Asymptotic Diffusion Research Articles

Universal superdiffusion of random walks in media with embedded fractal networks of low diffusivity.

Diffusion in composite media with high contrasts between diffusion coefficients in fractal sets of inclusions and in their embedding matrices is modeled by lattice random walks (RWs) with probabilities p<1 of hops from fractal sites and 1 from matrix sites. Superdiffusion is predicted in time intervals that depend on p and with diffusion exponents that depend on the dimensions of matrix (E) and fractal (D_{F}) as ν=1/(2+D_{F}-E). This contrasts with the nonuniversal subdiffusion of RWs confined to fractal media. Simulations with four fractals show the anomaly at several time decades for p≲10^{-3} and the crossover to the asymptotic normal diffusion. These results show that superdiffusion can be observed in isotropic RWs with finite moments of hop length distributions and allow the estimation of the dimension of the inclusion set from the diffusion exponent. However, displacements within single trajectories have normal scaling, which shows transient ergodicity breaking.

Diffusion of noiseless active particles in a planar convection array.

We investigated, both analytically and numerically, the dynamics of a noiseless overdamped active particle in a square lattice of planar counter-rotating convection rolls. Below a first threshold of the self-propulsion speed, a fraction of the simulated particle's trajectories spatially diffuse around the convection rolls, whereas the remaining trajectories remain trapped inside the injection roll. We detected two chaotic diffusion regimes: (i) below a second, higher threshold of the self-propulsion speed, the particle performs a random motion characterized by asymptotic normal diffusion. Long superdiffusive transients were observed for vanishing small self-propulsion speeds. (ii) above that threshold, the particle follows chaotic running trajectories with speed and orientation close to those of the self-propulsion vector at injection and its dynamics is superdiffusive. Chaotic diffusion disappears in the ballistic limit of extremely large self-propulsion speeds.

An asymptotically correct implicit-explicit time integration scheme for finite volume radiation-hydrodynamics.

Numerical radiation-hydrodynamics (RHD) for non-relativistic flows is a challenging problem because it encompasses processes acting over a very broad range of time-scales, and where the relative importance of these processes often varies by orders of magnitude across the computational domain. Here, we present a new implicit-explicit method for numerical RHD that has a number of desirable properties that have not previously been combined in a single method. Our scheme is based on moments and allows machine-precision conservation of energy and momentum, making it highly suitable for adaptive mesh refinement applications; it requires no more communication than hydrodynamics and includes no non-local iterative steps, making it highly suitable for massively parallel and Graphics Processing Unit (GPU)-based systems where communication is a bottleneck; and we show that it is asymptotically accurate in the streaming, static diffusion, and dynamic diffusion limits, including in the so-called asymptotic diffusion regime where the computational grid does not resolve the photon mean-free path. We implement our method in the GPU-accelerated RHD code quokka and show that it passes a wide range of numerical tests.

Thermodynamic uncertainty relations in the presence of non-linear friction and memory

A new thermodynamic uncertainty relation (TUR) is derived for systems described by linearly coupled Langevin equations in the presence of non-linear frictional forces. In our scheme, the main variable represents the velocity of a particle, while the other coupled variables describe memory effects which may arise from strongly correlated degrees of freedom with several time-scales and, in general, are associated with thermal baths at different temperatures. The new TUR gives a lower bound for the mean-squared displacement of the position of the particle, including its asymptotic diffusion coefficient. This bound, in several examples worked out here, appears to be a good analytical estimate of the real diffusion coefficient. The new TUR can be also applied in the absence of any external force (with or without thermal equilibrium between the baths), a case which usually goes beyond the scope of original TURs. We show applications to non-linear frictional models with memory, such as the Coulomb and the Prandtl-Tomlinson models, usually representative of friction at the nano-scale and within atomic-force microscopy experiments.

Asymptotic diffusion analysis of the retrial queuing system with feedback and batch Poisson arrival

The mathematical model of the retrial queuing system \(M^{[n]}/M/1\) with feedback and batch Poisson arrival is constructed. Customers arrive in groups. If the server is free, one of the arriving customers starts his service, the rest join the orbit. The retrial and service times are exponentially distributed. The customer whose service is completed leaves the system, or reservice, or goes to the orbit. The method of asymptotic diffusion analysis is proposed for finding the probability distribution of the number of customers in orbit. The asymptotic condition is growing average waiting time in orbit. The accuracy of the diffusion approximation is obtained.

Asymptotic Diffusion Method for Retrial Queues with State-Dependent Service Rate

In this paper, we consider a retrial queue with a state-dependent service rate as a mathematical model of a node of FANET communications. We suppose that the arrival process is Poisson, the delay duration is exponentially distributed, the orbit is unlimited, and there is multiple random access from the orbit. There is one server, and the service time of every call is distributed exponentially with a variable parameter depending on the number of calls in the orbit. The service rate has an infinite number of values. We propose the asymptotic diffusion method for the model study. The asymptotic diffusion approximation of the probability distribution of the number of calls in the orbit is derived. Some numerical examples are demonstrated.

Anomalous transport in periodic photonic chains with designed loss

Here we show that a coherent random walk in a perfectly periodic chain of bosonic modes with designed loss can exhibit a variety of different anomalous transfer regimes in dependence on the initial state of the chain. In particular, for any given finite initial time-interval there is a set of initial states leading to a hyperballistic transport regime. Also, there are initial states allowing one to achieve a subdiffusive regime or even localization for a given time-interval, or change an asymptotic long-time diffusion rate. We show how these anomalous transport regimes can be practically realized in a laser-written network of single-mode waveguides in balk glass or how a planar system of coupled single-mode waveguides can be realized with an integrated photonic platform.

Cooperation and competition in the collective drive by motor proteins: mean active force, fluctuations, and self-load.

We consider the dynamics of a bio-filament under the collective drive of motor proteins. They are attached irreversibly to a substrate and undergo stochastic attachment-detachment with the filament to produce a directed force on it. We establish the dependence of the mean directed force and force correlations on the parameters describing the individual motor proteins using analytical theory and direct numerical simulations. The effective Langevin description for the filament motion gives mean-squared displacement, asymptotic diffusion constant, and mobility leading to an effective temperature. Finally, we show how competition between motor protein extensions generates a self-load, describable in terms of the effective temperature, affecting the filament motion.

Asymptotic Diffusion Analysis of Retrial Queueing System M/M/1 with Impatient Customers, Collisions and Unreliable Servers

In this paper, a retrial queueing system of the M/M/1 type with Poisson flows of arrivals, impatient customers, collisions, and an unreliable service device is considered. To make the problem more realistic and, hence, more complicated, we include the breakdowns and repairs of the service in this research study. The retrial times of customers in the orbit, service time, impatience time of customers in the orbit, server’s lifetime (depending on whether it is idle or busy), and server recovery time are supposed to be exponentially distributed. The problem of finding the stationary probability distribution of the number of customers in orbit is solved by using the method of asymptotic diffusion analyses under the condition of a heavy load of the system and the patience of customers in orbit. Numerical results are presented that demonstrate the effectiveness of the obtained theoretical conclusions, and a comparative analysis of the method of asymptotic analysis and the method of asymptotic diffusion analysis for the considered problem is given.

Extension and dynamical phases in random walkers depositing and following chemical trails

Active walker models have proved to be extremely effective in understanding the evolution of a large class of systems in biology like ant trail formation and pedestrian trails. We propose a simple model of a random walker which modifies its local environment that in turn influences the motion of the walker at a later time. We perform direct numerical simulations of the walker in a discrete lattice with the walker actively depositing a chemical which attracts the walker trajectory and also evaporates in time. We propose a method to look at the structural transitions of the trajectory using radius of gyration for finite time walks. The extension over a definite time window shows a non-monotonic change with the deposition rate characteristic of a coil-globule transition. At certain regions of the parameter space of the chemical deposition and evaporation rates, the extensions of the walker shows a re-entrant behavior. The dynamics, characterised by the mean-squared displacement, shows deviation from diffusive scaling at intermediate time scales, returning to diffusive behavior asymptotically. A mean-field theory captures the variation of the asymptotic diffusivity.

Smoothness of the diffusion coefficients for particle systems in continuous space

For a class of particle systems in continuous space with local interactions, we show that the asymptotic diffusion matrix is an infinitely differentiable function of the density of particles. Our method allows us to identify relatively explicit descriptions of the derivatives of the diffusion matrix in terms of correctors.

Asymptotic streamline diffusion finite element method for singularly perturbed convection–diffusion differential difference equations

In this article, we presented an asymptotic streamline diffusion finite element method (SDFEM) for singularly perturbed convection–diffusion‐type differential difference equations. Using the asymptotic expansion approximation, the differential difference equations are converted into nondelay differential equations. A SDFEM is applied to the modified equation. Further, we proved that the method gives an almost second‐order convergence in maximum norm, in L2‐norm whereas first‐order convergence in H1‐norm. Numerical results are presented to validate the theoretical results.

Anomalous diffusion in single and coupled standard maps with extensive chaotic phase spaces

We investigate the long-term diffusion transport and chaos properties of single and coupled standard maps. We consider model parameters that are known to induce anomalous diffusion in the maps’ phase spaces, as opposed to normal diffusion which is associated with Gaussian distribution properties of the kinematic variables. This type of transport originates in the presence of the so-called accelerator modes, i.e. non-chaotic initial conditions which exhibit ballistic transport, which also affect the dynamics in their vicinity. We first systematically study the dynamics of single standard maps, investigating the impact of different ensembles of initial conditions on their behavior and asymptotic diffusion rates, as well as on the respective time-scales needed to acquire these rates. We consider sets of initial conditions in chaotic regions enclosing accelerator modes, which are not bounded by invariant tori. These types of chaotic initial conditions typically lead to normal diffusion transport. We then setup different arrangements of coupled standard maps and investigate their global diffusion properties and chaotic dynamics. Although individual maps bear accelerator modes causing anomalous transport, the global diffusion behavior of the coupled system turns out to depend on the specific configuration of the imposed coupling. Estimating the average diffusion properties for ensembles of initial conditions, as well as measuring the strength of chaos through computations of appropriate indicators, we find conditions and systems’ arrangements which systematically favor the suppression of anomalous transport and long-term convergence to normal diffusion rates.

Two-Group Radiative Transfer Benchmarks for the Non-Equilibrium Diffusion Model

Semi-analytic solutions for high-energy density radiation diffusion problems in slab geometry using a two-group model for the frequency (photon energy) variable are presented. To obtain these solutions we specify forms for the heat capacity and emissivity in the high energy group that are a function of the fraction of radiation emission in the low energy group in order to linearize the problem. This results in a linear system of equations that are solved via Laplace and Fourier transforms: the Laplace transform is inverted analytically and the inverse Fourier transform is computed using numerical integration. It is demonstrated that these solutions can be useful in verifying codes for solving the radiation diffusion equations in the high-energy density regime. Additionally, we include solutions for an optically thick problem that can be used to test the asymptotic diffusion limit of codes solving a transport model for radiative transfer.

An asymptotic streamline diffusion finite element method for singularly perturbed convection‐diffusion delay differential equations with point source

In this article, we presented an asymptotic SDFEM for singularly perturbed convection diffusion type differential difference equations with point source term. First, the solution is decomposed into two functions, among them one is the solution of delay differential equation and other one is the solution of differential equation with point source. Furthermore, using the asymptotic expansion approximation, the delay differential equation is modified as a nondelay differential equations. Streamline diffusion finite element methods are applied to approximate the solutions of the two problems. We prove that the present method gives an almost second-order convergence in maximum norm and square integrable norm, whereas first-order convergence in H 1 norm. Numerical results are presented to validate the theoretical results.

On a characteristic method for the SN neutron transport equation

We present a quadratic characteristic (QC) method for solving the SN neutron transport equation. In this QC method, the SN equation is integrated analytically on each gird point based on the method of characteristics, in which the scattering source and the cell boundary incoming angular flux are approximated with second-order polynomial functions, respectively. Numerical experiments show that the QC method is very robust and can achieve second-order accuracy and it possesses the asymptotic diffusion limit.

Multi-group discontinuous asymptotic [formula omitted] approximation in radiative Marshak waves experiments

We study the propagation of radiative heat (Marshak) waves, using modified P1-approximation equations. When the heat wave propagation is supersonic, i.e. propagates faster than the sound velocity, hydrodynamic motion is negligible, and the wave can be described by the radiative transfer Boltzmann equation, coupled with the material energy equation. However, the exact thermal radiative transfer problem is still difficult to solve and requires massive simulation capabilities. Hence, there still exists a need for adequate approximations that are comparatively easy to carry out. Classic approximations, such as the classic diffusion and classic P1, fail to describe the correct heat wave velocity, when the optical depth is not sufficiently high. Therefore, we use the recently developed discontinuous asymptotic P1 approximation, which is a time-dependent analogy for the adjustment of the discontinuous asymptotic diffusion for two different zones. This approximation was tested via several benchmarks, showing better results than other common approximations, and has also demonstrated a good agreement with a main Marshak wave experiment and its Monte-Carlo gray simulation. Here we derive energy expansion of the discontinuous asymptotic P1 approximation in slab geometry, and test it with numerous experimental results for propagating Marshak waves inside low density foams. The new approximation describes the heat wave propagation with good agreement. Furthermore, a comparison of the simulations to exact implicit Monte-Carlo slab-geometry multi-group simulations, in this wide range of experimental conditions, demonstrates the superiority of this approximation to others.

Simulations of hydrogen outgassing from a carbon fiber electrode

Outgassing remains a pertinent issue in high-power systems as it can lead to effects such as breakdown, surface flashover, and pulse shortening and is typically the first stage of deleterious plasma formation. In this context, experimental reports suggest that carbon fibers (CFs) may likely be a superior cathode material for low outgassing. Here, model-based assessments of outgassing from CFs are performed based on molecular dynamics simulations. Carbon fibers were generated based on interconnection of an array of graphene sheets resembling ladder-like structures. Our results of temperature-dependent diffusion coefficients for hydrogen in CFs are shown to exhibit Arrhenius behavior and have values smaller than copper by factors of 15.5 and 86.8 at 400 K and 1000 K, respectively. This points to even stronger improvements for operation at high temperatures, with the asymptotic diffusion constant ratio predicted to be ∼187. With reduced outgassing, higher temperature operation, and durability, our results support CF cathodes as an excellent choice for cathode material in high-power microwave and pulsed power systems.

Discontinuous Galerkin solutions for elliptic problems on polygonal grids using arbitrary-order Bernstein-Bézier functions

In this paper, we present a symmetric interior penalty discontinuous Galerkin finite element discretization for the numerical solution of second-order elliptic partial differential equations on general polygonal meshes using a reduced-space polygonal Bernstein-Bézier functions. They form a space of interpolatory functions (vice local polynomial functions), which satisfy the Lagrange property at their interpolation points. This can enable the use of efficient DGFEM physics-based diffusion preconditioners for the first-order linear transport equation on arbitrary polygons in the computationally-challenging asymptotic diffusion limit. On a polygonal element with n vertices and polynomial degree p, there are n vertex functions, (p−1) functions per edge, and (p−1)(p−2)/2 interior functions that span the {xayb}(a+b)≤p space of bivariate monomials. Numerical experiments highlighting the performance of the proposed method are presented through uniform and adaptive h, p, and hp refinement strategies. Appropriate error rates are observed including hp adaptivity yielding exponential convergence in the presence of singularities in the solution.

Empirical model of magnetic field line spreading in isotropic turbulence with varying mean field

In many astrophysical phenomena, understanding the diffusion of the magnetic eld line random walk (FLRW) is central to understand cosmic ray transport. In 3D uctuations, the behavior of the FLRW can be characterized by the Kubo number R = (b/B 0)(l ⫽⃥ /l ∧ ) [1], where the parameters b and B 0 are the rms magnetic uctuation and the large-scale mean eld, respectively. The parameters l ∥ and l ∧ are coherence scales parallel and perpendicular to B 0, respectively. For isotropic turbulence, in which l ∥ = l ∧ , Sonsrettee et al. [2] found that Corrsin-based theories can be applied to study the FLRW’s behavior for a whole range of R by varying b/B 0. Sonsrettee et al. [2] used Corrsin-based theory with three models of eld line spreading to examine the R-scaling of the asymptotic diffusion coefficients for the FLRW. The models are the diffusive decorrelation (DD) model [3{5], the random ballistic decorrelation (RBD) model [6], and the ordinary differential equation (ODE) model [7]. To improve the theory of the FLRW in isotropic turbulence with B 0 = 0, Sonsrettee [8] proposed the empirical (EMP) model of magnetic eld line spreading to determine the asymptotic diffusion coefficients. Benchmarked against the previous models, the EMP model is the best model to predict computer simulation results (with ≤ 0:9% error). In this work, we extend the previous works [2, 8] by formulating the EMP model to explore the R scaling FLRW behavior in isotropic magnetic turbulence by varying B 0. In the limit of very low R, we obtain the the closed-form solution of the FLRW for the EMP model. In order to develop the closed-form solution at any R, we employ the Padé approximants to the EMP model. The EMP model predicts that, with increasing R, the FLRW behavior transits from quasilinear diffusion to Bohm diffusion. This work shows that the theoretical results of the EMP model match the computer simulation results for the FLRW in Kolmogorov turbulence better than the other models significantly.