10 research outputs found
Finite speed of propagation in Degenerate Einstein Brownian Motion Model
We considered the generalization of Einstein's model of Brownian motion when
the key parameter of the time interval of free jumps degenerates. This
phenomenon manifests in two scenarios: a) flow of the fluid, which is highly
dispersing like a non-dense gas, and b) flow of fluid far away from the source
of flow, when the velocity of the flow is incomparably smaller than the
gradient of the pressure. First, we will show that both types of flows can be
modeled using the Einstein paradigm. We will investigate the question: What
features will particle flow exhibit if the time interval of the free jump is
inverse proportional to the density of the fluid and its gradient ? We will
show that in this scenario, the flow exhibits localization property, namely: if
at some moment of time in the region gradient of the pressure or
pressure itself is equal to zero, then for some time T during t interval there is no flow in the region. This directly links to
Barenblatt's finite speed of propagation property for the degenerate equation.
The method of proof is very different from Barenblatt's method and based on
Vespri - Tedeev technique.Comment: 14 page
Nonlinear Einstein paradigm of Brownian motion and localization property of solutions
We employ a generalization of Einstein's random walk paradigm for diffusion
to derive a class of multidimensional degenerate nonlinear parabolic equations
in non-divergence form. Specifically, in these equations, the diffusion
coefficient can depend on both the dependent variable and its gradient, and it
vanishes when either one of the latter does. It is known that solution of such
degenerate equations can exhibit finite speed of propagation (so-called
localization property of solutions). We give a proof of this property using a
De Giorgi--Ladyzhenskaya iteration procedure for non-divergence-from equations.
A mapping theorem is then established to a divergence-form version of the
governing equation for the case of one spatial dimension. Numerical results via
a finite-difference scheme are used to illustrate the main mathematical results
for this special case. For completeness, we also provide an explicit
construction of the one-dimensional self-similar solution with finite speed of
propagation function, in the sense of Kompaneets--Zel'dovich--Barenblatt. We
thus show how the finite speed of propagation quantitatively depends on the
model's parameters.Comment: 18 pages, 2 figure
Cavity-Enhanced Absorption Measurements Across Broad Absorbance and Reflectivity Ranges
Cavity-enhanced spectrometry constitutes
an important and highly
sensitive technique for absorbance measurements. The current practice
generally involves very high reflectivity mirrors and hence intense
light sources (typically lasers) to have enough light transmitted.
Available theory describes the situation only for high-finesse cavities
(high-reflectance mirrors) and generally for systems with very low
absorbances. We develop the general expression for absorbance regardless
of mirror reflectivity or the absorbance and show that in the limit
of high reflectivities and low absorbances it predicts the same numerical
values as that derived by O’Keefe (Chem. Phys. Lett. 1998, 293, 331−336; Chem. Phys.
Lett. 1999, 307, 343–349). Signal to noise in any photometric
system is also dependent on the amount of light reaching the detector
because of shot noise limitations. We show that a small aperture in
the entrance mirror greatly improves light throughput without significant
departure from the theoretically predicted amplification of absorbance;
such simple modifications result in real improvement of detection
limits, even with mirrors of modest reflectivity and inexpensive detectors.
This allows the merits of cavity enhancement measurements to be demonstrated
for pedagogic purposes