Comment on "Local accumulation times for source, diffusion, and degradation models in two and three dimensions" [J. Chem. Phys. 138, 104121 (2013)]

In a recent paper, Gordon, Muratov, and Shvartsman studied a partial differential equation (PDE) model describing radially symmetric diffusion and degradation in two and three dimensions. They paid particular attention to the local accumulation time (LAT), also known in the literature as the mean action time, which is a spatially dependent timescale that can be used to provide an estimate of the time required for the transient solution to effectively reach steady state. They presented exact results for three-dimensional applications and gave approximate results for the two-dimensional analogue. Here we make two generalizations of Gordon, Muratov, and Shvartsman’s work: (i) we present an exact expression for the LAT in any dimension and (ii) we present an exact expression for the variance of the distribution. The variance provides useful information regarding the spread about the mean that is not captured by the LAT. We conclude by describing further extensions of the model that were not considered by Gordon,Muratov, and Shvartsman. We have found that exact expressions for the LAT can also be derived for these important extensions...

Critical time scales for advection–diffusion–reaction processes

The concept of local accumulation time (LAT) was introduced by Berezhkovskii and coworkers in 2010–2011 to give a finite measure of the time required for the transient solution of a reaction–diffusion equation to approach the steady–state solution (Biophys J. 99, L59 (2010); Phys Rev E. 83, 051906 (2011)). Such a measure is referred to as a critical time. Here, we show that LAT is, in fact, identical to the concept of mean action time (MAT) that was first introduced by McNabb in 1991 (IMA J Appl Math. 47, 193 (1991)). Although McNabb’s initial argument was motivated by considering the mean particle lifetime (MPLT) for a linear death process, he applied the ideas to study diffusion. We extend the work of these authors by deriving expressions for the MAT for a general one–dimensional linear advection–diffusion–reaction problem. Using a combination of continuum and discrete approaches, we show that MAT and MPLT are equivalent for certain uniform–to-uniform transitions; these results provide a practical interpretation for MAT, by directly linking the stochastic microscopic processes to a meaningful macroscopic timescale. We find that for more general transitions, the equivalence between MAT and MPLT does not hold. Unlike other critical time definitions, we show that it is possible to evaluate the MAT without solving the underlying partial differential equation (pde). This makes MAT a simple and attractive quantity for practical situations. Finally, our work explores the accuracy of certain approximations derived using the MAT, showing that useful approximations for nonlinear kinetic processes can be obtained, again without treating the governing pde directly.

Snow accumulation time series and their climatic interpretation

EXTRACT (SEE PDF FOR FULL ABSTRACT): Several snow accumulation time series derived from ice cores and extending over 3 to 5 centuries are examined for spatial and temporal climatic information. ... A significant observation is the widespread depression of net snow accumulation during the latter part of the "Little Ice Age". This initially suggests sea surface temperatures were significantly depressed during the same period. However, prior to this, the available core records indicate generally higher than average precipitation rates. This also implies that influences such as shifted storm tracks or a dustier atmosphere may have been involved. Without additional spatial data coverage, these observations should properly be studied using a coupled (global) ocean/atmosphere GCM.

Moments of action provide insight into critical times for advection–diffusion–reaction processes

In 2010 Berezhkovskii and coworkers introduced the concept of local accumulation time (LAT) as a finite measure of the time required for the transient solution of a reaction diffusion equation to effectively reach steady state(Biophys J. 99, L59 (2010); Phys Rev E. 83, 051906 (2011)). Berezhkovskii’s approach is a particular application of the concept of mean action time (MAT) that was introduced previously by McNabb (IMA J Appl Math. 47, 193 (1991)). Here, we generalize these previous results by presenting a framework to calculate the MAT, as well as the higher moments, which we call the moments of action. The second moment is the variance of action time; the third moment is related to the skew of action time, and so on. We consider a general transition from some initial condition to an associated steady state for a one–dimensional linear advection–diffusion–reaction partial differential equation(PDE). Our results indicate that it is possible to solve for the moments of action exactly without requiring the transient solution of the PDE. We present specific examples that highlight potential weaknesses of previous studies that have considered the MAT alone without considering higher moments. Finally, we also provide a meaningful interpretation of the moments of action by presenting simulation results from a discrete random walk model together with some analysis of the particle lifetime distribution. This work shows that the moments of action are identical to the moments of the particle lifetime distribution for certain transitions.

Beyond fractional derivatives: local approximation of other convolution integrals

Dynamic systems involving convolution integrals with decaying kernels, of which fractionally damped systems form a special case, are non-local in time and hence infinite dimensional. Straightforward numerical solution of such systems up to time t needs O(t(2)) computations owing to the repeated evaluation of integrals over intervals that grow like t. Finite-dimensional and local approximations are thus desirable. We present here an approximation method which first rewrites the evolution equation as a coupled in finite-dimensional system with no convolution, and then uses Galerkin approximation with finite elements to obtain linear, finite-dimensional, constant coefficient approximations for the convolution. This paper is a broad generalization, based on a new insight, of our prior work with fractional order derivatives (Singh & Chatterjee 2006 Nonlinear Dyn. 45, 183-206). In particular, the decaying kernels we can address are now generalized to the Laplace transforms of known functions; of these, the power law kernel of fractional order differentiation is a special case. The approximation can be refined easily. The local nature of the approximation allows numerical solution up to time t with O(t) computations. Examples with several different kernels show excellent performance. A key feature of our approach is that the dynamic system in which the convolution integral appears is itself approximated using another system, as distinct from numerically approximating just the solution for the given initial values; this allows non-standard uses of the approximation, e. g. in stability analyses.

Changes in mid-troposphere snow accumulation on Mt. Logan, Yukon, over the last three centuries

EXTRACT (SEE PDF FOR FULL ABSTRACT): A net snow accumulation time series is presented. It is derived from a 102.5 m ice core retrieved from Mt. Logan at an altitude of 5340 m a.s.l. Annual increments are identified using stable isotopes, trace chemistry, and beta activity. ... The resulting time series of nearly 300 years seems to indicate a lower mean accumulation from AD 1700 to the mid-19th century than after that time. The last 100 years of the series correlates significantly with certain instrumental station records at mid-northern latitudes.

Direct observation of coherent spin transfer processes in an InGaAs/GaAs quantum well via two-color time-resolved Kerr rotation measurements

A two-color time-resolved Kerr rotation spectroscopy system was built, with a femtosecond Ti:sapphire laser and a photonic crystal fiber, to study coherent spin transfer processes in an InGaAs/GaAs quantum well sample. The femtosecond Ti:sapphire laser plays two roles: besides providing a pump beam with a tunable wavelength, it also excites the photonic crystal fiber to generate supercontinuum light ranging from 500 nm to 1600 nm, from which a probe beam with a desirable wavelength is selected with a suitable interference filter. With such a system, we studied spin transfer processes between two semiconductors of different gaps in an InGaAs/GaAs quantum well sample. We found that electron spins generated in the GaAs barrier were transferred coherently into the InGaAs quantum well. A model based on rate equations and Bloch-Torrey equations is used to describe the coherent spin transfer processes quantitatively. With this model, we obtain an effective electron spin accumulation time of 21 ps in the InGaAs quantum well.

Selective accumulation of differentiated FOXP3(+) CD4 (+) T cells in metastatic tumor lesions from melanoma patients compared to peripheral blood

Precise identification of regulatory T cells is crucial in the understanding of their role in human cancers. Here, we analyzed the frequency and phenotype of regulatory T cells (Tregs), in both healthy donors and melanoma patients, based on the expression of the transcription factor FOXP3, which, to date, is the most reliable marker for Tregs, at least in mice. We observed that FOXP3 expression is not confined to human CD25(+/high) CD4(+) T cells, and that these cells are not homogenously FOXP3(+). The circulating relative levels of FOXP3(+) CD4(+) T cells may fluctuate close to 2-fold over a short period of observation and are significantly higher in women than in men. Further, we showed that FOXP3(+) CD4(+) T cells are over-represented in peripheral blood of melanoma patients, as compared to healthy donors, and that they are even more enriched in tumor-infiltrated lymph nodes and at tumor sites, but not in normal lymph nodes. Interestingly, in melanoma patients, a significantly higher proportion of functional, antigen-experienced FOXP3(+) CD4(+) T was observed at tumor sites, compared to peripheral blood. Together, our data suggest that local accumulation and differentiation of Tregs is, at least in part, tumor-driven, and illustrate a reliable combination of markers for their monitoring in various clinical settings.

Coordinate and time-dependent diffusion dynamics in protein folding

Coordenação de Aperfeiçoamento de Pessoal de Nível Superior (CAPES)

Non-Markovian stochastic processes and their applications: from anomalous diffusion to time series analysis

This work provides a forward step in the study and comprehension of the relationships between stochastic processes and a certain class of integral-partial differential equation, which can be used in order to model anomalous diffusion and transport in statistical physics. In the first part, we brought the reader through the fundamental notions of probability and stochastic processes, stochastic integration and stochastic differential equations as well. In particular, within the study of H-sssi processes, we focused on fractional Brownian motion (fBm) and its discrete-time increment process, the fractional Gaussian noise (fGn), which provide examples of non-Markovian Gaussian processes. The fGn, together with stationary FARIMA processes, is widely used in the modeling and estimation of long-memory, or long-range dependence (LRD). Time series manifesting long-range dependence, are often observed in nature especially in physics, meteorology, climatology, but also in hydrology, geophysics, economy and many others. We deepely studied LRD, giving many real data examples, providing statistical analysis and introducing parametric methods of estimation. Then, we introduced the theory of fractional integrals and derivatives, which indeed turns out to be very appropriate for studying and modeling systems with long-memory properties. After having introduced the basics concepts, we provided many examples and applications. For instance, we investigated the relaxation equation with distributed order time-fractional derivatives, which describes models characterized by a strong memory component and can be used to model relaxation in complex systems, which deviates from the classical exponential Debye pattern. Then, we focused in the study of generalizations of the standard diffusion equation, by passing through the preliminary study of the fractional forward drift equation. Such generalizations have been obtained by using fractional integrals and derivatives of distributed orders. In order to find a connection between the anomalous diffusion described by these equations and the long-range dependence, we introduced and studied the generalized grey Brownian motion (ggBm), which is actually a parametric class of H-sssi processes, which have indeed marginal probability density function evolving in time according to a partial integro-differential equation of fractional type. The ggBm is of course Non-Markovian. All around the work, we have remarked many times that, starting from a master equation of a probability density function f(x,t), it is always possible to define an equivalence class of stochastic processes with the same marginal density function f(x,t). All these processes provide suitable stochastic models for the starting equation. Studying the ggBm, we just focused on a subclass made up of processes with stationary increments. The ggBm has been defined canonically in the so called grey noise space. However, we have been able to provide a characterization notwithstanding the underline probability space. We also pointed out that that the generalized grey Brownian motion is a direct generalization of a Gaussian process and in particular it generalizes Brownain motion and fractional Brownain motion as well. Finally, we introduced and analyzed a more general class of diffusion type equations related to certain non-Markovian stochastic processes. We started from the forward drift equation, which have been made non-local in time by the introduction of a suitable chosen memory kernel K(t). The resulting non-Markovian equation has been interpreted in a natural way as the evolution equation of the marginal density function of a random time process l(t). We then consider the subordinated process Y(t)=X(l(t)) where X(t) is a Markovian diffusion. The corresponding time-evolution of the marginal density function of Y(t) is governed by a non-Markovian Fokker-Planck equation which involves the same memory kernel K(t). We developed several applications and derived the exact solutions. Moreover, we considered different stochastic models for the given equations, providing path simulations.

Particle size distribution effects on preferential deposition areas in metal foam wrapped tube bundle

This paper presents a numerical model for understanding particle transport and deposition in metal foam heat exchangers. Two-dimensional steady and unsteady numerical simulations of a standard single row metal foam-wrapped tube bundle are performed for different particle size distributions, i.e. uniform and normal distributions. Effects of different particle sizes and fluid inlet velocities on the overall particle transport inside and outside the foam layer are also investigated. It was noted that the simplification made in the previously-published numerical works in the literature, e.g. uniform particle deposition in the foam, is not necessarily accurate at least for the cases considered here. The results highlight the preferential particle deposition areas both along the tube walls and inside the foam using a developed particle deposition likelihood matrix. This likelihood matrix is developed based on three criteria being particle local velocity, time spent in the foam, and volume fraction. It was noted that the particles tend to deposit near both front and rear stagnation points. The former is explained by the higher momentum and direct exposure of the particles to the foam while the latter only accommodate small particles which can be entrained in the recirculation region formed behind the foam-wrapped tubes.

4D GPR tracking of water infiltration in fractured high-porosity limestone

Three thousand liters of water were infiltrated from a 4 m diameter pond to track flow and transport inside fractured carbonates with 20-40 % porosity. Sixteen time-lapse 3D Ground Penetrating Radar (GPR) surveys with repetition intervals between 2 hrs and 5 days monitored the spreading of the water bulb in the subsurface. Based on local travel time shifts between repeated GPR survey pairs, localized changes of volumetric water content can be related to the processes of wetting, saturation and drainage. Deformation bands consisting of thin sub vertical sheets of crushed grains reduce the magnitude of water content changes but enhance flow in sheet parallel direction. This causes an earlier break through across a stratigraphic boundary compared to porous limestone without deformation bands. This experiment shows how time-lapse 3D GPR or 4D GPR can non-invasively track ongoing flow processes in rock-volumes of over 100 m3.

Dendritic structures produced on solidification of multicomponent aqueous solutions

Dendrite structures of ice produced on undirectional solidification of ternary and quaternary aqueous solutions have been studied. Upon freezing, solutions containing more than one solute produce plate-shaped dendrites of ice. The spacing between dendrites increase linearly with the distance from the chill surface and the square root of local solidification time (or square root of inverse freezing rate) for any fixed composition. For fixed freezing conditions, the dendrite spacings from multicomponent aqueous solutions were a function of the concentrations and diffusion coefficients of the individual solutes. The dendrite spacing produced by freezing of a solution was changed by the addition of a solute different from those already present. If the main diffusion coefficient of the added solute is higher than that of solutes already present, the dendrite spacing is increased and vice versa. The dendrite spacing in multi-component systems increases with the total solute concentration if the constituent solutes are present in equal amounts. The dendrite spacing obtained on freezing of these dilute multicomponent solutions can be expressed by regression equations of the type Image Full-size image (2K) where L is the dendrite spacing in microns, C1, C2 and C3 are concentrations of individual solutes, Θf is the total freezing time and A1 −A8 are constants. A Yates analysis of the dendrite spacings in a factorial design of quaternary solutions indicates that there are strong interactions between individual solutes in regard to their effect on the dendrite spacings. A mass transport analysis has been used to calculate the interdendritic supersaturation ΔC of the individual solutes, the supercooling in the interdendritic liquid ΔT, and the transverse growth velocity of the dendrites, VT. In ternary solutions if two solutes are present in equal amount the supersaturation of the solute with higher main diffusion coefficient is lower, and vice versa. If a solute with higher main diffusion coefficient is added to a binary solution, the interface growth velocity, the interdendritic supersaturation of the base solute and the interdendritic supercooling increase with the quantity of solute added.