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 $t_{0}$ in the region gradient of the pressure or pressure itself is equal to zero, then for some time T during t interval $[ t_{0}, t_0+T ]$ 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