806 research outputs found
Critical Behavior of a Heavy Particle in a Granular Fluid
Behavior analogous to a second order phase transition is observed for the
homogeneous cooling state of a heavy impurity particle in a granular fluid. The
order parameter is the ratio of impurity mean square velocity to that
of the fluid, with a conjugate field proportional to the mass ratio. A
parameter , measuring the fluid cooling rate relative to the
impurity--fluid collision rate, is the analogue of the inverse temperature. For
the fluid is ``normal'' with at , as in the case of a
system with elastic collisions. For an ``ordered'' state with occurs at , representing an extreme breakdown of equipartition.
Critical slowing and qualitative changes in the velocity distribution function
for the impurity particle near the transition are notedComment: 4 pages (4 figures included
Linear Response for Granular Fluids
The linear response of an isolated, homogeneous granular fluid to small
spatial perturbations is studied by methods of non-equilibrium statistical
mechanics. The long wavelength linear hydrodynamic equations are obtained, with
formally exact expressions for the susceptibilities and transport coefficients.
The latter are given in equivalent Einstein-Helfand and Green-Kubo forms. The
context of these results and their contrast with corresponding results for
normal fluids are discussed.Comment: Submitted to PR
Volume fluctuations and compressibility of a vibrated granular gas
The volume fluctuations in the steady state reached by a vibrated granular
gas of hard particles confined by a movable piston on the top are investigated
by means of event driven simulations. Also, a compressibility factor, measuring
the response in volume of the system to a change in the mass of the piston, is
introduced and measured. From the second moment of the volume fluctuations and
the compressibility factor, an effective temperature is defined, by using the
same relation as obeyed by equilibrium molecular systems. The interpretation of
this effective temperature and its relationship with the granular temperature
of the gas, and also with the velocity fluctuations of the movable piston, is
discussed. It is found that the ratio of the temperature based on the volume
fluctuations to the temperature based on the piston kinetic energy, obeys
simple dependencies on the inelasticity and on the piston-particle mass ratio
Gaussian Kinetic Model for Granular Gases
A kinetic model for the Boltzmann equation is proposed and explored as a
practical means to investigate the properties of a dilute granular gas. It is
shown that all spatially homogeneous initial distributions approach a universal
"homogeneous cooling solution" after a few collisions. The homogeneous cooling
solution (HCS) is studied in some detail and the exact solution is compared
with known results for the hard sphere Boltzmann equation. It is shown that all
qualitative features of the HCS, including the nature of over population at
large velocities, are reproduced semi-quantitatively by the kinetic model. It
is also shown that all the transport coefficients are in excellent agreement
with those from the Boltzmann equation. Also, the model is specialized to one
having a velocity independent collision frequency and the resulting HCS and
transport coefficients are compared to known results for the Maxwell Model. The
potential of the model for the study of more complex spatially inhomogeneous
states is discussed.Comment: to be submitted to Phys. Rev.
Spatial Correlations in Compressible Granular Flows
For a freely evolving granular fluid, the buildup of spatial correlations in
density and flow field is described using fluctuating hydrodynamics. The theory
for incompressible flows is extended to the general, compressible case,
including longitudinal velocity and density fluctuations, and yields
qualitatively different results for long range correlations. The structure
factor of density fluctuations shows a maximum at finite wavenumber, shifting
in time to smaller wavenumbers and corresponding to a growing correlation
length. It agrees well with two-dimensional molecular dynamics simulations.Comment: 12 pages, Latex, 3 figure
Hydrodynamic modes, Green-Kubo relations, and velocity correlations in dilute granular gases
It is shown that the hydrodynamic modes of a dilute granular gas of inelastic
hard spheres can be identified, and calculated in the long wavelength limit.
Assuming they dominate at long times, formal expressions for the Navier-Stokes
transport coefficients are derived. They can be expressed in a form that
generalizes the Green-Kubo relations for molecular systems, and it is shown
that they can also be evaluated by means of -particle simulation methods.
The form of the hydrodynamic modes to zeroth order in the gradients is used to
detect the presence of inherent velocity correlations in the homogeneous
cooling state, even in the low density limit. They manifest themselves in the
fluctuations of the total energy of the system. The theoretical predictions are
shown to be in agreement with molecular dynamics simulations. Relevant related
questions deserving further attention are pointed out
Diffusion in a Granular Fluid - Theory
Many important properties of granular fluids can be represented by a system
of hard spheres with inelastic collisions. Traditional methods of
nonequilibrium statistical mechanics are effective for analysis and description
of the inelastic case as well. This is illustrated here for diffusion of an
impurity particle in a fluid undergoing homogeneous cooling. An appropriate
scaling of the Liouville equation is described such that the homogeneous
cooling ensemble and associated time correlation functions map to those of a
stationary state. In this form the familiar methods of linear response can be
applied, leading to Green - Kubo and Einstein representations of diffusion in
terms of the velocity and mean square displacement correlation functions. These
correlation functions are evaluated approximately using a cumulant expansion
and from kinetic theory, providing the diffusion coefficient as a function of
the density and the restitution coefficients. Comparisons with results from
molecular dynamics simulation are given in the following companion paper
Effects of Velocity Correlation on Early Stage of Free Cooling Process of Inelastic Hard Sphere System
The free cooling process in the inelastic hard sphere system is studied by
analysing the data from large scale molecular dynamics simulations on a three
dimensional system. The initial energy decay, the velocity distribution
function, and the velocity correlation functions are calculated to be compared
with theoretical predictions. The energy decay rate in the homogeneous cooling
state is slightly but distinctively smaller than that expected from the
independent collision assumption. The form of the one particle velocity
distribution is found not to be stationary. These contradict to the predictions
of the kinetic theory based on the Enskog-Boltzmann equation and suggest that
the velocity correlation is already important in the early stage of homogeneous
cooling state. The energy decay rate is analysed in terms of the velocity
correlation.Comment: 9 pages (figures included). To be published in J. Phys. Soc. Jpn.
Vol. 73 No. 1 (2004) Added two references and removed one. Changed the name
of T_{L}. Added unit constants in Sec. 5 and
Kinetic Theory of Response Functions for the Hard Sphere Granular Fluid
The response functions for small spatial perturbations of a homogeneous
granular fluid have been described recently. In appropriate dimensionless
variables, they have the form of stationary state time correlation functions.
Here, these functions are expressed in terms of reduced single particle
functions that are expected to obey a linear kinetic equation. The functional
assumption required for such a kinetic equation, and a Markov approximation for
its implementation are discussed. If, in addition, static velocity correlations
are neglected, a granular fluid version of the linearized Enskog kinetic theory
is obtained. The derivation makes no a priori limitation on the density, space
and time scale, nor degree of inelasticity. As an illustration, recently
derived Helfand and Green-Kubo expressions for the Navier-Stokes order
transport coefficients are evaluated with this kinetic theory. The results are
in agreement with those obtained from the Chapman-Enskog solution to the
nonlinear Enskog kinetic equation.Comment: Submitted to J. Stat. Mec
Patterns and Long Range Correlations in Idealized Granular Flows
An initially homogeneous freely evolving fluid of inelastic hard spheres
develops inhomogeneities in the flow field (vortices) and in the density field
(clusters), driven by unstable fluctuations. Their spatial correlations, as
measured in molecular dynamics simulations, exhibit long range correlations;
the mean vortex diameter grows as the square root of time; there occur
transitions to macroscopic shearing states, etc.
The Cahn--Hilliard theory of spinodal decomposition offers a qualitative
understanding and quantitative estimates of the observed phenomena. When
intrinsic length scales are of the order of the system size, effects of
physical boundaries and periodic boundaries (finite size effects in
simulations) are important.Comment: 13 pages with 7 postscript figures, LaTeX (uses psfig). Submitted to
International Journal of Modern Physics
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