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Puthalath F, Biswas A, Prasad VV, Rajesh R. Lattice models for ballistic aggregation: Cluster-shape-dependent exponents. Phys Rev E 2023; 108:044127. [PMID: 37978638 DOI: 10.1103/physreve.108.044127] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 01/27/2023] [Accepted: 09/21/2023] [Indexed: 11/19/2023]
Abstract
We study ballistic aggregation on a two-dimensional square lattice, where particles move ballistically in between momentum and mass conserving coalescing collisions. Three models are studied based on the shapes of the aggregates: In the first the aggregates remain point particles, in the second they retain the fractal shape at the time of collision, and in the third they assume a spherical shape. The exponents describing the power-law temporal decay of number of particles and energy as well as dependence of velocity correlations on mass are determined using large-scale Monte Carlo simulations. It is shown that the exponents are universal only for the point-particle model. In the other two cases, the exponents are dependent on the initial number density and correlations vanish at high number densities. The fractal dimension for the second model is close to 1.49.
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Affiliation(s)
- Fahad Puthalath
- Institut für Materialphysik im Weltraum, Deutsches Zentrum für Luft- und Raumfahrt (DLR), 51170 Köln, Germany
- Institut für Theoretische Physik, Universität zu Köln, Zülpicher Strasse 77, 50937 Köln, Germany
| | - Apurba Biswas
- The Institute of Mathematical Sciences, C.I.T. Campus, Taramani, Chennai 600113, India
- Homi Bhabha National Institute, Training School Complex, Anushakti Nagar, Mumbai 400094, India
| | - V V Prasad
- Department of Physics, Cochin University of Science and Technology, Cochin 682022, India
| | - R Rajesh
- The Institute of Mathematical Sciences, C.I.T. Campus, Taramani, Chennai 600113, India
- Homi Bhabha National Institute, Training School Complex, Anushakti Nagar, Mumbai 400094, India
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2
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Chen Y, Zhang J. High-energy velocity tails in uniformly heated granular materials. Phys Rev E 2022; 106:L052903. [PMID: 36559423 DOI: 10.1103/physreve.106.l052903] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 07/04/2022] [Accepted: 11/17/2022] [Indexed: 06/17/2023]
Abstract
We experimentally investigate the velocity distributions of quasi-two-dimensional granular materials uniformly heated by an electromagnetic vibrator, where the translational velocity and the rotation of a single particle are Gaussian and independent. We observe the non-Gaussian distributions of particle velocity, with the density-independent high-energy tails characterized by an exponent of β=1.50±0.03 for volume fractions of 0.111≤ϕ≤0.832, covering a wide range of structures and dynamics. Surprisingly, our results are not only in excellent agreement with the prediction of the kinetic theories of granular gas but also hold for an extremely high-volume fraction of ϕ=0.832 where the granular material forms a crystalline solid and the kinetic theory of granular gas fails fantastically. Our experiment suggests that the density-independent high-energy velocity tails of β=1.50 are a characteristic of uniformly heated granular matter.
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Affiliation(s)
- Yangrui Chen
- School of Physics and Astronomy, Shanghai Jiao Tong University, Shanghai 200240, China
| | - Jie Zhang
- School of Physics and Astronomy, Shanghai Jiao Tong University, Shanghai 200240, China
- Institute of Natural Sciences, Shanghai Jiao Tong University, Shanghai 200240, China
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3
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Paul S, Das SK. Dimension dependence of clustering dynamics in models of ballistic aggregation and freely cooling granular gas. Phys Rev E 2018; 97:032902. [PMID: 29776153 DOI: 10.1103/physreve.97.032902] [Citation(s) in RCA: 7] [Impact Index Per Article: 1.2] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 09/06/2017] [Indexed: 11/07/2022]
Abstract
Via event-driven molecular dynamics simulations we study kinetics of clustering in assemblies of inelastic particles in various space dimensions. We consider two models, viz., the ballistic aggregation model (BAM) and the freely cooling granular gas model (GGM), for each of which we quantify the time dependence of kinetic energy and average mass of clusters (that form due to inelastic collisions). These quantities, for both the models, exhibit power-law behavior, at least in the long time limit. For the BAM, corresponding exponents exhibit strong dimension dependence and follow a hyperscaling relation. In addition, in the high packing fraction limit the behavior of these quantities become consistent with a scaling theory that predicts an inverse relation between energy and mass. On the other hand, in the case of the GGM we do not find any evidence for such a picture. In this case, even though the energy decay, irrespective of packing fraction, matches quantitatively with that for the high packing fraction picture of the BAM, it is inversely proportional to the growth of mass only in one dimension, and the growth appears to be rather insensitive to the choice of the dimension, unlike the BAM.
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Affiliation(s)
- Subhajit Paul
- Theoretical Sciences Unit, Jawaharlal Nehru Centre for Advanced Scientific Research, Jakkur P.O., Bangalore 560064, India
| | - Subir K Das
- Theoretical Sciences Unit, Jawaharlal Nehru Centre for Advanced Scientific Research, Jakkur P.O., Bangalore 560064, India
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Abstract
The kinetic energy of a force-free granular gas decays monotonously due to inelastic collisions of the particles. For a homogeneous granular gas of identical particles, the corresponding decay of granular temperature is quantified by Haff’s law. Here, we report that for a granular gas of aggregating particles, the granular temperature does not necessarily decay but may even increase. Surprisingly, the increase of temperature is accompanied by the continuous loss of total gas energy. This stunning effect arises from a subtle interplay between decaying kinetic energy and gradual reduction of the number of degrees of freedom associated with the particles’ dynamics. We derive a set of kinetic equations of Smoluchowski type for the concentrations of aggregates of different sizes and their energies. We find scaling solutions to these equations and a condition for the aggregation mechanism predicting growth of temperature. Numerical direct simulation Monte Carlo results confirm the theoretical predictions. Granular gases—dilute systems composed of dissipatively colliding particles—exhibit anomalous dynamics and numerous surprising phenomena. Here, Brilliantov et al. show that the aggregation mechanism can induce increase of the gas temperature despite the fact that the total kinetic energy decreases.
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Singh C, Mazza MG. Early-stage aggregation in three-dimensional charged granular gas. Phys Rev E 2018; 97:022904. [PMID: 29548210 DOI: 10.1103/physreve.97.022904] [Citation(s) in RCA: 3] [Impact Index Per Article: 0.5] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 10/27/2017] [Indexed: 06/08/2023]
Abstract
Neutral grains made of the same dielectric material can attain considerable charges due to collisions and generate long-range interactions. We perform molecular dynamic simulations in three dimensions for a dilute, freely cooling granular gas of viscoelastic particles that exchange charges during collisions. As compared to the case of clustering of viscoelastic particles solely due to dissipation, we find that the electrostatic interactions due to collisional charging alter the characteristic size, morphology, and growth rate of the clusters. The average cluster size grows with time as a power law, whose exponent is relatively larger in the charged gas than the neutral case. The growth of the average cluster size is found to be independent of the ratio of characteristic Coulomb to kinetic energy, or equivalently, of the typical Bjerrum length. However, this ratio alters the crossover time of the growth. Both simulations and mean-field calculations based on Smoluchowski's equation suggest that a suppression of particle diffusion due to the electrostatic interactions helps in the aggregation process.
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Affiliation(s)
- Chamkor Singh
- Max Planck Institute for Dynamics and Self-Organization (MPIDS), Am Faßberg 17, 37077, Göttingen, Germany
- Georg-August-Universität Göttingen, Friedrich-Hund-Platz 1, 37077 Göttingen, Germany
| | - Marco G Mazza
- Max Planck Institute for Dynamics and Self-Organization (MPIDS), Am Faßberg 17, 37077, Göttingen, Germany
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Paul S, Das SK. Ballistic aggregation in systems of inelastic particles: Cluster growth, structure, and aging. Phys Rev E 2018; 96:012105. [PMID: 29347235 DOI: 10.1103/physreve.96.012105] [Citation(s) in RCA: 12] [Impact Index Per Article: 2.0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 11/10/2016] [Indexed: 11/07/2022]
Abstract
We study far-from-equilibrium dynamics in models of freely cooling granular gas and ballistically aggregating compact clusters. For both the cases, from event-driven molecular dynamics simulations, we have presented detailed results on structure and dynamics in space dimensions d=1 and 2. Via appropriate analyses it has been confirmed that the ballistic aggregation mechanism applies in d=1 granular gases as well. Aging phenomena for this mechanism, in both the dimensions, have been studied via the two-time density autocorrelation function. This quantity is demonstrated to exhibit scaling property similar to that in the standard phase transition kinetics. The corresponding functional forms have been quantified and the outcomes have been discussed in connection with the structural properties. Our results on aging establish a more complete equivalence between the granular gas and the ballistic aggregation models in d=1.
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Affiliation(s)
- Subhajit Paul
- Theoretical Sciences Unit, Jawaharlal Nehru Centre for Advanced Scientific Research, Jakkur P.O., Bangalore 560064, India
| | - Subir K Das
- Theoretical Sciences Unit, Jawaharlal Nehru Centre for Advanced Scientific Research, Jakkur P.O., Bangalore 560064, India
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Giacco F, de Arcangelis L, Pica Ciamarra M, Lippiello E. Rattler-induced aging dynamics in jammed granular systems. SOFT MATTER 2017; 13:9132-9137. [PMID: 29184951 DOI: 10.1039/c7sm01976a] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 06/07/2023]
Abstract
Granular materials jam when developing a network of contact forces able to resist the applied stresses. Through numerical simulations of the dynamics of the jamming process, we show that the jamming transition does not occur when the kinetic energy vanishes. Rather, as the system jams, the kinetic energy becomes dominated by rattler particles, which scatter within their cages. The relaxation of the kinetic energy in the jammed configuration exhibits a double power-law decay, which we interpret in terms of the interplay between backbone and rattler particles.
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Affiliation(s)
- Ferdinando Giacco
- Department of Mathematics and Physics, University of Campania "Luigi Vanvitelli", Caserta, Italy.
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Joy JP, Pathak SN, Das D, Rajesh R. Shock propagation in locally driven granular systems. Phys Rev E 2017; 96:032908. [PMID: 29347038 DOI: 10.1103/physreve.96.032908] [Citation(s) in RCA: 2] [Impact Index Per Article: 0.3] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 06/29/2017] [Indexed: 06/07/2023]
Abstract
We study shock propagation in a system of initially stationary hard spheres that is driven by a continuous injection of particles at the origin. The disturbance created by the injection of energy spreads radially outward through collisions between particles. Using scaling arguments, we determine the exponent characterizing the power-law growth of this disturbance in all dimensions. The scaling functions describing the various physical quantities are determined using large-scale event-driven simulations in two and three dimensions for both elastic and inelastic systems. The results are shown to describe well the data from two different experiments on granular systems that are similarly driven.
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Affiliation(s)
- Jilmy P Joy
- The Institute of Mathematical Sciences, CIT Campus, Taramani, Chennai 600113, India
- Homi Bhabha National Institute, Training School Complex, Anushakti Nagar, Mumbai 400094, India
| | - Sudhir N Pathak
- Division of Physics and Applied Physics, School of Physical and Mathematical Sciences, Nanyang Technological University, Singapore 637371
| | - Dibyendu Das
- Department of Physics, Indian Institute of Technology, Bombay, Powai, Mumbai 400076, India
| | - R Rajesh
- The Institute of Mathematical Sciences, CIT Campus, Taramani, Chennai 600113, India
- Homi Bhabha National Institute, Training School Complex, Anushakti Nagar, Mumbai 400094, India
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Prasad VV, Das D, Sabhapandit S, Rajesh R. Velocity distribution of a driven inelastic one-component Maxwell gas. Phys Rev E 2017; 95:032909. [PMID: 28415251 DOI: 10.1103/physreve.95.032909] [Citation(s) in RCA: 2] [Impact Index Per Article: 0.3] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 01/13/2017] [Indexed: 06/07/2023]
Abstract
The nature of the velocity distribution of a driven granular gas, though well studied, is unknown as to whether it is universal or not, and, if universal, what it is. We determine the tails of the steady state velocity distribution of a driven inelastic Maxwell gas, which is a simple model of a granular gas where the rate of collision between particles is independent of the separation as well as the relative velocity. We show that the steady state velocity distribution is nonuniversal and depends strongly on the nature of driving. The asymptotic behavior of the velocity distribution is shown to be identical to that of a noninteracting model where the collisions between particles are ignored. For diffusive driving, where collisions with the wall are modeled by an additive noise, the tails of the velocity distribution is universal only if the noise distribution decays faster than exponential.
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Affiliation(s)
- V V Prasad
- Institute of Mathematical Sciences, C.I.T. Campus, Taramani, Chennai-600113, India
- Homi Bhabha National Institute, Training School Complex, Anushakti Nagar, Mumbai-400094, India
| | - Dibyendu Das
- Department of Physics, Indian Institute of Technology, Bombay, Powai, Mumbai-400076, India
| | | | - R Rajesh
- Institute of Mathematical Sciences, C.I.T. Campus, Taramani, Chennai-600113, India
- Homi Bhabha National Institute, Training School Complex, Anushakti Nagar, Mumbai-400094, India
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Prasad VV, Sabhapandit S, Dhar A. Driven inelastic Maxwell gases. PHYSICAL REVIEW. E, STATISTICAL, NONLINEAR, AND SOFT MATTER PHYSICS 2014; 90:062130. [PMID: 25615067 DOI: 10.1103/physreve.90.062130] [Citation(s) in RCA: 1] [Impact Index Per Article: 0.1] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 08/19/2014] [Indexed: 06/04/2023]
Abstract
We consider the inelastic Maxwell model, which consists of a collection of particles that are characterized by only their velocities and evolving through binary collisions and external driving. At any instant, a particle is equally likely to collide with any of the remaining particles. The system evolves in continuous time with mutual collisions and driving taken to be point processes with rates τ(c)(-1) and τ(w)(-1), respectively. The mutual collisions conserve momentum and are inelastic, with a coefficient of restitution r. The velocity change of a particle with velocity v, due to driving, is taken to be Δv=-(1+r(w))v+η, where r(w)∈[-1,1] and η is Gaussian white noise. For r(w)∈(0,1], this driving mechanism mimics the collision with a randomly moving wall, where r(w) is the coefficient of restitution. Another special limit of this driving is the so-called Ornstein-Uhlenbeck process given by dv/dt=-Γv+η. We show that while the equations for the n-particle velocity distribution functions (n=1,2,...) do not close, the joint evolution equations of the variance and the two-particle velocity correlation functions close. With the exact formula for the variance we find that, for r(w)≠-1, the system goes to a steady state. Also we obtain the exact tail of the velocity distribution in the steady state. On the other hand, for r(w)=-1, the system does not have a steady state. Similarly, the system goes to a steady state for the Ornstein-Uhlenbeck driving with Γ≠0, whereas for the purely diffusive driving (Γ=0), the system does not have a steady state.
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Affiliation(s)
- V V Prasad
- Raman Research Institute, Bangalore 560080, India
| | | | - Abhishek Dhar
- International Centre for Theoretical Sciences, TIFR, Bangalore 560012, India
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Mitrano PP, Garzó V, Hrenya CM. Instabilities in granular binary mixtures at moderate densities. PHYSICAL REVIEW. E, STATISTICAL, NONLINEAR, AND SOFT MATTER PHYSICS 2014; 89:020201. [PMID: 25353402 DOI: 10.1103/physreve.89.020201] [Citation(s) in RCA: 6] [Impact Index Per Article: 0.6] [Reference Citation Analysis] [Abstract] [MESH Headings] [Track Full Text] [Subscribe] [Scholar Register] [Received: 09/20/2013] [Indexed: 06/04/2023]
Abstract
A linear stability analysis of the Navier-Stokes (NS) granular hydrodynamic equations is performed to determine the critical length scale for the onset of vortices and clusters instabilities in granular dense binary mixtures. In contrast to previous attempts, our results (which are based on the solution to the inelastic Enskog equation to NS order) are not restricted to nearly elastic systems since they take into account the complete nonlinear dependence of the NS transport coefficients on the coefficients of restitution α(ij). The theoretical predictions for the critical length scales are compared to molecular dynamics (MD) simulations in flows of strong dissipation (α(ij) ≥ 0.7) and moderate solid volume fractions (ϕ ≤ 0.2). We find excellent agreement between MD and kinetic theory for the onset of velocity vortices, indicating the applicability of NS hydrodynamics to polydisperse flows even for strong inelasticity, finite density, and particle dissimilarity.
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Affiliation(s)
- Peter P Mitrano
- Department of Chemical and Biological Engineering, University of Colorado, Boulder, Colorado 80309, USA
| | - Vicente Garzó
- Departamento de Física, Universidad de Extremadura, E-06071 Badajoz, Spain
| | - Christine M Hrenya
- Department of Chemical and Biological Engineering, University of Colorado, Boulder, Colorado 80309, USA
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