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Bobylev, A. V.; Cercignani, Carlo

doi:10.1023/a:1014037804043

Cited by 59 publications

(11 citation statements)

References 15 publications

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“…In contrast, the total energy may be either finite or infinite because both σ > d + 2 and σ < d+2 are possible. The stationary states studied here appear to be fundamentally different than the infinite energy solutions of the elastic Boltzmann equation because they require dissipation and because they always involve infinite dissipation [45].…”

Section: Stationary Statesmentioning

confidence: 91%

Power-law velocity distributions in granular gases

Ben‐Naim

Machta

2005

Phys. Rev. E

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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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Section: Stationary Statesmentioning

confidence: 91%

Power-law velocity distributions in granular gases

Ben‐Naim

Machta

2005

Phys. Rev. E

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“…As it turned out, this conjecture was not supported by numerical and analytic results obtained for the physically most relevant initial distributions with a finite second moment in the limit as t → +∞ [20]. However, Bobylev and Cercignani have recently shown that the conjecture holds in systems of elastic Maxwell molecules in the limit t → −∞ for the so-called eternal solutions f (v, t) [31,32], which are characterized by a divergent second moment.…”

Section: Introductionmentioning

confidence: 94%

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Ernst

Brito

2002

Journal of Statistical Physics

107

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This paper deals with solutions of the nonlinear Boltzmann equation for spatially uniform freely cooling inelastic Maxwell models for large times and for large velocities, and the nonuniform convergence to these limits. We demonstrate how the velocity distribution approaches in the scaling limit to a similarity solution with a power law tail for general classes of initial conditions and derive a transcendental equation from which the exponents in the tails can be calculated. Moreover on the basis of the available analytic and numerical results for inelastic hard spheres and inelastic Maxwell models we formulate a conjecture on the approach of the velocity distribution function to a scaling form.

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“…For elastic collisions, the velocity distributions are Maxwellian, so the no-dissipation limit is singular. Interestingly, the energy may be either finite or infinite, depending on whether is larger or smaller than d 2 [26]. In either case, the integral underlying the dissipation rate is divergent, a general characteristic of the stationary distributions.…”

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confidence: 99%

Stationary States and Energy Cascades in Inelastic Gases

Ben‐Naim

Machta

2005

Phys. Rev. Lett.

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We find a general class of nontrivial stationary states in inelastic gases where, due to dissipation, energy is transferred from large velocity scales to small velocity scales. These steady states exist for arbitrary collision rules and arbitrary dimension. Their signature is a stationary velocity distribution f(v) with an algebraic high-energy tail, f(v) approximately v(-sigma). The exponent sigma is obtained analytically and it varies continuously with the spatial dimension, the homogeneity index characterizing the collision rate, and the restitution coefficient. We observe these stationary states in numerical simulations in which energy is injected into the system by infrequently boosting particles to high velocities. We propose that these states may be realized experimentally in driven granular systems.

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Cited by 59 publications

References 15 publications

Power-law velocity distributions in granular gases

Power-law velocity distributions in granular gases

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Stationary States and Energy Cascades in Inelastic Gases

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