On the rate of convergence to the normal law for solutions of the Burgers equation with singular initial data

We study the scaling limit of random fields which are the solutions of a nonlinear partial differential equation known as the Burgers equation, under stochastic initial condition. These are assumed to be a Gaussian process with long-range dependence. We present some results on the rate of convergence to the normal law.     

Entropy Production in Nonlinear,Thermally Driven Hamiltonian Systems

We consider a finite chain of nonlinear oscillators coupled at its ends to two infinite heat baths which are at different temperatures. Using our earlier results about the existence of a stationary state, we show rigorously that for arbitrary temperature differences and arbitrary couplings, such a system has a unique stationary state. (This extends our earlier results for small temperature differences.) In all these cases, any initial state will converge (at an unknown rate) to the stationary state. We show that this stationary state continually produces entropy. The rate of entropy production is strictly negative when the temperatures are unequal and is proportional to the mean energy flux through the system     

聚变反应时间历程的双峰结构测量

利用新研制的聚变反应速率测量系统,在神光Ⅲ原型装置上测量了间接驱动时充DT气体的玻璃球壳内爆靶丸的聚变反应速率的时间历程,获得了DT中子产额约为1010时的聚变反应速率随时间的变化过程,发现了聚变中子发射在时间上的双峰结构。利用辐射流体程序对聚变中子在时间上的双峰结构进行了数值模拟,发现双峰结构分别由冲击压缩过程和惯性压缩过程产生,靶丸壳层厚度不同时产生的聚变中子发射双峰强度比变化可能是由靶丸的初始表面调制度不同所致。通过理论模拟与实验结果的对比,验证了中子聚变反应历程的双峰结构。     

Invasion exponents in biological networks

This article is concerned with the characterization of invasion exponents in biological networks defined by a population of replicating elements: molecules, cells, higher organisms. We show that the outcome of competition between an invader and a resident population is a stochastic process, determined by the rate at which a population returns to its steady state after a random perturbation in the parameters that characterize the replicating elements. This return rate is defined by the macroscopic parameter evolutionary entropy, a measure of the diversity of the interaction between the individuals in the population. We also show that the evolutionary stability of a population, that is the invulnerability of a resident to the introduction of an invader competing for the available resources, are given by extremal states of entropy. These results which pertain to networks of interacting molecules, cells and higher organisms, are generalizations of results established for demographic networks, that is populations of replicating organisms parametrized by the ages at which they reproduce and die.     

Leak rate of seals: Effective-medium theory and comparison with experiment

Seals are extremely useful devices to prevent fluid leakage. We present an effective-medium theory of the leak rate of rubber seals, which is based on a recently developed contact mechanics theory. We compare the theory with experimental results for seals consisting of silicon rubber in contact with sandpaper and sand-blasted PMMA surfaces.     

Theory of rumour spreading in complex social networks

We introduce a general stochastic model for the spread of rumours, and derive mean-field equations that describe the dynamics of the model on complex social networks (in particular, those mediated by the Internet). We use analytical and numerical solutions of these equations to examine the threshold behaviour and dynamics of the model on several models of such networks: random graphs, uncorrelated scale-free networks and scale-free networks with assortative degree correlations. We show that in both homogeneous networks and random graphs the model exhibits a critical threshold in the rumour spreading rate below which a rumour cannot propagate in the system. In the case of scale-free networks, on the other hand, this threshold becomes vanishingly small in the limit of infinite system size. We find that the initial rate at which a rumour spreads is much higher in scale-free networks than in random graphs, and that the rate at which the spreading proceeds on scale-free networks is further increased when assortative degree correlations are introduced. The impact of degree correlations on the final fraction of nodes that ever hears a rumour, however, depends on the interplay between network topology and the rumour spreading rate. Our results show that scale-free social networks are prone to the spreading of rumours, just as they are to the spreading of infections. They are relevant to the spreading dynamics of chain emails, viral advertising and large-scale information dissemination algorithms on the Internet.     

The Role of Cell-Cell Adhesion in Wound Healing

We present a stochastic model which describes fronts of cells invading a wound. In the model cells can move, proliferate, and experience cell-cell adhesion. We find several qualitatively different regimes of front motion and analyze the transitions between them. Above a critical value of adhesion and for small proliferation large isolated clusters are formed ahead of the front. This is mapped onto the well-known ferromagnetic phase transition in the Ising model. For large adhesion, and larger proliferation the clusters become connected (at some fixed time). For adhesion below the critical value the results are similar to our previous work which neglected adhesion. The results are compared with experiments, and possible directions of future work are proposed. We consider as an example the experiment of Sheardown and Cheng⁽¹³⁾ on the wounding of rabbit corneas. It was shown in Ref. 2 that the typical ratio of proliferation rate and basic diffusion rate is of the order of 3 × 10⁻⁴.     

Chemical transients in closed chaotic flows: the role of effective dimensions

We investigate chemical activity in hydrodynamical flows in closed containers. In contrast to open flows, in closed flows the chemical field does not show a well-defined fractal property; nevertheless, there is a transient filamentary structure present. We show that the effect of the filamentary patterns on the chemical activity can be modeled by the use of time-dependent effective dimensions. We derive a new chemical rate equation, which turns out to be coupled to the dynamics of the effective dimension, and predicts an exponential convergence. Previous results concerning activity in open flows are special cases of this new rate equation.     

Controlling the size of ZnO quantum dots using the dressed photon–phonon-assisted sol–gel method

We developed a sol–gel method using the dressed photon–phonon (DPP) process. DPPs are selectively exited in nanoscale structures at photon energies that are lower than the bandgap energy, which allows one to increase the growth rate of smaller ZnO quantum dots (QDs). Thus, we obtained a smaller size variance of ZnO QDs. The growth rate was proportional to the power of the light used for DPP excitation. The results were confirmed using a rate equation that accounted for the concentration of the sol–gel solution.     

Diffusion-influenced reactions

We summarize three of our recent results on diffusion-influenced reactions in solutions. All deal with the concentration dependence of the reaction rate when the reactants must first diffuse together before reaction can occur. When one species (the sink species) is not dilute, the rate cannot be obtained by solution of a pair diffusion equation; the correlations among the sinks for the diffusing species must be accounted for. First, we consider fluorescence quenching when the quenchers are not dilute. For charged quenchers and fluorophores we discuss how the solution dielectric constant and ionic strength can strongly influence the deviations from the linear Stern-Volmer behavior (the dilute sink result) which arise due to the sink correlations. Second, we consider heterogeneous catalysis where a reactive species is adsorbed onto a surface and must surface diffuse to reactive sites (the sinks). We find that surface diffusion can be an important factor contributing to the rate of reaction; especially when surface diffusion is rapid relative to the adsorption/desorption rate. Third, we discuss diffusion influenced reactions with sinks which are long ellipsoids. Dilute long ellipsoids provide a large rate enhancement relative to a spherical sink; we show that this rate enhancement survives when nondilute ellipsoids are considered.     

Statistical prefactor and nucleation rate near and out of the critical point

The nucleation rate derived in the classical theory contains at least one undetermined parameter, which may be expressed in terms of the Langer first-principles theory. But the uncertainties in the accounting for fluctuation modes, which are either absorbed into the free energy of a critical cluster or not, result in different evaluations of the statistical prefactor and nucleation rate. We get the scaling approximations of the nucleation rate for the vapour condensation both near and out of the critical range. The results obtained deserve the experimental verification to resolve the theoretical uncertainty.     

Quasi Exact Model for the Anisotropy Driven Weibel Instability all over Fourier Space

The Weibel instability is prompted by a temperature anisotropy within a plasma. We investigate its growth rate for wave vectors making an arbitrary angle with the high temperature axis. We use a two temperatures waterbags model and derive stability conditions depending on both temperatures and wave vector orientation. It is found that the growth rate is maximum for wave vectors normal to the high temperature axis. Also, a critical angle is evidenced in the k space in which direction modes are unstable at high k although the growth rate decreases quickly in this direction. Exact results are derived in most cases. (© 2007 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

Computer-Assisted Bounds for the Rate of Decay of Correlations

The rate of decay of correlations quantitatively describes the rate at which a chaotic system “mixes” the state space. We present a new rigorous method to estimate a bound for this rate of mixing. The technique may be implemented on a computer and is applicable to both multidimensional expanding and hyperbolic systems. The bounds produced are significantly less conservative than current rigorous bounds. In some situations it is possible to approximate resonant eigenfunctions and to strengthen our bound to an estimate of the decay rate. Order of convergence results are stated. Received: 8 January 1997 / Accepted: 20 March 1997     

Obstacle Effects on One-Dimensional Translocation of ATPase

We apply a general random walk model to the study of the ATPase‘s one-dimensional translocation along obstacle biological environment, and show the effects of random obstacles on the ATPase translocation along single stranded DNA. We find that the obstacle environment can reduce the lifetime of ATPase lattice-bound state which results in the inhibition of ATPase activity. We also carry out the ranges of rate constant of ATPase unidirectonal translocation and bidirectional translocation. Our results are consistent with the experiments and relevant theoretical consideration, and can be used to explain some physiological phenomena.     

Quantum tunneling of ultracold atoms in optical traps

We review our theoretical advances in quantum tunneling of BoseEinstein condensates in optical traps and in microcavities. By employing a real physical system, the frequencies of the pseudo Goldstone modes in different phases between two optical traps are studied respectivdy, which are tile crucial feature of the non-Abelian Joseptmon effect. When the optical lattices are under gravity, we investigate the quantum tummling in the ＂Wannier-Stark localization＂ regime and ＂Lan（lau Zener tunneling＂ regime. We finally get the total decay rate and the rate is valid over the entire range of temperatures. At high temperatures, we show how the decay rate reduces to the appropriate results for the classical thermal activation. At hltermediate temperatures, the results of tile total decay rate are consistent with the thermally assisted tunneling. At low temperatures, we obtain the pure quantmn tunneling ultimately. And we study the alternating-current and direct-current （ac and de） photonic 3osephson effects in two weakly linked microcavities containing ultracold two-level atones, which allows for direct observation of the effects. This enables new investigations of the effect of maw-body physics in strongly coupled atom-cavity systems and provides a strategy for constructing novel interference devices of coherent photons. In addition, we propose the experimental protocols to observe these quantmn tunneling of Bose- Einstein condensates.     

On fully quantum kinetic equations for BEC,new theorem on nonpolynomial averages,and new special numbers enumerating one-cycle oriented graphs

We find the analytical formulas for a full nonlinear rate of condensation and for a collision integral in fully quantum kinetic equations for a homogeneous Bose-Einstein condensation which go beyond the previously known semiclassical kinetic equations where only a linear stimulated rate was taken into account. These formulas yield, in particular, a rate of spontaneous condensation and allow us to address a long-standing problem on the mechanism of a BEC phase transition governed by the nonperturbation effects of quantum fluctuations in a many-body system. The results are obtained with the help of a new general theorem on nonpolynomial averages in statistical physics that amounts to an exact summation of a full infinite set of Wick’s diagrams. We present in some details a derivation of these nonperturbation-in-fluctuations results as well as the general theorem based on the newly introduced special numbers enumerating one-cycle oriented graphs.     

Scaling and universality of the complexity of analog computation

We apply a probabilistic approach to study the computational complexity of analog computers which solve linear programming problems. We numerically analyze various ensembles of linear programming problems and obtain, for each of these ensembles, the probability distribution functions of certain quantities which measure the computational complexity, known as the convergence rate, the barrier and the computation time. We find that in the limit of very large problems these probability distributions are universal scaling functions. In other words, the probability distribution function for each of these three quantities becomes, in the limit of large problem size, a function of a single scaling variable, which is a certain composition of the quantity in question and the size of the system. Moreover, various ensembles studied seem to lead essentially to the same scaling functions, which depend only on the variance of the ensemble. These results extend analytical and numerical results obtained recently for the Gaussian ensemble, and support the conjecture that these scaling functions are universal.     

Effect of phase noise on parametric instabilities

We report an experimental study on the effect of an external phase noise on the parametric amplification of surface waves. We observe that both the instability growth rate and the wave amplitude above the instability onset are decreased in the presence of noise. We show that all the results can be understood with a deterministic amplitude equation for the wave in which the effect of noise is just to change the forcing term. All the data for the growth rate (respectively the wave amplitude), obtained for different forcing amplitudes and different intensities of the noise, can be collapsed on a single curve using this renormalized forcing in the presence of noise.     

Obstacle Effects on One-Dimensional Translocation of ATPase

We apply a general random walk model to the study of the ATPase's one-dimensional translocation along obstacle biological environment, and show the effects of random obstacles on the ATPase translocation along single-stranded DNA. We find that the obstacle environment can reduce the lifetime of ATPase lattice-bound state which results in the inhibition of ATPase activity. We also carry out the ranges of rate constant of ATPase unidirectional translocation and bidirectional translocation. Our results are consistent with the experiments and relevant theoretical consideration, and can be used to explain some physiological phenomena.