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For an effectively one-dimensional, semi-infinite disordered system connected to a reservoir of tracer particles kept at constant concentration, we provide the dynamics of the concentration profile.
Technically, we start with the Montroll-Weiss equation of a continuous time random walk with a scale-free waiting time density.
From this we pass to a formulation in terms of the fractional diffusion equation for the concentration profile C(x, t) in a semi-infinite space for the boundary condition C(0, t) = C-0, using a subordination approach.
From this we deduce the tracer flux and the so-called breakthrough curve (BTC) at a given distance from the tracer source.
In particular, BTCs are routinely measured in geophysical contexts but are also of interest in single-particle tracking experiments.
For the "residual' BTCs, given by 1- P(x, t), we demonstrate a long-time power-law behaviour that can be compared conveniently to experimental measurements.
For completeness we also derive expressions for the moments in this constant-concentration boundary condition.
Diffusive search for a static target is a common problem in statistical physics with numerous applications in chemistry and biology.
We look at this problem from a different perspective and investigate the statistics of encounters between the diffusing particle and the target. While an exact solution of this problem was recently derived in the form of a spectral expansion over the eigenbasis of the Dirichlet-to-Neumann operator, the latter is generally difficult to access for an arbitrary target.
In this paper, we present three complementary approaches to approximate the probability density of the rescaled number of encounters with a small target in a bounded confining domain. In particular, we derive a simple fully explicit approximation, which depends only on a few geometric characteristics such as the surface area and the harmonic capacity of the target, and the volume of the confining domain.
We discuss the advantages and limitations of three approaches and check their accuracy.
We also deduce an explicit approximation for the distribution of the first-crossing time, at which the number of encounters exceeds a prescribed threshold. Its relations to common first-passage time problems are discussed.
Geochemical homogeneity in shale is often assumed when tracing subsurface fluids and characterizing sedimentary basins. This study presents measurements of the bulk gas composition, stable isotopes, and noble gas volume fraction and isotopes for shale gas samples collected from gas wells in the Wufeng-Longmaxi Shale, the southern Sichuan Basin, China. The dryness [C-1 /(C-2 + C-3)] ranging from 166.3 to 251.2, combined with delta C-13(1) and delta DC1 that vary from -28.8 to -27.3 parts per thousand and - 153 to -145 parts per thousand, respectively, point to a late mature thermogenic origin of hydrocarbon gas. He-3/He-4 ratios of gas samples are around 0.01 times the air value suggesting dominantly crust-derived He. Ne-21/Ne-22 and Ar-40/Ar-36 ratios of many gas samples are higher than the corresponding air values indicating the mixing of crustal and atmospheric noble gases. Multiple dichotomous patterns are observed in noble gas signatures of forelimb and backlimb samples, and depression and crest samples. Ne-20/Ne-22 ratios of some crest samples are higher than that of depression samples in the backlimb, pointing to the presence of diffusion-driven fractionation that is likely caused by the long-distance migration from depression to crest. Elemental ratios of air-derived noble gas isotopes - Ne-22/Ar-36, Kr-84/Ar-36, and Xe-132/Ar-36 are compared to the recharge water values, suggesting the interactions of oil, gas, and water phases in the shale over geologic time. Forelimb samples generally display older ages than backlimb samples, indicating a larger flux of external radiogenic He-4 due to the higher density of deep faults in the forelimb area caused by the basementinvolved deformation. The basement-involved deformation also causes pore collapse especially in the forelimb leading to a lower porosity that results in a more pristine noble gas signature in the forelimb due to the reduced impact of younger recharge water.
Diffusion with stochastic resetting is a paradigm of resetting processes. Standard renewal or master equation approach are typically used to study steady state and other transport properties such as average, mean squared displacement etc.
What remains less explored is the two time point correlation functions whose evaluation is often daunting since it requires the implementation of the exact time dependent probability density functions of the resetting processes which are unknown for most of the problems.
We adopt a different approach that allows us to write a stochastic solution for a single trajectory undergoing resetting.
Moments and the autocorrelation functions between any two times along the trajectory can then be computed directly using the laws of total expectation. Estimation of autocorrelation functions turns out to be pivotal for investigating the ergodic properties of various observables for this canonical model.
In particular, we investigate two observables (i) sample mean which is widely used in economics and (ii) time-averaged-mean-squared-displacement (TAMSD) which is of acute interest in physics.
We find that both diffusion and drift-diffusion processes with resetting are ergodic at the mean level unlike their reset-free counterparts. In contrast, resetting renders ergodicity breaking in the TAMSD while both the stochastic processes are ergodic when resetting is absent. We quantify these behaviors with detailed analytical study and corroborate with extensive numerical simulations.
Our results can be verified in experimental set-ups that can track single particle trajectories and thus have strong implications in understanding the physics of resetting.
Lithium and boron are trace components of magmas, released during exsolution of a gas phase during volcanic activity.
In this study, we determine the diffusivity and isotopic fractionation of Li and B in hydrous silicate melts.
Two glasses were synthesized with the same rhyolitic composition (4.2 wt% water), having different Li and B contents; these were studied in diffusion-couple experiments that were performed using an internally heated pressure vessel, operated at 300 MPa in the temperature range 700-1250 degrees C for durations from 0 s to 24 h. From this we determined activation energies for Li and B diffusion of 57 +/- 4 kJ/mol and 152 +/- 15 kJ/mol with pre-exponential factors of 1.53 x 10(-7) m(2)/s and 3.80 x 10(-8) m(2)/s, respectively.
Lithium isotopic fractionation during diffusion gave beta values between 0.15 and 0.20, whereas B showed no clear isotopic fractionation.
Our Li diffusivities and isotopic fractionation results differ somewhat from earlier published values, but overall confirm that Li diffusivity increases with water content. Our results on B diffusion show that similarly to Li, B mobility increases in the presence of water.
By applying the Eyring relation, we confirm that B diffusivity is limited by viscous flow in silicate melts.
Our results on Li and B diffusion present a new tool for understanding degassing-related processes, offering a potential geospeedometer to measure volcanic ascent rates.
The field of movement ecology has seen a rapid increase in high-resolution data in recent years, leading to the development of numerous statistical and numerical methods to analyse relocation trajectories. Data are often collected at the level of the individual and for long periods that may encompass a range of behaviours.
Here, we use the power spectral density (PSD) to characterise the random movement patterns of a black-winged kite (Elanus caeruleus) and a white stork (Ciconia ciconia). The tracks are first segmented and clustered into different behaviours (movement modes), and for each mode we measure the PSD and the ageing properties of the process.
For the foraging kite we find 1/f noise, previously reported in ecological systems mainly in the context of population dynamics, but not for movement data. We further suggest plausible models for each of the behavioural modes by comparing both the measured PSD exponents and the distribution of the single-trajectory PSD to known theoretical results and simulations.
Quantifying root water uptake is essential to understanding plant water use and responses to different environmental conditions. However, non-destructive measurement of water transport and related hydraulics in the soil-root system remains a challenge.
Neutron imaging, with its high sensitivity to hydrogen, has become an unparalleled tool to visualize and quantify root water uptake in vivo. In combination with isotopes (e.g., deuterated water) and a diffusion-convection model, root water uptake and hydraulic redistribution in root and soil can be quantified.
Here, we review recent advances in utilizing neutron imaging to visualize and quantify root water uptake, hydraulic redistribution in roots and soil, and root hydraulic properties of different plant species.
Under uniform soil moisture distributions, neutron radiographic studies have shown that water uptake was not uniform along the root and depended on both root type and age. For both tap (e.g., lupine [Lupinus albus L.]) and fibrous (e.g., maize [Zea mays L.]) root systems, water was mainly taken up through lateral roots. In mature maize, the location of water uptake shifted from seminal roots and their laterals to crown/nodal roots and their laterals.
Under non-uniform soil moisture distributions, part of the water taken up during the daytime maintained the growth of crown/nodal roots in the upper, drier soil layers. Ultra-fast neutron tomography provides new insights into 3D water movement in soil and roots. We discuss the limitations of using neutron imaging and propose future directions to utilize neutron imaging to advance our understanding of root water uptake and soil-root interactions.
We study the diffusive motion of a particle in a subharmonic potential of the form U(x) = |x|( c ) (0 < c < 2) driven by long-range correlated, stationary fractional Gaussian noise xi ( alpha )(t) with 0 < alpha <= 2. In the absence of the potential the particle exhibits free fractional Brownian motion with anomalous diffusion exponent alpha. While for an harmonic external potential the dynamics converges to a Gaussian stationary state, from extensive numerical analysis we here demonstrate that stationary states for shallower than harmonic potentials exist only as long as the relation c > 2(1 - 1/alpha) holds. We analyse the motion in terms of the mean squared displacement and (when it exists) the stationary probability density function. Moreover we discuss analogies of non-stationarity of Levy flights in shallow external potentials.
Transport properties of potential host rocks for nuclear waste disposal are typically determined in laboratory or in-situ experiments under geochemically controlled and constant conditions. Such a homogeneous assumption is no longer applicable on the host rock scale as can be seen from the pore water profiles of the potential host rock Opalinus Clay at Mont Terri (Switzerland). The embedding aquifers are the hydro-geological boundaries, that established gradients in the 210 m thick low permeable section through diffusive exchange over millions of years. Present-day pore water profiles were confirmed by a data-driven as well as by a conceptual scenario. Based on the modelled profiles, the influence of the geochemical gradient on uranium migration was quantified by comparing the distances after one million years with results of common homogeneous models. Considering the heterogeneous system, uranium migrated up to 24 m farther through the formation depending on the source term position within the gradient and on the partial pressure of carbon dioxide pCO2 of the system. Migration lengths were almost equal for single- and multicomponent diffusion. Differences can predominantly be attributed to changes in the sorption capacity, whereby pCO2 governs how strong uranium migration is affected by the geochemical gradient. Thus, the governing parameters for uranium migration in the Opalinus Clay can be ordered in descending priority: pCO2, geochemical gradients, mineralogical heterogeneity.</p>
We analyse mobile-immobile transport of particles that switch between the mobile and immobile phases with finite rates. Despite this seemingly simple assumption of Poissonian switching, we unveil a rich transport dynamics including significant transient anomalous diffusion and non-Gaussian displacement distributions. Our discussion is based on experimental parameters for tau proteins in neuronal cells, but the results obtained here are expected to be of relevance for a broad class of processes in complex systems. Specifically, we obtain that, when the mean binding time is significantly longer than the mean mobile time, transient anomalous diffusion is observed at short and intermediate time scales, with a strong dependence on the fraction of initially mobile and immobile particles. We unveil a Laplace distribution of particle displacements at relevant intermediate time scales. For any initial fraction of mobile particles, the respective mean squared displacement (MSD) displays a plateau. Moreover, we demonstrate a short-time cubic time dependence of the MSD for immobile tracers when initially all particles are immobile.