131 research outputs found
Transport coefficients for an inelastic gas around uniform shear flow: Linear stability analysis
The inelastic Boltzmann equation for a granular gas is applied to spatially
inhomogeneous states close to the uniform shear flow. A normal solution is
obtained via a Chapman-Enskog-like expansion around a local shear flow
distribution. The heat and momentum fluxes are determined to first order in the
deviations of the hydrodynamic field gradients from their values in the
reference state. The corresponding transport coefficients are determined from a
set of coupled linear integral equations which are approximately solved by
using a kinetic model of the Boltzmann equation. The main new ingredient in
this expansion is that the reference state (zeroth-order
approximation) retains all the hydrodynamic orders in the shear rate. In
addition, since the collisional cooling cannot be compensated locally for
viscous heating, the distribution depends on time through its
dependence on temperature. This means that in general, for a given degree of
inelasticity, the complete nonlinear dependence of the transport coefficients
on the shear rate requires the analysis of the {\em unsteady} hydrodynamic
behavior. To simplify the analysis, the steady state conditions have been
considered here in order to perform a linear stability analysis of the
hydrodynamic equations with respect to the uniform shear flow state. Conditions
for instabilities at long wavelengths are identified and discussed.Comment: 7 figures; previous stability analysis modifie
Fractional Fokker-Planck Equation for Fractal Media
We consider the fractional generalizations of equation that defines the
medium mass. We prove that the fractional integrals can be used to describe the
media with noninteger mass dimensions. Using fractional integrals, we derive
the fractional generalization of the Chapman-Kolmogorov equation (Smolukhovski
equation). In this paper fractional Fokker-Planck equation for fractal media is
derived from the fractional Chapman-Kolmogorov equation. Using the Fourier
transform, we get the Fokker-Planck-Zaslavsky equations that have fractional
coordinate derivatives. The Fokker-Planck equation for the fractal media is an
equation with fractional derivatives in the dual space.Comment: 17 page
Dynamics of a metastable state nonlinearly coupled to a heat bath driven by an external noise
Based on a system-reservoir model, where the system is nonlinearly coupled to
a heat bath and the heat bath is modulated by an external stationary Gaussian
noise, we derive the generalized Langevin equation with space dependent
friction and multiplicative noise and construct the corresponding Fokker-Planck
equation, valid for short correlation time, with space dependent diffusion
coefficient to study the escape rate from a metastable state in the moderate to
large damping regime. By considering the dynamics in a model cubic potential we
analyze the result numerically which are in good agreement with the theoretical
prediction. It has been shown numerically that the enhancement of rate is
possible by properly tuning the correlation time of the external noise.Comment: 13 pages, 5 figures, Revtex4. To appear in Physical Review
Kramers-Kronig Relations For The Dielectric Function And The Static Conductivity Of Coulomb Systems
The mutual influence of singularities of the dielectric permittivity e(q,w)
in a Coulomb system in two limiting cases w tends to zero, q tends to zero, and
opposite q tends to zero, w tends to zero is established. It is shown that the
dielectric permittivity e(q,w) satisfies the Kramers-Kronig relations, which
possesses the singularity due to a finite value of the static conductivity.
This singularity is associated with the long "tails" of the time correlation
functions.Comment: 9 pages, 0 figure
Density expansion for transport coefficients: Long-wavelength versus Fermi surface nonanalyticities
The expansion of the conductivity in 2-d quantum Lorentz models in terms of
the scatterer density n is considered. We show that nonanalyticities in the
density expansion due to scattering processes with small and large momentum
transfers, respectively, have different functional forms. Some of the latter
are not logarithmic, but rather of power-law nature, in sharp contrast to the
3-d case. In a 2-d model with point-like scatterers we find that the leading
nonanalytic correction to the Boltzmann conductivity, apart from the frequency
dependent weak-localization term, is of order n^{3/2}.Comment: 4 pp., REVTeX, epsf, 3 eps figs, final version as publishe
Exact limiting relation between the structure factors in neutron and x-ray scattering
The ratio of the static matter structure factor measured in experiments on
coherent X-ray scattering to the static structure factor measured in
experiments on neutron scattering is considered. It is shown theoretically that
this ratio in the long-wavelength limit is equal to the nucleus charge at
arbitrary thermodynamic parameters of a pure substance (the system of nuclei
and electrons, where interaction between particles is pure Coulomb) in a
disordered equilibrium state. This result is the exact relation of the quantum
statistical mechanics. The experimental verification of this relation can be
done in the long wavelength X-ray and neutron experiments.Comment: 7 pages, no figure
Multiplicative cross-correlated noise induced escape rate from a metastable state
We present an analytical framework to study the escape rate from a metastable
state under the influence of two external multiplicative cross-correlated noise
processes. Starting from a phenomenological stationary Langevin description
with multiplicative noise processes, we have investigated the Kramers' theory
for activated rate processes in a nonequilibrium open system (one-dimensional
in nature) driven by two external cross-correlated noise processes which are
Gaussian, stationary and delta correlated. Based on the Fokker-Planck
description in phase space, we then derive the escape rate from a metastable
state in the moderate to large friction limit to study the effect of degree of
correlation on the same. By employing numerical simulation in the presence of
external cross-correlated additive and multiplicative noises we check the
validity of our analytical formalism for constant dissipation, which shows a
satisfactory agreement between both the approaches for the specific choice of
noise processes. It is evident both from analytical development and the
corresponding numerical simulation that the enhancement of rate is possible by
increasing the degree of correlation of the external fluctuations.Comment: 9 pages, 1 figures, RevTex
The Boltzmann Entropy for Dense Fluids Not in Local Equilibrium
We investigate, via computer simulations, the time evolution of the
(Boltzmann) entropy of a dense fluid not in local equilibrium. The
macrovariables describing the system are the (empirical) particle density
f=\{f(\un{x},\un{v})\} and the total energy . We find that is
monotone increasing in time even when its kinetic part is decreasing. We argue
that for isolated Hamiltonian systems monotonicity of
should hold generally for ``typical'' (the overwhelming majority of) initial
microstates (phase-points) belonging to the initial macrostate ,
satisfying . This is a direct consequence of Liouville's theorem
when evolves autonomously.Comment: 8 pages, 5 figures. Submitted to PR
Molecular random walks and invariance group of the Bogolyubov equation
Statistics of molecular random walks in a fluid is considered with the help
of the Bogolyubov equation for generating functional of distribution functions.
An invariance group of solutions to this equation as functions of the fluid
density is discovered. It results in many exact relations between probability
distribution of the path of a test particle and its irreducible correlations
with the fluid. As the consequence, significant restrictions do arise on
possible shapes of the path distribution. In particular, the hypothetical
Gaussian form of its long-range asymptotic proves to be forbidden (even in the
Boltzmann-Grad limit). Instead, a diffusive asymptotic is allowed which
possesses power-law long tail (cut off by ballistic flight length).Comment: 23 pages, no figures, LaTeX AMSART, author's translation from Russian
of the paper accepted to the TMPh (``Theoretical and mathematical physics''
Fractional Liouville and BBGKI Equations
We consider the fractional generalizations of Liouville equation. The
normalization condition, phase volume, and average values are generalized for
fractional case.The interpretation of fractional analog of phase space as a
space with fractal dimension and as a space with fractional measure are
discussed. The fractional analogs of the Hamiltonian systems are considered as
a special class of non-Hamiltonian systems. The fractional generalization of
the reduced distribution functions are suggested. The fractional analogs of the
BBGKI equations are derived from the fractional Liouville equation.Comment: 20 page
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