Narrow Results

The diffusion of a Brownian particle exploring a meta-stable potential energy landscape with fluctuations is studied theoretically and numerically. The inverse harmonic part of the potential is replaced by a Gaussian approximated curve in the neighborhood of the saddle point and a new type of Monte-Carlo simulation algorithm is presented to simulate the random fluctuations of the potential energy. The probability of successful barrier surmounting and terminal arrival was calculated and compared with previous results. It is shown that, the replacement we have made is reasonable for the study of diffusion, which is primarily affected by the fluctuation of the barrier height, while other variations have little influence on the diffusion process. The Gaussian permutation and random Gaussian algorithms proposed here are expected to bring new ideas to the research of more complex problems.

The dynamical behaviors for a delay-diffusion housefly equation with two kinds of Dirichlet boundary conditions are considered in this paper. The existence and uniqueness of the steady state solutions are investigated, and the stability of the constant steady state solutions is obtained by using qualitative theory. The existence of Hopf bifurcation near the positive constant steady state solution is discussed and the expressions which can identify the bifurcation properties, including the stability of the bifurcating periodic solution and the bifurcation direction, are presented.

Micro-environmental acidity is a common feature of the tumor. One of the causes behind tumor acidity is lactate production by hypoxic cells of the tumor. Hypoxia is a direct result of the establishment of oxygen gradients. It is commonly observed in the tumor in an in vitro experimental setup and also in vivo situation. Here, we propose a mathematical model to analyze the production of lactate by hypoxic cells, and it is used as an alternative fuel by normoxic cells in tumor tissue in vitro and in vivo conditions. Also, we study the effects of unequal oxygen concentrations at the tumor boundaries on lactate status in the tumor. The effects of necrotic core on lactate accumulation is examined. The results are in good agreement with experimental data and are in align with the theoretical findings of previous studies. The analytical results show that lactate levels are elevated in an in vivo tumor compared to that in an in vitro tumor. Also, during the onset of necrotic core formation, the effects of necrotic core on lactate levels are noticed. Knowledge of the lactate status in a patient’s tumor may be helpful in choosing the appropriate and effective medicines for cancer treatment.

The propagation and reflection of thermo-elastic waves through diffusive nonlocal isotropic medium had been studied in this paper. The Green–Lindsay model of thermo-elasticity is incorporated in the context of Temperature Rate Dependent Theory. Using Helmholtz vector decomposition rule, the system of governing equations has been transformed to their respective components. The dispersion relation in frequency indicates the existence of three coupled waves and one independent wave propagating through the medium. The coupled waves are affected by non-locality, temperature field and diffusivity in the medium; anyhow, un-coupled shear vertical wave is only affected by the non-local parameter. The reflection of $P$-wave is also studied at the free boundary of the solid and their corresponding amplitude ratios are computed using set of suitable boundary conditions. The obtained results are further discussed graphically for significant physical parameters of interest. The results in the literature are obtained as a special case after ignoring the diffusivity in the solid.

Potential step chronoamperospectrometry (PSCAS) was carried out to analyze electron transfer in the redox reaction processes of 5,10,15,20-tetraphenylporphyrinatocobalt(II) (Co^IITPP(-2)) incorporated in a Nafion film. The reactions of Co^IITPP(-2) to [Co^IIITPP(-2)]⁺ and of [Co^IIITPP(-2)]⁺ to Co^IITPP(-2) took place through a diffusion mechanism, as confirmed by the first-order initial reaction rate with respect to the complex concentration in the matrix. However, the reaction of [Co^IIITPP(-2)]⁺ to [Co^IIITPP(-1)]²⁺ occurred by an electron-hopping mechanism, as confirmed by the second-order initial reaction rate with respect to the complex concentration. The fraction of electroactive complex (R_ct) increased with the sample time after the potential step until it reached saturation. In the reactions of Co^IITPP(-2) to [Co^IIITPP(-2)]⁺ and of [Co^IIITPP(-2)]⁺ to Co^IITPP(-2), R_ct approached 1.0, while in the reaction of [Co^IIITPP(-2)]⁺ to [Co^IIITPP(-1)]²⁺, only about 0.3 was reached. The apparent diffusion coefficient (D_app) decreased in the order of [Co^IIITPP(-2)]⁺ to Co^IITPP(-2) > Co^IITPP(-2) to [Co^IIITPP(-2)]⁺>[Co^IIITPP(-2)]⁺ to [Co^IIITPP(-1)]²⁺. The different behavior of these redox reactions was ascribed to the microenvironment of the redox species in the matrix, interaction of the redox centers, especially the product with the framework, and counter ion migration.

In living cells, the diffusion of molecules is fundamental for the proper functioning of many intracellular processes and for their correct response to external or internal signaling. In this paper, we consider the diffusion of a pivotal protein involved in bacterial chemotaxis, and show how to model and simulate the formation of concentration gradients of this protein along the bacterial cell. For this purpose, we exploit the framework of P systems to represent the partition of the bacterial volume into separate but communicating virtual volumes, and make use of communication rules to mimic the diffusive events. We simulate this system by means of the multivolume stochastic simulation algorithm τ-DPP, and present the effects of different diffusion coefficients and of a varying number of virtual volumes over the response of the chemotactic pathway.

The problems of mapping and load balancing applications on arbitrary networks are considered. A novel diffusion algorithm is presented to solve the mapping problem. It complements the well known diffusion algorithms for load balancing which have enjoyed success on massively parallel computers (MPPs). Mapping is more difficult on interconnection networks than on MPPs because of the variations which occur in network topology. Popular mapping algorithms for MPPs which depend on recursive topologies are not applicable to irregular networks. The most celebrated of these MPP algorithms use information from the Laplacian matrix of a graph of communicating processes. The diffusion algorithm presented in this paper is also derived from this Laplacian matrix. The diffusion algorithm works on arbitrary network topologies and is dramatically faster than the celebrated MPP algorithms. It is delay and fault tolerant. Time to convergence depends on initial conditions and is insensitive to problem scale. This excellent scalability, among other features, makes the diffusion algorithm a viable candidate for dynamically mapping and load balancing not only existing MPP systems but also large distributed systems like the Internet, small cluster computers, and networks of workstations.

We construct and study the stochastic force field generated by a Poisson distribution of sources at finite density, $x_1,x_2,\dots ,$ in ${ℝ}^3$ each of them yielding a long range potential $Q_i\mathrm{\Phi }(x-x_i)$ with possibly different charges $Q_i\in ℝ$. The potential $\mathrm{\Phi }$ is assumed to behave typically as $|x{|}^{-s}$ for large $\left|x\right|$, with $s>1/2$. We will denote the resulting random field as “generalized Holtsmark field”. We then consider the dynamics of one tagged particle in such random force fields, in several scaling limits where the mean free path is much larger than the average distance between the scatterers. We estimate the diffusive time scale and identify conditions for the vanishing of correlations. These results are used to obtain appropriate kinetic descriptions in terms of a linear Boltzmann or Landau evolution equation depending on the specific choices of the interaction potential.

In this pedagogical review, we summarize the mathematical basis and practical hints for the explicit analytical computation of spectral sums that involve the eigenvalues of the Laplace operator in simple domains such as $d$-dimensional balls (with $d=1,2,3$), an annulus, a spherical shell, right circular cylinders, rectangles and rectangular cuboids. Such sums appear as spectral expansions of heat kernels, survival probabilities, first-passage time densities, and reaction rates in many diffusion-oriented applications. As the eigenvalues are determined by zeros of an appropriate linear combination of a Bessel function and its derivative, there are powerful analytical tools for computing such spectral sums. We discuss three main strategies: representations of meromorphic functions as sums of partial fractions, Fourier–Bessel and Dini series, and direct evaluation of the Laplace-transformed heat kernels. The major emphasis is put on a pedagogic introduction, the practical aspects of these strategies, their advantages and limitations. The review gathers many summation formulas for spectral sums that are dispersed in the literature.

Fossil fragments of bone and teeth that are exposed to a humid environment take up fluorine from the surrounding soil and accumulate it in their mineral phase. In cortical parts of long bone diaphysis a fluorine concentration profile can be observed, which carries information on the exposure duration of the buried object in its shape. The distribution of fluorine in a sample however is strongly influenced by environmentally induced processes of bone diagenesis, i.e. alteration in the structure and composition of bone mineral and degradation of the organic components that may make the time information indistinct.

PIGE (Proton Induced Gamma-ray Emission) is a precise and fast analytical technique to determine the quantitative fluorine content and its distribution in cross sections of bone and tooth specimen non-destructively. The simultaneous detection of Ca by PIXE (Proton Induced X-ray Emission) provides additional information on the sample topography. Cracks, alteration haloes and the porosity, which is typical for human bone samples, are parameters which have direct influence on the fluorine uptake and transport during burial. This contribution outlines the combined approach of using PIGE and PIXE measurement to understand some aspects of the complex environmental impact that impedes exposure age dating by fluorine diffusion profiling.

We have developed an analytical model for polycrystalline-based organic thin-film transistors (OTFTs) that employs, as far as possible, new concepts on carrier injection to the conventional polysilicon model. The drain current equations, both in diffusion and drift regimes, predict the voltage and temperature dependencies on the various device and circuit parameters. Interestingly, upon direct comparison with previously developed disordered model, similarities between the two are not thought to be coincidental. The effect of gate voltage on surface potential is affected by the Fermi level pinning in the grain boundary, which is assumed to consist of mainly disordered material. This work also highlights the problem of using drift mobility, as an organic circuit design parameter, and consequently alternative quantities are proposed for simpler circuits such as an inverter. Upon validation of the model, relatively good fits are obtained with the experimental data on TIPS-based TFTs. The divergence at low drain voltages are thought to be associated with short channel and/or high contact resistance effects.

Diffusion on 2D site percolation clusters at p = 0.7, 0.8, and 0.9 above p_c on the square lattice in the presence of two crossed bias fields, a local bias B and a global bias E, has been investigated. The global bias E is applied in a fixed global direction whereas the local bias B imposes a rotational constraint on the motion of the diffusing particle. The rms displacement R_t ~ t^k in the presence of both biases is studied. Depending on the strength of E and B, the behavior of the random walker changes from diffusion to drift to no-drift or trapping. There is always diffusion for finite B with no global bias. A crossover from drift to no-drift at a critical global bias E_c is observed in the presence of local bias B for all disordered lattices. At the crossover, value of the rms exponent changes from k = 1 to k < 1, the drift velocity v_t changes from constant in time t to decreasing power law nature, and the "relaxation" time τ has a maximum rate of change with respect to the global bias E. The value of critical bias E_c depends on the disorder p as well as on the strength of local bias B. Phase diagrams for diffusion, drift, and no-drift are obtained as a function of bias fields E and B for these systems.

The vacancy mechanism is simulated by means of Monte Carlo (MC) method. In this model, the impurity diffusion occurs by migration of substitutional atoms B via the exchange with vacancies whose frequencies near a solute atom differ from a free vacancy. Whenever a defect leaves the lattice, periodic boundary conditions are made to bring it into the lattice. The solute concentration profiles are given using a technique developed by Murch which has been shown to be equivalent to a finite source. The fit of these profiles allows the comparison between our results and analytical solutions. The parameters extracted from a Gaussian function fit which agrees well with numerical profiles are in very good quantitative agreement with theoretical predictions.

Tracer diffusion and fluid transport are studied in a model for a geomarine system in which fluid constituents move from regions of high to low concentration. An interacting lattice gas is used to model the system. Collective diffusion of fluid particles in lattice gas is consistent with the solution of the continuum diffusion equation for the concentration profile. Comparison of these results validates the applicability and provides a calibration for arbitrary (time and length) units of the lattice gas. Unlike diffusive motion in an unsteady-state regime, both fluid and tracer exhibit a drift-like transport in a steady-state regime. The transverse components of fluid and tracer displacements differ significantly. While the average tracer motion becomes nondiffusive in the long time regime, the collective motion exhibits an onset of oscillation.

The method employed by Chandrasekhar for the astronomical bodies is brought down here to study the diffusion in Silicon. A continuous position probability density for the diffusing particle, ω(r, t) representing the position of the diffusing particle at any time t, is used in the evaluation of the diffusion constant. The results agree reasonably well with the available experimental and theoretically reported values. The existence of "traps" in the semiconducting systems has been clearly brought out by this simulation technique.

We present the results from simulation studies of evaporation of a single fluid in a capillary porous medium. Employing a three-dimensional site-bond correlated network model to represent a porous medium, namely Clashac sandstone, we analyze different aspects of the phase distribution by evaporation of a single fluid in the porous medium. As a direct consequence of the porous medium utilized, we analyze the influence of a strongly disordered porous media with a broad range of pore and throat size distributions in the evaporation process. Experimental data togheter with throat and pore size distributions were used to build and match the network model, allowing us to determine the porosimetric curve for the Clashac sandstone for different degrees of correlation. Also, the correlation length was obtained from the percolation theory. In our case study the evaporation process modeled was insensitive to the different degrees of correlation that might occur between pores and throats. In addition, it was observed that the evaporation pattern was the same for all analyzed networks above the correlation length.

A formulation of bit-string models of language evolution, based on differential equations for the population speaking each language, is introduced and preliminarily studied. Connections with replicator dynamics and diffusion processes are pointed out. The stability of the dominance state, where most of the population speaks a single language, is analyzed within a mean-field-like approximation, while the homogeneous state, where the population is evenly distributed among languages, can be studied. This analysis discloses the existence of a bistability region, where dominance coexists with homogeneity as possible asymptotic states. Numerical resolution of the differential system validates these findings.

We present a new algorithm to implement Dirichlet boundary conditions for diffusive processes in arbitrarily complex geometries. In this approach, the boundary conditions around the diffusing object is replaced by the fictitious phase transition of a pure substance where the energy cost of the phase transition largely overwhelms the amount of energy stored in the system. The computing cost of this treatment of the boundary condition is independent of the topology of the boundary. Moreover, the implementation of this new approach is straightforward and follows naturally from enthalpy-based numerical methods. This algorithm is compatible with a wide variety of discretization methods, finite differences, finite volume, lattice Boltzmann methods and finite elements, to cite a few. We show, here, using both lattice Boltzmann and finite-volume methods that our model is in excellent agreement with analytical solutions for high symmetry geometries. We also illustrate the advantages of the algorithm to handle more complex geometries.

Why does diffusion sometimes show cascade phenomena but at other times is impeded? In addressing this question, we considered a threshold model of diffusion, focusing on the formation of a critical mass, which enables diffusion to be self-sustaining. Performing an agent-based simulation, we found that the diffusion model produces only two outcomes: Almost perfect adoption or relatively few adoptions. In order to explain the difference, we considered the various properties of network structures and found that the manner in which thresholds are arrayed over a network is the most critical factor determining the size of a cascade. On the basis of the results, we derived a threshold arrangement method effective for generation of a critical mass and calculated the size required for perfect adoption.

The dissolution and diffusion behaviors of H in the four low-Miller-index W surfaces ((110), (112), (100) and (111)) are systematically studied by the density functional theory approach to understand the orientation dependence of the H bubble distribution on surface. The results show that H accumulation on surface is influenced by H diffusion barrier as well as vacancy and H formation. The barriers of diffusion towards surfaces are larger than that in bulk. It indicates that H is prone to diffuse into the deep in bulk once H dissolves in surface. H is preferred to accumulate on the W(111) surface due to the lower formation energies of vacancy and H comparing to that in bulk. However, W(110) is the resistant surface for forming H bubble due to the higher formation energies of vacancy and H. The results are helpful for understanding the orientation dependence of surface damages on W surface and designing new plasma-facing materials.