Anomalous dispersion in correlated porous media: a coupled continuous time random walk approach

We study the causes of anomalous dispersion in Darcy-scale porous media characterized by spatially heterogeneous hydraulic properties. Spatial variability in hydraulic conductivity leads to spatial variability in the flow properties through Darcy’s law and thus impacts on solute and particle transport. We consider purely advective transport in heterogeneity scenarios characterized by broad distributions of heterogeneity length scales and point values. Particle transport is characterized in terms of the stochastic properties of equidistantly sampled Lagrangian velocities, which are determined by the flow and conductivity statistics. The persistence length scales of flow and transport velocities are imprinted in the spatial disorder and reflect the distribution of heterogeneity length scales. Particle transitions over the velocity length scales are kinematically coupled with the transition time through velocity. We show that the average particle motion follows a coupled continuous time random walk (CTRW), which is fully parameterized by the distribution of flow velocities and the medium geometry in terms of the heterogeneity length scales. The coupled CTRW provides a systematic framework for the investigation of the origins of anomalous dispersion in terms of heterogeneity correlation and the distribution of conductivity point values. We derive analytical expressions for the asymptotic scaling of the moments of the spatial particle distribution and first arrival time distribution (FATD), and perform numerical particle tracking simulations of the coupled CTRW to capture the full average transport behavior. Broad distributions of heterogeneity point values and lengths scales may lead to very similar dispersion behaviors in terms of the spatial variance. Their mechanisms, however are very different, which manifests in the distributions of particle positions and arrival times, which plays a central role for the prediction of the fate of dissolved substances in heterogeneous natural and engineered porous materials.     

On the acoustic boundary condition in the presence of flow

The boundary condition on the acoustic perturbation velocity at an impermeable surface in a flow is considered for the cases in which the surface generates a sound field by vibration or is acoustically deformed by an incident sound field. It is shown that in general the condition is not equivalent to the requirement of continuity of acoustic particle displacement in the direction normal to the unperturbed surface.     

Extremal behavior of a coupled continuous time random walk

Coupled continuous time random walks (CTRWs) model normal and anomalous diffusion of random walkers by taking the sum of random jump lengths dependent on the random waiting times immediately preceding each jump. They are used to simulate diffusion-like processes in econophysics such as stock market fluctuations, where jumps represent financial market microstructure like log returns. In this and many other applications, the magnitude of the largest observations (e.g. a stock market crash) is of considerable importance in quantifying risk. We use a stochastic process called a coupled continuous time random maxima (CTRM) to determine the density governing the maximum jump length of a particle undergoing a CTRW. CTRM are similar to continuous time random walks but track maxima instead of sums. The many ways in which observations can depend on waiting times can produce an equally large number of CTRM governing density shapes. We compare densities governing coupled CTRM with their uncoupled counterparts for three simple observation/wait dependence structures.     

Excitable dynamics in the presence of time delay

The spiking properties of a subcritical Hopf oscillator with a time delayed nonlinear feedback is investigated. Finite time delay is found to significantly affect both the statistics and the fine structure of the spiking behavior. These dynamical changes are explained in terms of the fundamental modifications occurring in the bifurcation scenario of the system. Our mathematical model can find useful applications in understanding the dynamical behavior of various real life excitable systems where propagation delay effects are ubiquitous.     

Vacuum energy and Casimir force in the presence of a dimensional parameter in the boundary condition

The Hamiltonian for a scalar field that satisfies the boundary condition ?? _n ? = (1/δ)? must include a surface potential energy. The corresponding term in the Casimir energy $\tilde E_C$ proves to be a leading one when the dimension of the region is l～δ. The energy $\tilde E_C$ does not involve arbitrariness associated with regularization and is an unambiguously determined function of the field mass m, the size l, and the penetration depth δ. The inclusion of the surface term is of importance for ensuring that the derivative ?? $\tilde E_C$ /?l is equal to the ll component of the vacuum energy-momentum tensor. The Casimir energy $\tilde E_C$ is related to its volume component E _C by a Legendre transformation where the quantity conjugate to 1/δ is the product of the vacuum surface energy and δ. If δ is negative and if ?/mc>|δ|, there exists a critical value l=l _c (δ) above which (l>l _c ) the vacuum is unstable; if a self-interaction of the form ? ⁴ is taken into account, this will lead to a phase transition accompanied by the formation of a condensate of the field ?. If δ=+0 or ∞ and if the dimensionalities are even, it is possible to construct a vacuum energy-momentum tensor (not only energy) that is finite over the entire space. Specially chosen counterterms leave unchanged the analytic dependence of the vacuum energy on the dimensionality of space and the character of the coordinate dependence of the energy density for x>?/mc.     

Analysis of wave fluctuation phenomena in random media in presence of a reflecting boundary

Modelling of wave propagation for modern high-frequency radio links must take into account the fluctuation phenomena caused by random changes of the medium's parameters. There are two factors that cause signals from transmitter to receiver to arrive via multiple random paths. The first is a result of scattering by boundaries and various obstacles, while the second is related to random changes in the refractive index of the medium. Similar phenomena are observed in underwater acoustic propagation. In this work the influence of both factors is demonstrated in a simple example when the propagation takes place above a perfectly reflecting surface in the presence of a randomly fluctuating medium.     

A random walker on a ratchet potential: effect of a non Gaussian noise

We analyze the effect of a colored non Gaussian noise on a model of a random walker moving along a ratchet potential. Such a model was motivated by the transport properties of motor proteins, like kinesin and myosin. Previous studies have been realized assuming white noises. However, for real situations, in general we could expect that those noises be correlated and non Gaussian. Among other aspects, in addition to a maximum in the current as the noise intensity is varied, we have also found another optimal value of the current when departing from Gaussian behavior. We show the relevant effects that arise when departing from Gaussian behavior, particularly related to current's enhancement, and discuss its relevance for both biological and technological situations.     

Numerical study of premixed flame dynamics in a closed tube: Effect of wall boundary condition

Numerical simulations were conducted to study the dynamics of premixed flames propagating in a closed tube by solving the fully compressible reactive Navier–Stokes equations using a high-order numerical method on a dynamically adapting grid. A simplified chemical-diffusive model was used to describe the reactions and energy release in a stoichiometric hydrogen-air mixture. The influence of wall boundary condition on the flame dynamics was explored by considering three different types of condition on the walls: adiabatic no-slip, adiabatic free-slip, and isothermal. The calculations show that the wall boundary condition has a significant effect on the generation and amplification of pressure waves and consequently on the flame dynamics. In the early stages of flame propagation, the flame behaves in a similar manner for different boundary conditions, that is, the flame develops a tulip shape that further evolves into a distorted tulip flame (DTF) through Rayleigh-Taylor instability arising from acoustic-flame interaction. Significant differences, however, arise after DTF formation in the late stages, especially when the primary acoustic wave is amplified to form a shock wave in the adiabatic free-slip and isothermal cases. The shock-flame interactions facilitate the formation of a series of increasingly corrugated flames by triggering the Richtmyer–Meshkov instabilities. The way how the lateral flame fronts touch the tube sidewalls to generate the primary acoustics and the heat conduction through the tube sidewalls play an important role in the generation and amplification of the pressure waves.     

Origins and applications of the Montroll-Weiss continuous time random walk

The Continuous Time Random Walk (CTRW) was introduced by Montroll and Weiss in 1965 in a purely mathematical paper. Its antecedents and later applications beginning in 1973 are discussed, especially for the case of fractal time where the mean waiting time between jumps is infinite.     

无标度立体Koch网络上随机游走的平均吸收时间

作为一种基本的动力学过程,复杂网络上的随机游走是当前学术界研究的热点问题,其中精确计算带有陷阱的随机游走过程的平均吸收时间(mean trapping time,MTT)是该领域的一个难点.这里的MTT定义为从网络上任意一个节点出发首次到达设定陷阱的平均时间.本文研究了无标度立体Koch网络上带有一个陷阱的随机游走问题,解析计算了陷阱置于网络中度最大的节点这一情形的网络MTT指标.通过重正化群方法,利用网络递归生成的模式,给出了立体Koch网络上MTT的精确解,所得计算结果与数值解一致,并且从所得结果可以看出,立体Koch网络的MTT随着网络节点数N呈线性增长.最后,将所得结果与之前研究的完全图、规则网络、Sierpinski网络和T分形网络进行比较,结果表明Koch网络具有较高的传输效率.     

Exact determination of all ground states of random field systems in polynomial time

An algorithm is developed which allows calculating all ground states of ferromagnetic and unfrustrated antiferromagnetic Ising systems with arbitrary site-dependent fields by transforming the system into an equivalent network and calculating the maximal flow. By a trial and error scheme a minimum cut is constructed which corresponds to a spin configuration. In this way each ground state is calculated with a finite probability. The algorithm is applied to site-diluted antiferromagnets in external magnetic fields. It is found that in this case its time complexity is approximately quadratic in the lattice size. As an application we calculate the distribution of overlaps between ground states of the site-diluted antiferromagnet in a strong magnetic field and we analyse the fractal structure of these ground states.     

Exact field-driven interface dynamics in the two-dimensional stochastic Ising model with helicoidal boundary conditions

We investigate the interface dynamics of the two-dimensional stochastic Ising model in an external field under helicoidal boundary conditions. At sufficiently low temperatures and fields, the dynamics of the interface is described by an exactly solvable high-spin asymmetric quantum Hamiltonian that is the infinitesimal generator of the zero range process. Generally, the critical dynamics of the interface fluctuations is in the Kardar–Parisi–Zhang universality class of critical behavior. We remark that a whole family of RSOS interface models similar to the Ising interface model investigated here can be described by exactly solvable restricted high-spin quantum XXZ$XXZ$-type Hamiltonians.     

Exact scaling for the mean first-passage time of random walks on a generalized Koch network with a trap

In this paper, we study the scaling for the mean first-passage time (MFPT) of the random walks on a generalized Koch network with a trap. Through the network construction, where the initial state is transformed from a triangle to a polygon, we obtain the exact scaling for the MFPT. We show that the MFPT grows linearly with the number of nodes and the dimensions of the polygon in the large limit of the network order. In addition, we determine the exponents of scaling efficiency characterizing the random walks. Our results are the generalizations of those derived for the Koch network, which shed light on the analysis of random walks over various fractal networks.     

Exact scaling for the mean first-passage time of random walks on a generalized Koch network with a trap

In this paper,we study the scaling for the mean first-passage time(MFPT) of the random walks on a generalized Koch network with a trap.Through the network construction,where the initial state is transformed from a triangle to a polygon,we obtain the exact scaling for the MFPT.We show that the MFPT grows linearly with the number of nodes and the dimensions of the polygon in the large limit of the network order.In addition,we determine the exponents of scaling efficiency characterizing the random walks.Our results are the generalizations of those derived for the Koch network,which shed light on the analysis of random walks over various fractal networks.     

Statistical properties of the acoustic field of a moving source in a random inhomogeneous medium in the presence of a rough boundary

Institute of Applied Physics, Russian Academy of Sciences, Nizhegorodskii State University. Translated from Izvestiya Vysshikh Uchebnykh Zavedenii, Radiofizika, Vol. 35, No. 2, pp. 161–170, February, 1992.