Statistical analysis of aggregation in freeway traffic

We restudy the master-equation approach to aggregation in freeway traffic based on the theory of birth--death process, in which the clustering behaviour in one-lane freeway traffic model is investigated. The transition probabilities for the jump processes are reconstructed by using Greenshields' model, and the equation of the mean size of the cluster at any time t is derived from the birth--death equation. Numerical experiments show the clustering behaviours varying with time very well.     

A Statistical Theory to Aggregation in One-dimensional Freeway Traffic

The master equation of the car cluster in freeway traffic is reinvestigated on the basis of the linear assumption of the transition, in which the dynamic behavior of the cluster in a one-lane freeway traffic flow is studied. The expression of the mean size of the cluster versus t is derived in term of the solution of the birth-death equation. We also obtain the analytical expression of the probability distribution P(n,t). Numerical simulations testify the results derived as well.     

Asymptotic results on occupancy times of random walks

In this note we derive, using Wald's theorem asymptotic results on mean occupancy time of an interval for random walks with arbitrary transition probabilities. We show that our results are consistent with those obtained (by Weiss, Ref. 2) via the master equation approach, by demonstrating that the resulting infinite series can be summed exactly.     

On the master equation with time-dependent transition probabilities

The asymptotic behaviour of solutions of the master equation with time-dependent transition probabilities by means of the Kullback information is investigated. We assume that there are no absorbing states.     

The DLCA-RLCA transition arising in 2D-aggregation: simulations and mean field theory

A sticking probability model based on the average cluster lifetime is employed for deducing a kernel capable to describe the kinetics of computer simulated irreversible aggregation processes in two dimensions. The deduced kernel describes not only the time evolution of the cluster size distribution for diffusion limited aggregation (DLCA) and reaction limited aggregation (RLCA) but also for the entire transition region between both regimes. The model predicts a crossover to diffusion limited cluster aggregation for all sticking probabilities at long aggregation times. The time needed for reaching the DLCA limit increases for decreasing sticking probability. Received 16 April 2001 and Received in final form 24 May 2001     

Kinetic lattice gas model: Langmuir,Ising and interaction kinetics

The kinetic lattice gas model is formulated properly to account for adsorption, desorption, and diffusion at surfaces. We examine three choices for the transition probabilities in the master equation, which we term Langmuir, Ising and interaction kinetics, and show how they lead to different sticking coefficients and desorption rates.     

Reformulation of the path probability method and its application to crystal growth models

The microscopic master equation of a system is derived within the framework of the path probability method (PPM). Then, by extending Morita's method in equilibrium statistical mechanics, the path probability function constructed microscopically can be systematically decomposed to result in the conventional path probability function of cluster approximation when correlations larger than the chosen basic cluster are neglected. In order to critically compare the master equation method with the PPM, the triangle approximation is treated by both methods for crystal growth models. It is found that the PPM gives physically satisfactory kinetic equations, while the master equation (supplemented with a cluster probability in the superposition approximation) does not. The triangle PPM calculation considerably improves the result of the pair approximation for crystal growth velocity in the solid-on-solid model, and compares well with Monte Carlo results.     

Decay of correlations. IV. Necessary and sufficient conditions for a rapid decay of correlations

We considerN-particle systems whose probability distributions obey the master equation. For these systems, we derive the necessary and sufficient conditions under which the reducedn-particle (n) probabilities also obey master equations and under which the Ursell functions decay to their equilibrium values faster than the probability distributions. These conditions impose restrictions on the form of the transition rate matrix and thus on the form of its eigenfunctions. We first consider systems in which the eigenfunctions of theN-particle transition rate matrix are completely factorized and demonstrate that for such systems, the reduced probabilities obey master equations and the Ursell functions decay rapidly if certain additional conditions are imposed. As an example of such a system, we discuss a random walk ofN pairwise interacting walkers. We then demonstrate that for systems whoseN-particle transition matrix can be written as a sum of one-particle, two-particle, etc. contributions, and for which the reduced probabilities obey master equations, the reduced master equations become, in the thermodynamic limit, those for independent particles, which have been discussed by us previously. As an example of suchN-particle systems, we discuss the relaxation of a gas of interacting harmonic oscillators.Supported in part (grants to D.B. and K.E.S.) by the Advanced Research Projects Agency of the Department of Defense as monitored by the U.S. Office of Naval Research under Contract N00014-69-A-0200-6018, and in part (grant to I.O.) by the National Science Foundation.     

Information Dimension of Stochastic Processes on Networks: Relating Entropy Production to Spectral Properties

We consider discrete stochastic processes, modeled by classical master equations, on networks. The temporal growth of the lack of information about the system is captured by its non-equilibrium entropy, defined via the transition probabilities between different nodes of the network. We derive a relation between the entropy and the spectrum of the master equation’s transfer matrix. Our findings indicate that the temporal growth of the entropy is proportional to the logarithm of time if the spectral density shows scaling. In analogy to chaos theory, the proportionality factor is called (stochastic) information dimension and gives a global characterization of the dynamics on the network. These general results are corroborated by examples of regular and of fractal networks.     

非平衡系统的概率模型及Master方程的建立

针对非平衡态统计中出现的非线性及多元线性Master方程,明确提出了一般的概率假定;应用概率论的方法建立了一般的非线性及多元线性Master方程;讨论了它们与宏观动力学方程的一致性问题;最后应用多元线性Master方程讨论了Brusselator的涨落,并对其耗散结构的产生作了初步解释。 关键词：     

Analysis of ultrasonic anomaly in V3Si

V₃Si exhibits an ultrasonic anomaly when cooled well below its martensitic and superconducting transition temperatures (T _m andT _c), and a magnetic field is applied on to the sample. The anomaly is thought to be due to reorientation of microdomains formed belowT _m, to energetically favourable configurations. The effect disappears when the domains are stabilised in new configurations in the presence of the magnetic field. An analysis of these results is presented in this paper by relating the ultrasonic attenuation coefficient to strain fluctuations, arising here from domain reorientations. The treatment is based on a master equation for the probability matrix whose elements yield the probabilities of transitions between domain configurations, in the presence of both the magnetic field and the stress wave. Arguments for the validity of this master equation, when the oscillatory stress is weak, are given in a longish appendix. The derived results are used to analyse, in qualitative terms, the observed experimental facts. Also, new measurements are suggested which may help interpret the experimental data in a satisfactory manner.     

Some comments on nonequilibrium phase transitions in chemical systems

Two new approaches for investigating critical fluctuations near an instability point of unstable chemical models are proposed. The master equation approach is used. For a homogeneous system without the effect of diffusion, three single-component chemical systems exhibiting critical behavior are considered. The cumulant functions are expanded in a small parameter-the inverse size of the system-and singular perturbation solutions of the master equation are developed. Exponents describing the divergence of the second-order variance are found to be classical. For a system including diffusion effects, spatial correlations for a quasi-one-dimensional case are investigated by considering scale transformation behavior within the multivariate master equation formalism.This work was supported in part by NSF grants MPS-7411925 and CHE 76-05583.     

Sudden spreading of infections in an epidemic model with a finite seed fraction

We study a simple case of the susceptible-weakened-infected-removed model in regular random graphs in a situation where an epidemic starts from a finite fraction of initially infected nodes (seeds). Previous studies have shown that, assuming a single seed, this model exhibits a kind of discontinuous transition at a certain value of infection rate. Performing Monte Carlo simulations and evaluating approximate master equations, we find that the present model has two critical infection rates for the case with a finite seed fraction. At the first critical rate the system shows a percolation transition of clusters composed of removed nodes, and at the second critical rate, which is larger than the first one, a giant cluster suddenly grows and the order parameter jumps even though it has been already rising. Numerical evaluation of the master equations shows that such sudden epidemic spreading does occur if the degree of the underlying network is large and the seed fraction is small.     

重离子碰撞中超重核的形成机制(英文)

在双核模型框架下,用数值解主方程方法计算了超重核的熔合几率。 明确描述了包含能量、角动量和碎片形变弛豫的相对运动,并与核子扩散过程相耦合。因此,用微观方法推导出的核子跃迁几率是与时间相关的。所计算的以Pb为靶的冷熔合超重核形成截面和以48Ca为弹核的热熔合超重核形成激发函数与已知的实验值在合理的范围内符合。In the dinuclear system conception, the master equation is solved numerically to calculate the fusion probabilities of super heavy nuclei. The relative motion concerning the energy, the angular momentum and the fragment deformation relaxations is explicitly treated to couple with the diffusion process. The nucleon transition probabilities, which are derived microscopically, are related with the energy dissipation of the relative motion, thus they are time dependent. The formation cross sections of the super heavy nuclei from Pb based cold fusion and excitation functions from 48Ca induced hot fusion are reasonably consistent with known experimental data.     

Density viscous continuum traffic flow model

A new traffic flow model called density viscous continuum model is developed to describe traffic more reasonably. The two delay time scales are taken into consideration, differing from the model proposed by Xue and Dai [Phys. Rev. E 68 (2003) 066123]. Moreover the relative density is added to the motion equation from which the viscous term can be derived, so we obtain the macroscopic continuum model from microscopic car following model successfully. The condition for stable traffic flow is derived. Nonlinear analysis shows that the density fluctuation in traffic flow induces density waves. Near the onset of instability, a small disturbance could lead to solitons determined by the Korteweg-de-Vries (KdV) equation, and the soliton solution is derived. The results show that local cluster effects can be obtained from the new model and are consistent with the diverse nonlinear dynamical phenomena observed in the freeway traffic.     

Mean field model for synchronization of coupled two-state units and the effect of memory

A prototypical model for a mean field second order transition is presented, which is based on an ensemble of coupled two-states units. This system is used as a basic model to study the effect of memory. To wit, we distinguish two types of memories: weak and strong, depending on the feasibility of linearizing the generalized mean field master equation. For weak memory we find static solutions that behave much like those of the memoryless (Markovian) system. The latter exhibits a pitchfork bifurcation as the control parameter is increased, with two stable and one unstable solution. The former exhibits an imperfect pitchfork bifurcation to states with the same behaviors. In both cases, the stability of the static solutions is analyzed via the usual linearization around the equilibrium solution. For strong memories we again find an imperfect pitchfork bifurcation, with two stable and one unstable branch. However, it is no longer possible to analyze these behaviors via the usual linearization, which is local in time, because a strong memory requires knowledge of the system for its entire past. Finally, we are pleased to dedicate this publication to Helmut Brand on the occasion of his 60^th birthday.     

Unstable state dynamics: A systematic evaluation of the master equation

The master equation for diffusion in a bistable potential is evaluated systematically in terms of the small-noise parameter for the case where the system is initially at the unstable state. The expansion is valid for all times, that is in the initial and intermediate as well as in the final regime. The theory does not involve free fitting parameters and is easily generalized to more complicated processes.