Resonance phenomena and long-term chaotic advection in volume-preserving systems

Creating chaotic advection is the most efficient strategy to achieve mixing on microscale or in very viscous fluids. In this paper, we present a quantitative theory of the long-time resonant mixing in 3D near-integrable flows. We use the flow between two coaxial elliptic counter-rotating cylinders as a demonstrative model, where multiple scatterings on resonance result in mixing by causing the jumps of adiabatic invariants. We improve the existing estimates of the width of the mixing domain. We show that the resulting mixing both on short and long time scales can be described in terms of a single diffusion-type equation with a diffusion coefficient depending on the averaged effect of multiple passages through resonances. We discuss the exact location of the boundaries of the chaotic domain and show how it affects the properties of mixing.     

A stochastic multiscale framework for modeling flow through random heterogeneous porous media

Flow through porous media is ubiquitous, occurring from large geological scales down to the microscopic scales. Several critical engineering phenomena like contaminant spread, nuclear waste disposal and oil recovery rely on accurate analysis and prediction of these multiscale phenomena. Such analysis is complicated by inherent uncertainties as well as the limited information available to characterize the system. Any realistic modeling of these transport phenomena has to resolve two key issues: (i) the multi-length scale variations in permeability that these systems exhibit, and (ii) the inherently limited information available to quantify these property variations that necessitates posing these phenomena as stochastic processes.A stochastic variational multiscale formulation is developed to incorporate uncertain multiscale features. A stochastic analogue to a mixed multiscale finite element framework is used to formulate the physical stochastic multiscale process. Recent developments in linear and non-linear model reduction techniques are used to convert the limited information available about the permeability variation into a viable stochastic input model. An adaptive sparse grid collocation strategy is used to efficiently solve the resulting stochastic partial differential equations (SPDEs). The framework is applied to analyze flow through random heterogeneous media when only limited statistics about the permeability variation are given.     

Diffusion phenomena and thermodynamics

The established theory of linear diffusion phenomena is reformulated in the framework of the non-Hamiltonian dynamical systems theory (Liouville's approach), through the introduction of a suitable characteristic velocity for the processes under consideration. The dissipative features of diffusion processes are studied in the Riemannian manifold which has flux densities and generalized forces as coordinates, and a tensor metrics defined by the phenomenological coefficients. Some relationships between diffusion phenomena and thermodynamics are explored. Finally, a preliminary extension of the proposed approach to the case of nonlinear constitutive equations is presented, and the effects of non-linearity on the processes dynamics are briefly discussed.     

Highly efficient inelastic four-wave mixing using dual induced transparency and coherently prepared states

We investigate a life time broadened and coherently prepared five-state system for multi-wave mixing processes. We show that very efficient wave mixing occurs, producing an unconventional mixing wave that has the characteristics of both conventional four-wave mixing (FWM) and stimulated hyper-Raman (SHR) emission. In addition, we show interesting multiple simultaneous multi-photon interference effects at large propagation distances and demonstrate more than 10 orders of magnitude suppression of populations of the probe wave terminal state and the near three-photon resonance mixing wave generating state. These new type of multi-photon interference based induced transparency effects, which are critically dependent on two distinctive relaxation processes involving both an external supplied and an internally generated fields, are fundamentally different from the conventional three-state electromagnetically induced transparency effect which does not depend on propagation. As a consequence, both the probe and the wave-mixing field to propagate nearly free of absorption and distortions in a highly dispersive medium.     

Coherence resonance and stochastic synchronization in a nonlinear circuit near a subcritical Hopf bifurcation

We analyze noise-induced phenomena in nonlinear dynamical systems near a subcritical Hopf bifurcation. We investigate qualitative changes of probability distributions (stochastic bifurcations), coherence resonance, and stochastic synchronization. These effects are studied in dynamical systems for which a subcritical Hopf bifurcation occurs. We perform analytical calculations, numerical simulations and experiments on an electronic circuit. For the generalized Van der Pol model we uncover the similarities between the behavior of a self-sustained oscillator characterized by a subcritical Hopf bifurcation and an excitable system. The analogy is manifested through coherence resonance and stochastic synchronization. In particular, we show both experimentally and numerically that stochastic oscillations that appear due to noise in a system with hard excitation, can be partially synchronized even outside the oscillatory regime of the deterministic system.     

Brownian flights over a fractal nest and first-passage statistics on irregular surfaces

The diffusive motion of Brownian particles near irregular interfaces plays a crucial role in various transport phenomena in nature and industry. Most diffusion-reaction processes in confining interfacial systems involve a sequence of Brownian flights in the bulk, connecting successive hits with the interface (Brownian bridges). The statistics of times and displacements separating two interface encounters are then determinant in the overall transport. We present a theoretical and numerical analysis of this complex first-passage problem. We show that the bridge statistics is directly related to the Minkowski content of the surface within the usual diffusion length. In the case of self-similar or self-affine interfaces, we show and check numerically that the bridge statistics follows power laws with exponents depending directly on the surface fractal dimension.     

Non-Markov stochastic processes satisfying equations usually associated with a Markov process

There are non-Markov Ito processes that satisfy the Fokker-Planck, backward time Kolmogorov, and Chapman-Kolmogorov equations. These processes are non-Markov in that they may remember an initial condition formed at the start of the ensemble. Some may even admit 1-point densities that satisfy a nonlinear 1-point diffusion equation. However, these processes are linear, the Fokker-Planck equation for the conditional density (the 2-point density) is linear. The memory may be in the drift coefficient (representing a flow), in the diffusion coefficient, or in both. We illustrate the phenomena via exactly solvable examples. In the last section we show how such memory may appear in cooperative phenomena.     

Probing single jumps of surface atoms

Jumps of single atoms can be followed on their time and space scale (nanoseconds and Angstroms) by applying nuclear resonance scattering of synchrotron radiation. Here we develop the theory for jump diffusion in two-dimensional systems. Two types of phenomena are noteworthy: apparent acceleration of the nuclear decay and relaxation of hyperfine interactions, in particular, electric quadrupole interactions. The latter effect becomes for the first time one of significance and well measurable due to the inherent anisotropy of the surface. We show how, by way of motional narrowing, to distinguish between the motion of the probe atom itself and the motion of adatoms.     

Multiscale theory of fluctuating interfaces: renormalization of atomistic models

We describe a framework for the multiscale analysis of atomistic surface processes which we apply to a model of homoepitaxial growth with deposition according to the Wolf-Villain model and concurrent surface diffusion. Coarse graining is accomplished by calculating renormalization-group (RG) trajectories from initial conditions determined by the regularized atomistic theory. All of the crossover and asymptotic scaling regimes known from computer simulations are obtained, but we also find that two-dimensional substrates show an intriguing transition from smooth to mounded morphologies along the RG trajectory.     

Lattice limit cycle dynamics: Influence of long-distance reactive and diffusive mixing

The properties of global oscillations produced by coupled reactive stochastic discrete systems on a 2D lattice support are studied, taking into account the competitive influence of local and global mixing processes. Two types of global mixing are considered: reactive and diffusive. It is shown that in the case of diffusive mixing the increase in the diffusive coupling leads to a corresponding increase in the amplitude of the global oscillations. In the case of reactive mixing the competition of local-to-global effects leads to unexpected complex phenomena. Kinetic Monte Carlo simulations demonstrate that the amplitude of oscillations as a function of the mixing-reactive coupling presents an optimal value, which is attributed to the competitive effects between the local and global processes.     

Phase separation in binary systems with competing internal and external multiplicative noises

We have considered phase separation processes in binary stochastic systems with thermal diffusion and ballistic mixing representing irradiation influence. Introducing fluctuations of thermal flux and an external source of atom relocations due to ballistic diffusion into dynamics of a globally conserved field, we have shown that there are two competing mechanisms of phase transitions type of “order-disorder”: thermally assisted diffusion and irradiation induced atomic exchange. We have studied dynamics of the structure function at early stages of decomposition. In the framework of the mean field theory we have derived the effective Fokker-Planck equation to describe phase separation processes. It was shown that the ordering processes can be controlled by both regular and stochastic parts of external source influence. A reentrant behavior of a mean field order parameter versus the external noise intensity and fluctuations correlation radius is found.     

Geometry of intensive scalar dissipation events in turbulence

The maxima of the scalar dissipation rate in turbulence appear in the form of sheets and correspond to the potentially most intensive scalar mixing events. Their cross section extension determines a locally varying diffusion scale of the mixing process and extends the classical Batchelor picture of one mean diffusion scale. The distribution of the local diffusion scales is analyzed for different Reynolds and Schmidt numbers with a fast multiscale technique applied to very high-resolution simulation data. The scales always take values across the whole Batchelor range and beyond. Furthermore, their distribution is traced back to the distribution of the contractive short-time Lyapunov exponent of the flow.     

Corroboration of a multiscale approach with all atom calculations in analysis of dislocation nucleation from surface steps

We present a comparative study of the influence of atomic-scale surface steps on dislocation nucleation at crystal surfaces based on an all atom method and a hierarchal multiscale approach. The multiscale approach is based on the variational boundary integral formulation of the Peiersl–Nabarro dislocation model in which interatomic layer potentials derived from atomic calculations of generalized stacking fault energy surfaces are incorporated. We have studied nucleation of screw dislocations in two bcc material systems, molybdenum and tantalum, subjected to simple shear stress. Compared to dislocation nucleation from perfectly flat surfaces, the presence of atomic scale surface steps rapidly reduces the critical stress for dislocation nucleation by almost an order of magnitude as the step height increases. In addition, they may influence the slip planes on which dislocation nucleation occurs. The results of the all atom method and the multiscale approach are in good agreement, even for steps with height of only a single atomic layer. Such corroboration supports the further use of the multiscale approach to study dislocation nucleation phenomena in more realistic geometries of technological importance, which are beyond the reach of all current atom simulations.     

Effect of parametric modulation on pattern formation in reaction diffusion systems

Reaction diffusion systems can exhibite both spatial and temporal patterns. We show that the effect of spatial variation of the removal rate can have significant effect on the stability boundaries. In particular there can be a case of parametric resonance. Received 11 March 1998     

Exact propagator of the Fokker-Planck equation with a space-dependent diffusion coefficient and a time-dependent mean-reverting force

We have investigated the algebraic structure of the Fokker-Planck equation with a variable diffusion coefficient and a time-dependent mean-reverting force. Such a model could be useful to study the general problem of a Brownian walker with a space-dependent diffusion coefficient. We also show that this model is related to the Fokker-Planck equation with a constant diffusion coefficient and a time-dependent anharmonic potential of the form V(x, t) = ?a(t)x ² + b ln x, which has been widely applied to model different physical and biological phenomena, e.g. the study of neuron models and stochastic resonance in monostable nonlinear oscillators. Using the Lie algebraic approach we have derived the exact diffusion propagators for the Fokker-Planck equations associated with different boundary conditions, namely (i) the case of a single absorbing barrier, and (ii) the case of two absorbing barriers. These exact diffusion propagators enable us to study the time evolution of the corresponding stochastic systems. Received 23 October 2001 and Received in final form 24 December 2001     

Controlling the heat flow: now it is possible

We discuss the problem of heat conduction in 1D nonlinear chains in relation to the dynamical properties of the system. We provide convincing numerical evidence for the validity of Fourier law of heat conduction in linear mixing systems. Therefore, deterministic diffusion and normal heat transport which are usually associated with full hyperbolicity, actually take place in systems without exponential instability. We then show that, acting on the parameter which controls the strength of the on site potential inside a segment of the chain, we induce a transition from conducting to insulating behavior in the whole system. The control of heat conduction by nonlinearity opens the possibility to propose new devices such as a thermal rectifier.     

滚筒内非等粒径二元颗粒体系增混机理研究

提出了在圆形滚筒内设置十字形内构件的增混方式,并采用离散单元方法对设置不同大小内构件的滚筒内非等粒径二元颗粒体系的混合进行了数值模拟试验.通过模拟结果重点分析了内构件对混合的影响,讨论了内构件的尺寸对混合效果的作用,分析和探讨了滚筒内构件对二元颗粒体系的增混机理.研究发现,当滚筒内无内构件时,对流、扩散和离析三种作用机制对颗粒体系的混合和分离都起到了重要作用;当滚筒内含内构件时,颗粒的混合则只受到颗粒对流和扩散机制的作用,而颗粒的离析效应得到了很大程度的抑制.十字形内构件很大程度上会破坏滚筒内的自由表面流,从而使发生在自由表面流中的颗粒分离不能发生,最终可有效地增加颗粒之间的混合.对于采用在滚筒内设置十字形内构件的方式来增加颗粒间的混合,存在一个最优的内构件尺寸,内构件过小或过大都不利于颗粒间的混合. 关键词： 分离 混合 离散单元法     

Aspects of stochastic resonance in reaction–diffusion systems: The nonequilibrium-potential approach

We analyze several aspects of the phenomenon of stochastic resonance in reaction–diffusion systems, exploiting the nonequilibrium potential's framework. The generalization of this formalism (sketched in the appendix) to extended systems is first carried out in the context of a simplified scalar model, for which stationary patterns can be found analytically. We first show how system-size stochastic resonance arises naturally in this framework, and then how the phenomenon of array-enhanced stochastic resonance can be further enhanced by letting the diffusion coefficient depend on the field. A yet less trivial generalization is exemplified by a stylized version of the FitzHugh–Nagumo system, a paradigm of the activator–inhibitor class. After discussing for this system the second aspect enumerated above, we derive from it–through an adiabatic-like elimination of the inhibitor field–an effective scalar model that includes a nonlocal contribution. Studying the role played by the range of the nonlocal kernel and its effect on stochastic resonance, we find an optimal range that maximizes the system's response.