Non-local Transport Processes Research Articles

Structured population models on Polish spaces: A unified approach including graphs, Riemannian manifolds and measure spaces to describe dynamics of heterogeneous populations

This paper presents a mathematical framework for modeling the dynamics of heterogeneous populations. Models describing local and non-local growth and transport processes appear in a variety of applications, such as crowd dynamics, tissue regeneration, cancer development and coagulation-fragmentation processes. The diverse applications pose a common challenge to mathematicians due to the multiscale nature of the structures that underlie the system’s self-organization and control. Similar abstract mathematical problems arise when formulating problems in the language of measure evolution on a multi-faceted state space. Motivated by these observations, we propose a general mathematical framework for nonlinear structured population models on abstract metric spaces, which are only assumed to be separable and complete. We exploit the structure of the space of non-negative Radon measures with the dual bounded Lipschitz distance (flat metric), which is a generalization of the Wasserstein distance, capable of addressing non-conservative problems. The formulation of models on general metric spaces allows considering infinite-dimensional state spaces or graphs and coupling discrete and continuous state transitions. This opens up exciting possibilities for modeling single-cell data, crowd dynamics or coagulation-fragmentation processes.

Quantifying nonlocal bedload transport: A regional-based nonlocal model for bedload transport from local to global scales

Recent studies have emphasized the importance of nonlocal models in characterizing bedload transport in natural rivers, particularly in mixed-size gravel beds or steep hillslopes. Nonlocality denotes that a quantity (flux) at a specific location x is dependent on the conditions in the surrounding area, as opposed to solely at the location itself. This concept applies in bedload transport even in planar flumes, where particles are entrained at an upstream position and travel a finite distance, ultimately contributing nonlocally to the sediment flux. However, existing bedload transport models, such as the advection–diffusion equation (ADE) or the fractional derivative equation (FDE) models, are inadequate in characterizing the nonlocal transport behavior of bedload at a regional scale. Large errors may arise from the lack of an accurate description of the nonlocal bedload transport processes at regional scales. This study proposes a regional-based nonlocal bedload transport model, which is conceptualized from the probabilistic Exner-based equations and the peridynamic (PD) differential operator. The PD model encapsulates the nonlocal motion of bedload sediments on the basis of the PD differential operator, by utilizing a pre-defined weight function and influence domain. Comparisons demonstrate that the PD model serves as a generalized tool connecting the local and the global models with different PD functions and influence domains. Its variability on kernel function and influence domain, enable it conveniently describe sub-, super-, and normal diffusion behaviors of bedload transport.

Scattering description of Andreev molecules

An Andreev molecule is a system of closely spaced superconducting weak links accommodating overlapping Andreev Bound States. Recent theoretical proposals have considered one-dimensional Andreev molecules with a single conduction channel. Here we apply the scattering formalism and extend the analysis to multiple conduction channels, a situation encountered in epitaxial superconductor/semiconductor weak links. We obtain the multi-channel bound state energy spectrum and quantify the contribution of the microscopic non-local transport processes leading to the formation of Andreev molecules.

High order numerical schemes for solving fractional powers of elliptic operators

In many recent applications when new materials and technologies are developed it is important to describe and simulate new nonlinear and nonlocal diffusion transport processes. A general class of such models deals with nonlocal fractional power elliptic operators. In order to solve these problems numerically it is proposed (Vabishchevich, 2015) to consider equivalent local nonstationary initial value pseudo-parabolic problems. Previously such problems were solved by using the standard implicit backward and symmetrical Euler methods. In this paper we use the one-parameter family of three-level finite difference schemes for solving the initial value problem for the first order nonstationary pseudo-parabolic problem. The fourth-order approximation scheme is developed by selecting the optimal value of the weight parameter. The results of the theoretical analysis are supplemented by results of extensive computational experiments.

Turbulent particle pair diffusion: Numerical simulations.

A theory for turbulent particle pair diffusion in the inertial subrange [Malik NA, PLoS ONE 13(10):e0202940 (2018)] is investigated numerically using a Lagrangian diffusion model, Kinematic Simulations [Kraichnan RH, Phys. Fluids 13:22 (1970); Malik NA, PLoS ONE 12(12):e0189917 (2017)]. All predictions of the theory are observed in flow fields with generalised energy spectra of the type, E(k) ∼ k−p. Most importantly, two non-Richardson regimes are observed: for short inertial subrange of size 102 the simulations yield quasi-local regimes for the pair diffusion coefficient, ; and for asymptotically infinite inertial subrange the simulations yield non-local regimes , with γ intermediate between the purely local scaling γl = (1 + p)/2 and the purely non-local scaling γnl = 2. For intermittent turbulence spectra, E(k) ∼ k−1.72, the simulations yield , in agreement with the revised 1926 dataset [Richardson LF, Proc. Roy. Soc. Lond. A 100:709 (1926); Malik NA, PLoS ONE 13(10):e0202940 (2018)]. These results lend support to the physical picture proposed in the new theory that turbulent diffusion in the inertial subrange is governed by both local and non-local diffusion transport processes.

Spin dependence of crossed Andreev reflection and electron tunneling induced by Majorana fermions

We investigate spin dependence of the nonlocal transport induced by Majorana fermions in a one-dimensional ferromagnet–ferromagnetic-insulator–superconductor–ferromagnetic-insulator–ferromagnet junction on the edge of a two-dimensional topological insulator. The results show that coupled Majorana fermions lead to the nonlocal transport processes including electron tunneling and crossed Andreev reflection, which can be tuned by adjusting the spin polarizations of the Majorana fermions. By manipulating the bands in the two ferromagnets, the nonlocal transport can be selected as either pure electron tunneling or pure crossed Andreev reflection, the transmission probability of which could be 100%. Furthermore, the pure electron tunneling and the pure crossed Andreev reflection are well controlled by the spin directions of the electron states in the two ferromagnets.

A Framework for Convection and Boundary Layer Parameterization Derived from Conditional Filtering

Abstract A new theoretical framework is derived for parameterization of subgrid physical processes in atmospheric models; the application to parameterization of convection and boundary layer fluxes is a particular focus. The derivation is based on conditional filtering, which uses a set of quasi-Lagrangian labels to pick out different regions of the fluid, such as convective updrafts and environment, before applying a spatial filter. This results in a set of coupled prognostic equations for the different fluid components, including subfilter-scale flux terms and entrainment/detrainment terms. The framework can accommodate different types of approaches to parameterization, such as local turbulence approaches and mass flux approaches. It provides a natural way to distinguish between local and nonlocal transport processes and makes a clearer conceptual link to schemes based on coherent structures such as convective plumes or thermals than the straightforward application of a filter without the quasi-Lagrangian labels. The framework should facilitate the unification of different approaches to parameterization by highlighting the different approximations made and by helping to ensure that budgets of energy, entropy, and momentum are handled consistently and without double counting. The framework also points to various ways in which traditional parameterizations might be extended, for example, by including additional prognostic variables. One possibility is to allow the large-scale dynamics of all the fluid components to be handled by the dynamical core. This has the potential to improve several aspects of convection–dynamics coupling, such as dynamical memory, the location of compensating subsidence, and the propagation of convection to neighboring grid columns.

Nonlocal Transport Processes and the Fractional Cattaneo-Vernotte Equation

The Cattaneo-Vernotte equation is a generalization of the heat and particle diffusion equations; this mathematical model combines waves and diffusion with a finite velocity of propagation. In disordered systems the diffusion can be anomalous. In these kinds of systems, the mean-square displacement is proportional to a fractional power of time not equal to one. The anomalous diffusion concept is naturally obtained from diffusion equations using the fractional calculus approach. In this paper we present an alternative representation of the Cattaneo-Vernotte equation using the fractional calculus approach; the spatial-time derivatives of fractional order are approximated using the Caputo-type derivative in the range(0,2]. In this alternative representation we introduce the appropriate fractional dimensional parameters which characterize consistently the existence of the fractional space-time derivatives into the fractional Cattaneo-Vernotte equation. Finally, consider the Dirichlet conditions, the Fourier method was used to find the full solution of the fractional Cattaneo-Vernotte equation in analytic way, and Caputo and Riesz fractional derivatives are considered. The advantage of our representation appears according to the comparison between our model and models presented in the literature, which are not acceptable physically due to the dimensional incompatibility of the solutions. The classical cases are recovered when the fractional derivative exponents are equal to1.

Local electrical tuning of the nonlocal signals in a Cooper pair splitter

A Cooper pair splitter consists of a central superconducting contact, S, from which electrons are injected into two parallel, spatially separated quantum dots (QDs). This geometry as well as electron interactions can lead to correlated electrical currents due to the spatial separation of spin-singlet Cooper pairs from S. We present experiments on such a device with a series of bottom gates, which allows for spatially resolved tuning of the tunnel couplings between the QDs and the electrical contacts and between the two QDs. Our main findings are gate-induced transitions between positive conductance correlation in the QDs due to Cooper pair splitting and negative correlations due to QD dynamics. Using a semiclassical rate equation model we show that the experimental findings are consistent with in situ electrical tuning of the local and nonlocal quantum transport processes. In particular, we illustrate how the competition between Cooper pair splitting and local processes can be optimized in such hybrid nanostructures.

Controllable nonlocal transport of Majorana fermions with the aid of two quantum dots

Majorana fermions (MFs) are appearing in pairs and nonlocally distributed at the two ends of the wire. The two MFs can combine together to form a nonlocal fermion state. Due to the nonlocal nature of Majorana fermions in the topological superconductor, there are two types of nonlocal transport processes supported by MFs: crossed Andreev reflection and electron transmission. In this paper, we study electron transport through a normal lead-quantum dot-topological superconductor-quantum dot-normal lead (N-QD-TS-QD-N) junction. We show that these two nonlocal processes can be directly controlled by gating the energy level of QDs. For instance, when both energies of QDs (labeled as ${\ensuremath{\epsilon}}_{1}$ and ${\ensuremath{\epsilon}}_{2})$ are equal to the coupling energy of MFs (denoted as ${E}_{M})$, the electron in the left lead can transport to the right lead via MFs while when ${\ensuremath{\epsilon}}_{1}=\ensuremath{-}{\ensuremath{\epsilon}}_{2}={E}_{M}$ is satisfied, the electron in the left lead can combine one electron in the right lead to form a Cooper pair and tunnel into the topological superconductor via MFs.

Nonlocal spectroscopy of Andreev bound states

We experimentally investigate Andreev bound states (ABSs) in a carbon nanotube quantum dot (QD) connected to a superconducting Nb lead (S). A weakly coupled normal metal contact acts as a tunnel probe that measures the energy dispersion of the ABSs. Moreover, we study the response of the ABS to nonlocal transport processes, namely, Cooper pair splitting and elastic co-tunnelling, which are enabled by a second QD fabricated on the same nanotube on the opposite side of S. We find an appreciable nonlocal conductance with a rich structure, including a sign reversal at the ground-state transition from the ABS singlet to a degenerate magnetic doublet. We describe our device by a simple rate equation model that captures the key features of our observations and demonstrates that the sign of the nonlocal conductance is a measure for the charge distribution of the ABS, given by the respective Bogoliubov-de Gennes amplitudes $u$ and $v$.

Influence of surfactant transport suppression on dynamic contact angle hysteresis

The influence of local and nonlocal transport processes of cetyltrimethylammonium bromide (CTAB) molecules on dynamic contact angles and contact angle hysteresis was studied in a rotating drum setup. The influence of long-range surfactant transport was analyzed by hindering selectively the surface or the bulk transport via movable barriers. With increasing hindrance of the surfactant transport, the receding contact angle decreased at all withdrawing velocities in the presence of CTAB. The control experiment with pure water was unaffected by the presence of the barriers. Dynamic contact angles are, therefore, not only influenced by short-range effects like Marangoni stresses close to the contact line, but also by long-range transport processes (like diffusion and advection) between the regions close to the receding and advancing contact lines.

Molecular weight and chemical reactivity of dissolved trace metals (Cd, Cu, Ni) in surface waters from the Mississippi River to Gulf of Mexico

It is generally assumed that estuarine mixing is continuous for metals from terrestrial sources, gradually decreasing towards the open ocean endmember. Here we show that, chemical reactivity, determined by ion exchange method, and molecular weight distributions, obtained using cross-flow ultrafiltration, of dissolved Cd, Cu, and Ni in the surface waters of the Gulf of Mexico varied systematically across the estuarine mixing zone of the Mississippi River. Most size or chemical affinity fractions of dissolved metals (<0.4 μm) were linearly related to salinity (10.8–36.6), suggesting that the distribution of these elements was mainly controlled by continuous mixing processes. Dissolved concentrations across the salinity gradient ranged for Cd: 87–187 pM; Cu: 1.4–18.3 nM; and Ni: 2.6–18.8 nM, with highest values near the Mississippi river mouth, and lowest concentrations in a warm core ring in the Gulf of Mexico. Dissolved Cd was mostly present as a truly dissolved (<10 kDa, 97 ± 1%) and cationic fraction (Chelex-100 extractable, 94 ± 4%). A novel observation across the estuarine mixing zone was that colloidal metal concentrations were identical to either inert (for Cu, Ni) or AMPG-labile anionic (Cu, Cd) fractions. The difference in behavior between Cu and the other two metals might indicate differences in the biopolymeric nature of the metal–organic chelates. In particular, the anionic-organic Cd fractions accounted for just 3 ± 1%, on average. However, for Cu, it was 24 ± 4%, and for Ni, it was 9 ± 6%. The fractions of the total dissolved metal fractions that were “inert” averaged 31 ± 10% for Cu and 29 ± 12% for Ni. Small but noticeable amounts (6 ± 3%) of dissolved inert Cd fractions were also present. Apparent non-local transport processes, likely associated with cross-shelf sediment resuspension processes, could have been responsible for the relatively high concentrations of ‘inert’ and ‘anionic’ metal fractions in high salinity coastal waters, and accounting for the persistence of metals bound to humic substances in the Gulf of Mexico.

The fractional diffusion model with an absorption term and modified Fick’s law for non-local transport processes

A generalized non-local Fick’s law on fractal-dimension is derived. Using modified Fick’s law a time–space fractional diffusion model with a fractional oscillator term is built. The solution is obtained in terms of Mittag–Leffler function using finite Hankel integral transformation and Laplace transformation. In addition, numerical simulations are discussed. The results show that the effect range of time-fractional derivative ν on probability density is greater than that of fractional oscillator parameter β . The effect range of ν on probability density is opposite to that of β . This paper provides a new analytical tool to develop fluid mechanics, heat conduction and other engineering science.

Food supply impacts sediment reworking by Nereis diversicolor

This study assessed the effects of food supply on sediment reworking by Nereis diversicolor. We hypothesized that food supply would enhance sediment reworking and that the frequency of food supply would affect the intensity of bioturbation. Mesocosm experiments consisted of four treatments: (1) without worms (control cores), (2) with worms and no food supply, (3) with worms and daily food supply, (4) with worms and weekly food supply. Fluorescent particles, used as tracers, were spread over the sediment surface. Sediment reworking was quantified after 28 days based on the tracer distribution profiles. Results showed that sediment reworking by N. diversicolor was exclusively due to non-local transport processes. Food supply greatly increased non-local transport coefficients (more than 3 times) in comparison with those measured in the absence of a food supply. However, the intensity of sediment bioturbation by these worms was unaffected by the frequency of food supply. This study showed that environmental conditions affecting the quantity of food supply at the water-sediment interface could strongly influence bioturbation process.

The fractional Fick's law for non-local transport processes

Fick's law is extensively adopted as a model for standard diffusion processes. However, requiring separation of scales, it is not suitable for describing non-local transport processes. We discuss a generalized non-local Fick's law derived from the space-fractional diffusion equation generating the Lévy–Feller statistics. This means that the fundamental solutions can be interpreted as Lévy stable probability densities (in the Feller parameterization) with index α (1< α⩽2) and skewness θ (| θ|⩽2− α). We explore the possibility of defining an equivalent local diffusivity by displaying a few numerical case studies concerning the relevant quantities (flux and gradient). It turns out that the presence of asymmetry ( θ≠0) plays a fundamental role: it produces shift of the maximum location of the probability density function and gives raise to phenomena of counter-gradient transport.

A generalized Fick's law to describe non-local transport effects

Fick's law is extensively used as a model for turbulent diffusion processes. It requires separation of scales between those of the process driving the diffusion mechanism and the scale of variation of the mean quantity being diffused. This makes the model unsuitable for the description of non-local transport processes like those occurring in some turbulent flows.A generalized Fick's law is proposed using a fractional derivative operator which accounts for non-local phenomena in virtue of its integral nature. This generalization is suggested as a model for two typical phenomena, like those observed in the convective boundary layer (CBL), which cannot be reduced to a local formulation: the inadequacy in the flux-gradient relationship when considering bottom-up dispersion (in particular, the counter-gradient transport) and the vertical drift of the location of the maximum concentration even in the absence of the mean velocity field.The solution of the generalized diffusion equation qualitatively reproduces the above described features, supporting the fractional derivative description of turbulent transport in complex flows. A quantitative approach requires extensive investigation in order to deal with the details of real cases.

Non-linear and non-local transport processes in heterogeneous media: from long-range correlated percolation to fracture and materials breakdown

We review and discuss recent progress in modelling non-linear and non-local transport processes in heterogeneous media. The non-locality that we consider is caused by long-range correlations that either exist in the morphology of the media, or are caused by the transport processes themselves. The interplay between the non-linearity and non-locality is discussed in depth with the aim of establishing that, often non-linearity and non-locality are “two sides of the same coin”, such that one may have no meaning without the presence of the other one. First, we discuss linear and scalar, but non-local transport processes and, in particular, consider those in percolation systems with long-range correlations. It appears that there are significant differences between percolative transport processes in which the long-range correlations (or the covariance function) decrease with the distance r between two points, and those in which they increase as r does. Application of this problem to flow and transport in geological formations is discussed. We then consider linear vector percolation, one type of which, the rigidity percolation, provides an example of a non-local vector transport in heterogeneous media. Applications of vector percolation to modelling elastic properties of glasses, composite solids and rock, mechanical and viscoelastic properties of polymers, and vibrations and dynamical properties of heterogeneous materials are discussed. Non-linear and non-local scalar transport processes are discussed next, including various breakdown phenomena in disordered composites, power-law transport, piecewise linear transport characterized by a threshold, and non-linear processes that arise as a result of imposing a large external potential gradient on a heterogeneous medium. Their relevance to flow of non-Newtonian fluids in porous media, to electrical currents and dielectric breakdown in composite solids and doped polycrystalline semiconductors, and several other problems is then discussed. Finally, we discuss non-linear and non-local vector transport, the most important example of which is mechanical fracture of disordered solids. We discuss exactly solvable models of fracture, quasi-static and dynamic lattice models, molecular dynamics simulations, and continuum formulation of the problem. In all cases, the predictions of the models are compared with the relevant experimental data.