Maxwell Models for Granular Gases

Maxwell Models for Granular Gases

Description

A pseudo Maxwell model of a gas is described by a Boltzmann equation for the evolution of the single particle velocity distribution, for a specific form of the collisional kernel. It can be considered as the evolution of particles interacting with a power law potential (with power: -2(d-1), where d is the spatial dimension), or just as an approximation of the real hard spheres Boltzmann equation. The equation (in the 1d case) reads:

img38.png

where

img39.png

and r is the restitution coefficient of the inelastic collsion:

img-collision.gif

Our work

We found a scaling solution of the scalar inelastic Maxwell Model, which displays large (algebraic) tail. This solution can not be obtained through the study of the evolution of the moments, since most of them diverge, neither through a Taylor expansion of the characteristic function. Its expression is:

$P(v,t)= v_0(t)^{-1} f\left(\frac {v}{v_0(t)} \right)$

where the scaling function is:

$f(x)=\frac 1{2\pi} \frac 1{\left(1+x^2\right)^2}$

Our numerical simulations provide evidences that this scaling solution is reached asymptotically from a broad class of initial distributions, as shown in this figure:

img35.png

Interestingly the solution is independent of the restitution coefficient r.

You can also see the animated evolution of the rescaled velocity distribution with different starting condition:

References to our papers

  • The scaling solution of the 1d model is reported in:

A.Baldassarri, U.M.Bettolo Marconi, A.Puglisi Influence of correlations on the velocity statistics of scalar granular gases, published on Europhys. Lett., 58, pp.14-20 (2002) ( pre-print available).

where an original lattice version of inelastic Maxwell model is shown to agree with a real gas of inelastic rods.

  • A two dimensional version of the same lattice model has been studied in:

A. Baldassarri, U. Marini Bettolo Marconi, A. Puglisi Cooling of a lattice granular fluid as an ordering process, accepted on Phys.Rev.E (pre-print available)

  • A brief review of these works has appeared

A. Baldassarri, U. Marini Bettolo Marconi, A. Puglisi, Kinetic Models of Inelastic Gases, Mathematical Models & Methods in Applied Science (M3AS ) Vol. 12 N.7 (2002) (pre-print available).

A. Baldassarri, U. Marini Bettolo Marconi, A. Puglisi, Velocity fluctuations in cooling granular gases, cond-mat/0302418 to appear for the Springer Verlag Ed. (2003) (pre-print)

Other References

  • A complete review on Maxwell models for elastic gases is:

M.H. Ernst Nonlinear model-Boltzmann Equations and Exact solutions Phys.Rep. 78 (1981) 2-171.

  • A simple stochastic model by Ulam has a master equation equivalent to that of a Maxwell Model:

S.Ulam, On the operations of pair production, transmutations and generalized random walk Adv.Appl.Math. 1, 7 (1980).

  • Recently an inelastic Maxwell model has been addressed, and the moments of the velocity distribution has been computed:

E. Ben-Naim, P. L. Krapivsky, Multiscaling in inelastic collisions, Phys. Rev. E 61 R5 (2000).

  • Several elastic limits have been discussed in:

A. V. Bobylev, J. A. Carrillo, I. M. Gamba, J. Stat. Phys. 98, 743 (2000).

  • For higher dimensions, the algebraic tails persist and the exponent have been evaluated in the following papers:

P. L. Krapivsky, E. Ben-Naim, Nontrivial Velocity Distributions in Inelastic Gases J. Phys. A 35, L147 (2002) (cond-mat/0111044)

Matthieu H. Ernst, Ricardo Brito, Velocity Tails for Inelastic Maxwell Models (cond-mat/0111093)

M. H. Ernst, R. Brito, Scaling Solutions of Inelastic Boltzmann Equations with Over-populated High Granular.Energy Tails (cond-mat/0112417)

E. Ben-Naim, P.L. Krapivsky, Scaling, Granular.Multiscaling, and Nontrivial Exponents in Inelastic Collision Processes (cond-mat/0202332)

  • Impurities and mixtures:

E. Ben-Naim, P.L. Krapivsky, Impurity in a Granular Fluid ( cond-mat/0203099)

U. Marini Bettolo Marconi and A. Puglisi, A mean field model of free cooling inelastic mixtures ( cond-mat/0112336)

U. Marini Bettolo Marconi and A. Puglisi, Steady state properties of a mean field model of driven inelastic mixtures ( cond-mat/0202267)

  • Driven IMM

M. H. Ernst, R. Brito, Driven inelastic Maxwell models with high energy tails Phys. Rev. E 65, 040301(R) (2002)

  • Transport coefficients have been computed for IMM:

Andres Santos, Transport coefficients of d-dimensional inelastic Maxwell models ( cond-mat/0204071)