1,392 research outputs found
Discretization of the velocity space in solution of the Boltzmann equation
We point out an equivalence between the discrete velocity method of solving
the Boltzmann equation, of which the lattice Boltzmann equation method is a
special example, and the approximations to the Boltzmann equation by a Hermite
polynomial expansion. Discretizing the Boltzmann equation with a BGK collision
term at the velocities that correspond to the nodes of a Hermite quadrature is
shown to be equivalent to truncating the Hermite expansion of the distribution
function to the corresponding order. The truncated part of the distribution has
no contribution to the moments of low orders and is negligible at small Mach
numbers. Higher order approximations to the Boltzmann equation can be achieved
by using more velocities in the quadrature
Microscopic Derivation of Causal Diffusion Equation using Projection Operator Method
We derive a coarse-grained equation of motion of a number density by applying
the projection operator method to a non-relativistic model. The derived
equation is an integrodifferential equation and contains the memory effect. The
equation is consistent with causality and the sum rule associated with the
number conservation in the low momentum limit, in contrast to usual acausal
diffusion equations given by using the Fick's law. After employing the Markov
approximation, we find that the equation has the similar form to the causal
diffusion equation. Our result suggests that current-current correlations are
not necessarily adequate as the definition of diffusion constants.Comment: 10 pages, 1 figure, Final version published in Phys. Rev.
A causal statistical family of dissipative divergence type fluids
In this paper we investigate some properties, including causality, of a
particular class of relativistic dissipative fluid theories of divergence type.
This set is defined as those theories coming from a statistical description of
matter, in the sense that the three tensor fields appearing in the theory can
be expressed as the three first momenta of a suitable distribution function. In
this set of theories the causality condition for the resulting system of
hyperbolic partial differential equations is very simple and allow to identify
a subclass of manifestly causal theories, which are so for all states outside
equilibrium for which the theory preserves this statistical interpretation
condition. This subclass includes the usual equilibrium distributions, namely
Boltzmann, Bose or Fermi distributions, according to the statistics used,
suitably generalized outside equilibrium. Therefore this gives a simple proof
that they are causal in a neighborhood of equilibrium. We also find a bigger
set of dissipative divergence type theories which are only pseudo-statistical,
in the sense that the third rank tensor of the fluid theory has the symmetry
and trace properties of a third momentum of an statistical distribution, but
the energy-momentum tensor, while having the form of a second momentum
distribution, it is so for a different distribution function. This set also
contains a subclass (including the one already mentioned) of manifestly causal
theories.Comment: LaTex, documentstyle{article
Incorporating Forcing Terms in Cascaded Lattice-Boltzmann Approach by Method of Central Moments
Cascaded lattice-Boltzmann method (Cascaded-LBM) employs a new class of
collision operators aiming to improve numerical stability. It achieves this and
distinguishes from other collision operators, such as in the standard single or
multiple relaxation time approaches, by performing relaxation process due to
collisions in terms of moments shifted by the local hydrodynamic fluid
velocity, i.e. central moments, in an ascending order-by-order at different
relaxation rates. In this paper, we propose and derive source terms in the
Cascaded-LBM to represent the effect of external or internal forces on the
dynamics of fluid motion. This is essentially achieved by matching the
continuous form of the central moments of the source or forcing terms with its
discrete version. Different forms of continuous central moments of sources,
including one that is obtained from a local Maxwellian, are considered in this
regard. As a result, the forcing terms obtained in this new formulation are
Galilean invariant by construction. The method of central moments along with
the associated orthogonal properties of the moment basis completely determines
the expressions for the source terms as a function of the force and macroscopic
velocity fields. In contrast to the existing forcing schemes, it is found that
they involve higher order terms in velocity space. It is shown that the
proposed approach implies "generalization" of both local equilibrium and source
terms in the usual lattice frame of reference, which depend on the ratio of the
relaxation times of moments of different orders. An analysis by means of the
Chapman-Enskog multiscale expansion shows that the Cascaded-LBM with forcing
terms is consistent with the Navier-Stokes equations. Computational experiments
with canonical problems involving different types of forces demonstrate its
accuracy.Comment: 55 pages, 4 figure
Strong Shock Waves and Nonequilibrium Response in a One-dimensional Gas: a Boltzmann Equation Approach
We investigate the nonequilibrium behavior of a one-dimensional binary fluid
on the basis of Boltzmann equation, using an infinitely strong shock wave as
probe. Density, velocity and temperature profiles are obtained as a function of
the mixture mass ratio \mu. We show that temperature overshoots near the shock
layer, and that heavy particles are denser, slower and cooler than light
particles in the strong nonequilibrium region around the shock. The shock width
w(\mu), which characterizes the size of this region, decreases as w(\mu) ~
\mu^{1/3} for \mu-->0. In this limit, two very different length scales control
the fluid structure, with heavy particles equilibrating much faster than light
ones. Hydrodynamic fields relax exponentially toward equilibrium, \phi(x) ~
exp[-x/\lambda]. The scale separation is also apparent here, with two typical
scales, \lambda_1 and \lambda_2, such that \lambda_1 ~ \mu^{1/2} as \mu-->0$,
while \lambda_2, which is the slow scale controlling the fluid's asymptotic
relaxation, increases to a constant value in this limit. These results are
discussed at the light of recent numerical studies on the nonequilibrium
behavior of similar 1d binary fluids.Comment: 9 pages, 8 figs, published versio
Explicit coercivity estimates for the linearized Boltzmann and Landau operators
We prove explicit coercivity estimates for the linearized Boltzmann and
Landau operators, for a general class of interactions including any
inverse-power law interactions, and hard spheres. The functional spaces of
these coercivity estimates depend on the collision kernel of these operators.
They cover the spectral gap estimates for the linearized Boltzmann operator
with Maxwell molecules, improve these estimates for hard potentials, and are
the first explicit coercivity estimates for soft potentials (including in
particular the case of Coulombian interactions). We also prove a regularity
property for the linearized Boltzmann operator with non locally integrable
collision kernels, and we deduce from it a new proof of the compactness of its
resolvent for hard potentials without angular cutoff.Comment: 32 page
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