1,082 research outputs found
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.
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
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
Blow-up of the hyperbolic Burgers equation
The memory effects on microscopic kinetic systems have been sometimes
modelled by means of the introduction of second order time derivatives in the
macroscopic hydrodynamic equations. One prototypical example is the hyperbolic
modification of the Burgers equation, that has been introduced to clarify the
interplay of hyperbolicity and nonlinear hydrodynamic evolution. Previous
studies suggested the finite time blow-up of this equation, and here we present
a rigorous proof of this fact
An Ecological Perspective of Intergenerational Trauma: Clinical Implications
In this paper, the authors present information about both intergenerational trauma and an ecological case conceptualization model to assist counselors as they develop treatment plans and determine appropriate interventions. Bronfenbrenner’s Ecological model is introduced as a way to help professional counselors in a variety of settings explore a more holistic understanding of presenting problems. The authors use a case illustration to highlight how to implement an ecological framework with a client with Colombian heritage to better understand and address intergenerational trauma as an important aspect of treatment planning. The paper includes clinical examples, clinical resources, and implications for professional counselors, so they can intentionally consider intergenerational trauma while working with a variety of clients
Quantitative lower bounds for the full Boltzmann equation, Part I: Periodic boundary conditions
We prove the appearance of an explicit lower bound on the solution to the
full Boltzmann equation in the torus for a broad family of collision kernels
including in particular long-range interaction models, under the assumption of
some uniform bounds on some hydrodynamic quantities. This lower bound is
independent of time and space. When the collision kernel satisfies Grad's
cutoff assumption, the lower bound is a global Maxwellian and its asymptotic
behavior in velocity is optimal, whereas for non-cutoff collision kernels the
lower bound we obtain decreases exponentially but faster than the Maxwellian.
Our results cover solutions constructed in a spatially homogeneous setting, as
well as small-time or close-to-equilibrium solutions to the full Boltzmann
equation in the torus. The constants are explicit and depend on the a priori
bounds on the solution.Comment: 37 page
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
Relativistic Dissipative Hydrodynamics: A Minimal Causal Theory
We present a new formalism for the theory of relativistic dissipative
hydrodynamics. Here, we look for the minimal structure of such a theory which
satisfies the covariance and causality by introducing the memory effect in
irreversible currents. Our theory has a much simpler structure and thus has
several advantages for practical purposes compared to the Israel-Stewart theory
(IS). It can readily be applied to the full three-dimensional hydrodynamical
calculations. We apply our formalism to the Bjorken model and the results are
shown to be analogous to the IS.Comment: 25 pages, 2 figures, Phys. Rev. C in pres
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