 0.2] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 04/07/2005] [Indexed: 05/04/2023]
Abstract
The kinetic theory of granular gases is studied for spatially homogeneous systems. At large velocities, the equation governing the velocity distribution becomes linear, and it admits stationary solutions with a power-law tail, f (v) approximately v(-sigma) . This behavior holds in arbitrary dimension for arbitrary collision rates including both hard spheres and Maxwell molecules. Numerical simulations show that driven steady states with the same power-law tail can be realized by injecting energy into the system at very high energies. In one dimension, we also obtain self-similar time-dependent solutions where the velocities collapse to zero. At small velocities there is a steady state and a power-law tail but at large velocities, the behavior is time dependent with a stretched exponential decay.
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Affiliation(s)
- E Ben-Naim
- Theoretical Division and Center for Nonlinear Studies, Los Alamos National Laboratory, Los Alamos, New Mexico 87545, USA.
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Coppex F, Droz M, Trizac E. Maxwell and very-hard-particle models for probabilistic ballistic annihilation: hydrodynamic description. PHYSICAL REVIEW. E, STATISTICAL, NONLINEAR, AND SOFT MATTER PHYSICS 2005; 72:021105. [PMID: 16196544 DOI: 10.1103/physreve.72.021105] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 03/04/2005] [Indexed: 05/04/2023]
Abstract
The hydrodynamic description of probabilistic ballistic annihilation, for which no conservation laws hold, is an intricate problem with hard spherelike dynamics for which no exact solution exists. We consequently focus on simplified approaches, the Maxwell and very-hard-particle (VHP) models, which allows us to compute analytically upper and lower bounds for several quantities. The purpose is to test the possibility of describing such a far from equilibrium dynamics with simplified kinetic models. The motivation is also in turn to assess the relevance of some singular features appearing within the original model and the approximations invoked to study it. The scaling exponents are first obtained from the (simplified) Boltzmann equation, and are confronted against direct Monte Carlo simulations. Then, the Chapman-Enskog method is used to obtain constitutive relations and transport coefficients. The corresponding Navier-Stokes equations for the hydrodynamic fields are derived for both Maxwell and VHP models. We finally perform a linear stability analysis around the homogeneous solution, which illustrates the importance of dissipation in the possible development of spatial inhomogeneities.
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Affiliation(s)
- François Coppex
- Department of Theoretical Physics, University of Genève, CH-1211 Genève 4, Switzerland
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Coppex F, Droz M, Trizac E. Hydrodynamics of probabilistic ballistic annihilation. PHYSICAL REVIEW. E, STATISTICAL, NONLINEAR, AND SOFT MATTER PHYSICS 2004; 70:061102. [PMID: 15697336 DOI: 10.1103/physreve.70.061102] [Citation(s) in RCA: 1] [Impact Index Per Article: 0.1] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 07/25/2004] [Indexed: 05/24/2023]
Abstract
We consider a dilute gas of hard spheres in dimension d> or =2 that upon collision either annihilate with probability p or undergo an elastic scattering with probability 1-p . For such a system neither mass, momentum, nor kinetic energy is a conserved quantity. We establish the hydrodynamic equations from the Boltzmann equation description. Within the Chapman-Enskog scheme, we determine the transport coefficients up to Navier-Stokes order, and give the closed set of equations for the hydrodynamic fields chosen for the above coarse-grained description (density, momentum, and kinetic temperature). Linear stability analysis is performed, and the conditions of stability for the local fields are discussed.
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Affiliation(s)
- François Coppex
- Department of Theoretical Physics, University of Genève, CH-1211 Genève 4, Switzerland
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Coppex F, Droz M, Trizac E. Probabilistic ballistic annihilation with continuous velocity distributions. PHYSICAL REVIEW. E, STATISTICAL, NONLINEAR, AND SOFT MATTER PHYSICS 2004; 69:011303. [PMID: 14995610 DOI: 10.1103/physreve.69.011303] [Citation(s) in RCA: 2] [Impact Index Per Article: 0.1] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 09/04/2003] [Indexed: 05/24/2023]
Abstract
We investigate the problem of ballistically controlled reactions where particles either annihilate upon collision with probability p, or undergo an elastic shock with probability 1-p. Restricting to homogeneous systems, we provide in the scaling regime that emerges in the long time limit, analytical expressions for the exponents describing the time decay of the density and the root-mean-square velocity, as continuous functions of the probability p and of a parameter related to the dissipation of energy. We work at the level of molecular chaos (nonlinear Boltzmann equation), and using a systematic Sonine polynomials expansion of the velocity distribution, we obtain in arbitrary dimension the first non-Gaussian correction and the corresponding expressions for the decay exponents. We implement Monte Carlo simulations in two dimensions, which are in excellent agreement with our analytical predictions. For p<1, numerical simulations lead to the conjecture that unlike for pure annihilation (p=1), the velocity distribution becomes universal, i.e., does not depend on the initial conditions.
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Affiliation(s)
- François Coppex
- Department of Physics, University of Genève, CH-1211 Genève 4, Switzerland
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Coppex F, Droz M, Piasecki J, Trizac E, Wittwer P. Some exact results for Boltzmann's annihilation dynamics. PHYSICAL REVIEW. E, STATISTICAL, NONLINEAR, AND SOFT MATTER PHYSICS 2003; 67:021103. [PMID: 12636649 DOI: 10.1103/physreve.67.021103] [Citation(s) in RCA: 1] [Impact Index Per Article: 0.0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 10/17/2002] [Indexed: 05/24/2023]
Abstract
The problem of ballistic annihilation for a spatially homogeneous system is revisited within Boltzmann's kinetic theory in two and three dimensions. Analytical results are derived for the time evolution of the particle density for some isotropic discrete bimodal velocity modulus distributions. According to the allowed values of the velocity modulus, different behaviors are obtained: power law decay with nonuniversal exponents depending continuously upon the ratio of the two velocities, or exponential decay. When one of the two velocities is equal to zero, the model describes the problem of ballistic annihilation in the presence of static traps. The analytical predictions are shown to be in agreement with the results of two-dimensional molecular dynamics simulations.
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Affiliation(s)
- François Coppex
- Department of Physics, University of Genève, CH-1211 Genève 4, Switzerland
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Piasecki J, Trizac E, Droz M. Dynamics of ballistic annihilation. PHYSICAL REVIEW. E, STATISTICAL, NONLINEAR, AND SOFT MATTER PHYSICS 2002; 66:066111. [PMID: 12513351 DOI: 10.1103/physreve.66.066111] [Citation(s) in RCA: 5] [Impact Index Per Article: 0.2] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 05/23/2002] [Indexed: 05/24/2023]
Abstract
The problem of ballistically controlled annihilation is revisited for general initial velocity distributions and an arbitrary dimension. An analytical derivation of the hierarchy equations obeyed by the reduced distributions is given, and a scaling analysis of the corresponding spatially homogeneous system is performed. This approach points to the relevance of the nonlinear Boltzmann equation for dimensions larger than 1 and provides expressions for the exponents describing the decay of the particle density n(t) proportional, variant t(-xi) and the root-mean-square velocity v proportional, variant t(-gamma) in terms of a parameter related to the dissipation of kinetic energy. The Boltzmann equation is then solved perturbatively within a systematic expansion in Sonine polynomials. Analytical expressions for the exponents xi and gamma are obtained in arbitrary dimension as a function of the parameter mu characterizing the small velocity behavior of the initial velocity distribution. Moreover, the leading non-Gaussian corrections to the scaled velocity distribution are computed. These expressions for the scaling exponents are in good agreement with the values reported in the literature for continuous velocity distributions in d=1. For the two-dimensional case, we implement Monte Carlo and molecular dynamics simulations that turn out to be in excellent agreement with the analytical predictions.
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Affiliation(s)
- Jarosław Piasecki
- Institute of Theoretical Physics, University of Warsaw, Hoza 69, Poland
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Trizac E. Kinetics and scaling in ballistic annihilation. PHYSICAL REVIEW LETTERS 2002; 88:160601. [PMID: 11955221 DOI: 10.1103/physrevlett.88.160601] [Citation(s) in RCA: 4] [Impact Index Per Article: 0.2] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 01/23/2002] [Indexed: 05/23/2023]
Abstract
We study the simplest irreversible ballistically controlled reaction, whereby particles having an initial continuous velocity distribution annihilate upon colliding. In the framework of the Boltzmann equation, expressions for the exponents characterizing the density and typical velocity decay are explicitly worked out in arbitrary dimension. These predictions are in excellent agreement with the complementary results of extensive Monte Carlo and molecular dynamics simulations. We finally discuss the definition of universality classes indexed by a continuous parameter for this far from equilibrium dynamics with no conservation laws.
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Affiliation(s)
- Emmanuel Trizac
- Laboratoire de Physique Théorique, UMR 8627 du CNRS, Bâtiment 210, Université de Paris-Sud, 91405 Orsay Cedex, France.
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