Dynamics of annihilation. I. Linearized Boltzmann equation and hydrodynamics

Soria, M. I. García de; Maynar, P.; Schehr, Grégory; Barrat, Alain; Trizac, Emmanuel

doi:10.1103/physreve.77.051127

Cited by 11 publications

(21 citation statements)

References 24 publications

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“…In general, the shape of the initial distribution of velocities is not altered, and it only "shrinks" with the thermal velocity. A similar behaviour was found for elastic Maxwell molecules with annihilation starting from the Boltzmann equation [52].…”

Section: The Homogeneous Cooling Statesupporting

confidence: 76%

Lattice Models for Granular-Like Velocity Fields: Hydrodynamic Description

et al. 2016

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A recently introduced model describing—on a 1d lattice—the velocity field of a granular fluid is discussed in detail. The dynamics of the velocity field occurs through next-neighbours inelastic collisions which conserve momentum but dissipate energy. The dynamics is described through the corresponding Master Equation for the time evolution of the probability distribution. In the continuum limit, equations for the average velocity and temperature fields with fluctuating currents are derived, which are analogous to hydrodynamic equations of granular fluids when restricted to the shear modes. Therefore, the homogeneous cooling state, with its linear instability, and other relevant regimes such as the uniform shear flow and the Couette flow states are described. The evolution in time and space of the single particle probability distribution, in all those regimes, is also discussed, showing that the local equilibrium is not valid in general. The noise for the momentum and energy currents, which are correlated, are white and Gaussian. The same is true for the noise of the energy sink, which is usually negligible

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Section: The Homogeneous Cooling Statesupporting

confidence: 76%

Lattice Models for Granular-Like Velocity Fields: Hydrodynamic Description

et al. 2016

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“…On the other hand, much less is known in the case of driven granular fluids. To the best of our knowledge, the only derivation of Green-Kubo formula for Navier-Stokes transport coefficients of a granular dilute gas heated by the stochastic thermostat has been recently carried out by García de Soria et al [19]. Starting from the Boltzmann equation, these authors obtain explicit expressions for the transport coefficients as a function of the inelasticity and the spatial dimension.…”

Section: Introductionmentioning

confidence: 99%

Transport properties for driven granular fluids in situations close to homogeneous steady states

2013

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The transport coefficients of a granular fluid driven by a stochastic bath with friction are obtained by solving the inelastic Enskog kinetic equation from the Chapman-Enskog method. The heat and momentum fluxes as well as the cooling rate are determined to first order in the deviations of the hydrodynamic field gradients from their values in the homogeneous steady state. Since the collisional cooling cannot be compensated locally for the heat produced by the external driving force, the reference distribution f (0) (zeroth-order approximation) depends on time through its dependence on temperature. This fact gives rise to conceptual and practical difficulties not present in the undriven case. On the other hand, to simplify the analysis and given that we are interested in computing transport in the first order of deviations from the reference state, the steady-state conditions are considered to get explicit forms for the transport coefficients and the cooling rate. A comparison with recent Langevin dynamics simulations for driven granular fluids shows an excellent agreement for the kinematic viscosity although some discrepancies are observed for the longitudinal viscosity and the thermal diffusivity at large densities. Finally, a linear stability analysis of the hydrodynamic equations with respect to the homogeneous steady state is performed. As expected, no instabilities are found thanks to the presence of the external bath.

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“…For the annihilation model, the hydrodynamic equations have been derived using the Chapmann-Enskog method [11] by the usual assumption of the existence of a "normal solution", whose space and time dependence occurs only through the hydrodynamic fields [6]. Recently, the hydrodynamic equations linearized around the homogeneous decay state have been derived relaxing such an assumption [12]. Nevertheless, it must be assumed that there is scale separation, i.e that the spectrum of the linearized Boltzmann collision operator is such that the eigenvalues associated to the hydrodynamic excitations are separated from the faster "kinetic eigenvalues".…”

Section: Introductionmentioning

confidence: 99%

“…Nevertheless, it must be assumed that there is scale separation, i.e that the spectrum of the linearized Boltzmann collision operator is such that the eigenvalues associated to the hydrodynamic excitations are separated from the faster "kinetic eigenvalues". Although this property is valid for elastic collisions [13], it has not been proven for the probabilistic ballistic annihilation model in general, but only for Maxwell molecules [14] and for p smaller than a given threshold [12].…”

Section: Introductionmentioning

confidence: 99%

Brownian motion under annihilation dynamics

2008

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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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Dynamics of annihilation. I. Linearized Boltzmann equation and hydrodynamics

Cited by 11 publications

References 24 publications

Lattice Models for Granular-Like Velocity Fields: Hydrodynamic Description

Lattice Models for Granular-Like Velocity Fields: Hydrodynamic Description

Transport properties for driven granular fluids in situations close to homogeneous steady states

Brownian motion under annihilation dynamics

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