Mobility and diffusion of intruders in granular suspensions. Einstein relation

Abstract

The Enskog kinetic equation is considered to determine the mobility $\lambda$ and diffusion $D$ transport coefficients of intruders immersed in a granular gas of inelastic hard spheres (grains). Intruders and grains are in contact with a thermal bath, which plays the role of a background gas. As usual, the influence of the latter on the dynamics of intruders and grains is accounted for via a viscous drag force plus a stochastic Langevin-like term proportional to the background temperature $T_{\text{b}}$. The transport coefficients $\lambda$ and $D$ are determined by solving the kinetic equation by means of the Chapman--Enskog method adapted to dissipative dynamics. Both transport coefficients are given in terms of the solutions of two integral equations which are approximately solved up to the second order in a Sonine polynomial expansion. Theoretical results are compared against numerical solutions of the inelastic Enskog equation by means of the direct simulation Monte Carlo (DSMC) method. Good agreement between theory and simulations is in general found, specially in the case of the second Sonine approximation. The knowledge of the coefficients $\lambda$ and $D$ allow us to assess the departure of the Einstein relation $\epsilon =D/(T_{\text{b}}\lambda )$ from 1. As expected from previous results for driven granular gases, it is shown that the origin of the deviation of $\epsilon$ from 1 is only due to the non-Maxwellian behavior of reference state of intruders (measured by the cumulant $c_0$) when the bath temperature $T_{\text{b}}$ is replaced by the intruder temperature $T_0$ in the Einstein relation. Since the magnitude of $c_0$ is in general very small, deviations of the (modified) Einstein relation ${\epsilon }_0=D/(T_0\lambda )$ from 1 cannot be detected in computer simulations of dilute granular gases. This conclusion agrees well with previous computer simulation results.

arXiv comment21 pages, 7 figures

References in corpus (38)

• Fluctuation-Dissipation: Response Theory in Statistical Physics
• Stationary state volume fluctuations in a granular medium
• Enskog Theory for Polydisperse Granular Mixtures. I. Navier-Stokes order Transport
• Shear thickening in granular suspensions: inter-particle friction and dynamically correlated clusters
• Randomly Driven Granular Fluids: collisional statistics and short scale structure
• NMR Experiments on a Three-Dimensional Vibrofluidized Granular Medium
• Kinetic Temperatures for a Granular Mixture
• Approach to jamming in an air-fluidized granular bed
• Enskog Theory for Polydisperse Granular Mixtures II. Sonine Polynomial Approximation
• Measurements of Grain Motion in a Dense, Three-Dimensional Granular Fluid
• Statistical mechanics of fluidized granular media: short range velocity correlations
• Energy partition and segregation for an intruder in a vibrated granular system under gravity
• Fluctuation-Dissipation relations in Driven Granular Gases
• Energy non-equipartition in systems of inelastic, rough spheres
• Diffusion in a Granular Fluid - Simulation
• The rheology of dense, polydisperse granular fluids under shear
• Violation of the Einstein relation in Granular Fluids: the role of correlations
• Temperature probes in binary granular gases
• Green-Kubo Expressions for a Granular Gas
• Nonequilibrium Brownian motion beyond the effective temperature
• Transport properties for driven granular fluids in situations close to homogeneous steady states
• Navier-Stokes transport coefficients of $d$-dimensional granular binary mixtures at low density
• Mass transport of impurities in a moderately dense granular gas
• An exactly solvable model for driven dissipative systems
• Active Microrheology of Driven Granular Particles
• On the Einstein relation in a heated granular gas
• Revisiting ignited-quenched transition and the non-Newtonian rheology of a sheared dilute gas-solid suspension
• Long Range Correlation in Granular Shear Flow II: Theoretical Implications
• Enskog kinetic theory for multicomponent granular suspensions
• What is the temperature of a granular medium?
• Homogeneous states in driven granular mixtures: Enskog kinetic theory versus molecular dynamics simulations
• Fluctuation-dissipation relations in driven dissipative systems
• Kinetic theory of granular particles immersed in a molecular gas
• Time-dependent homogeneous states of binary granular suspensions
• Transport coefficients for granular suspensions at moderate densities
• Diffusion of intruders in granular suspensions: Enskog theory and random walk interpretation
• Tracer diffusion coefficients in a moderately dense granular suspension. Stability analysis and thermal diffusion segregation
• A note on the violation of the Einstein relation in a driven moderately dense granular gas

Cited by in corpus (1)

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