Bond percolation on a finite lattice: The one-state Potts model reconsidered

Bond percolation on a finite lattice is studied by looking at the Kac mean field model. The investigation utilizes the one-state Potts model connection established by Kasteleyn and Fortuin. To deal with special problems associated with the finite extent of the system we re-cast the partition function, which allows us to investigate the percolation transition in detail. This fundamental new formulation clears up certain ambiguities present in previous treatments and indicates a possible new direction in the study of other replica-type models.     

Global solutions of the Boltzmann equation on a lattice

The nonlinear Boltzmann equation with a discretized spatial variable is studied in a Banach space of absolutely integrable functions of the velocity variables. Conservation laws and positivity are utilized to extend weak local solutions to a global solution. This is shown to be a strong solution by analytic semigroup techniques.Supported by National Science Foundation Grant ENG-7515882.     

Thermodynamics of rotating self-gravitating systems

We investigate the statistical equilibrium properties of a system of classical particles interacting via Newtonian gravity, enclosed in a three-dimensional spherical volume. Within a mean-field approximation, we derive an equation for the density profiles maximizing the microcanonical entropy and solve it numerically. At low angular momenta, i.e. for a slowly rotating system, the well-known gravitational collapse “transition” is recovered. At higher angular momenta, instead, rotational symmetry can spontaneously break down giving rise to more complex equilibrium configurations, such as double-clusters (“double stars”). We analyze the thermodynamics of the system and the stability of the different equilibrium configurations against rotational symmetry breaking, and provide the global phase diagram. Received 8 July 2002 Published online 15 October 2002 RID="a" ID="a"e-mail: demartino@hmi.de     

A discrete velocity model of a gas: Global in time solutions of the BBGKY hierarchy

In the present paper we study the evolution of a system of hard disks moving in the plane with a finite number of velocities as in the framework of a discrete velocity model of the Enskog equation, proposed in previous papers. Starting from the BBGKY hierarchy of such a system we give existence and uniqueness results for the initial value problem in suitable Banach spaces. In particular, the main result presented is the global in time weak solution to the BBGKY hierarchy for local equilibrium initial data, in the thermodynamic limit.     

Global existence of solutions for a model Boltzmann equation

A model recently introduced by Ianiro and Lebowitz is shown to have a global solution for initial data having a finiteH-functional and belonging toL ¹ (L _x ). Methods previously introduced by Tartar to deal with discrete velocity models are used.     

EZ模型中的有限尺寸效应

研究EZ模型中的有限尺寸效应.当经纪人数目N足够大及发生交易的概率a1/N，发现有限尺 寸效应是重要的.此时，系统几乎变成包含所有经纪人的单一集团.而对较小集团，尺寸分布 仍然服从幂函数律，但是指数因涨落效应而改变.但当a1/N时，可以论证涨落效应不重要 ，因而平均场理论是严格成立的. 关键词： EZ模型 有限尺寸效应 涨落 平均场理论     

Self-consistent mean field approximation and application in three-flavor NJL model

In this study, we apply a self-consistent mean field approximation of the three-flavor Nambu–Jona-Lasinio(NJL) model and compare it with the two-flavor NJL model. The self-consistent mean field approximation introduces a new parameter, α, that cannot be fixed in advance by the mean field approach itself. Due to the lack of experimental data, the parameter, α, is undetermined. Hence, it is regarded as a free parameter and its influence on the chiral phase transition of strong interaction matter is studied based on this self-consistent mean field approximation. α affects numerous properties of the chiral phase transitions, such as the position of the phase transition point and the order of phase transition. Additionally, increasing α will decrease the number densities of different quarks and increase the chemical potential at which the number density of the strange quark is non-zero. Finally, we observed that α affects the equation of state(EOS) of the quark matter, and the sound velocity can be calculated to determine the stiffness of the EOS, which provides a good basis for studying the neutron star mass-radius relationship.     

Algebraic dynamics solutions and algebraic dynamics algorithm for nonlinear partial differential evolution equations of dynamical systems

Using functional derivative technique in quantum field theory, the algebraic dynamics approach for solution of ordinary differential evolution equations was generalized to treat partial differential evolution equations. The partial differential evolution equations were lifted to the corresponding functional partial differential equations in functional space by introducing the time translation operator. The functional partial differential evolution equations were solved by algebraic dynamics. The algebraic dynamics solutions are analytical in Taylor series in terms of both initial functions and time. Based on the exact analytical solutions, a new numerical algorithm—algebraic dynamics algorithm was proposed for partial differential evolution equations. The difficulty of and the way out for the algorithm were discussed. The application of the approach to and computer numerical experiments on the nonlinear Burgers equation and meteorological advection equation indicate that the algebraic dynamics approach and algebraic dynamics algorithm are effective to the solution of nonlinear partial differential evolution equations both analytically and numerically. Supported by the National Natural Science Foundation of China (Grant Nos. 10375039, 10775100 and 90503008), the Doctoral Program Foundation of the Ministry of Education of China, and the Center of Nuclear Physics of HIRFL of China     

A novel model and behavior analysis for a swarm of multi-agent systems with finite velocity

Inspired by the fact that in most existing swarm models of multi-agent systems the velocity of an agent can be infinite,which is not in accordance with the real applications, we propose a novel swarm model of multi-agent systems where the velocity of an agent is finite. The Lyapunov function method and LaSalle’s invariance principle are employed to show that by using the proposed model all of the agents eventually enter into a bounded region around the swarm center and finally tend to a stationary state. Numerical simulations are provided to demonstrate the effectiveness of the theoretical results.     

Diffusive corrections to asymptotics of a strong-field quantum transport equation

The asymptotic analysis of a linear high-field Wigner-BGK equation is developed by a modified Chapman-Enskog procedure. By an expansion of the unknown Wigner function in powers of the Knudsen number ?, evolution equations are derived for the terms of zeroth and first order in ?. In particular, a quantum drift-diffusion equation for the position density of electrons, with an ?-order correction on the field terms, is obtained. Well-posedness and regularity of the approximate problems are established, and a rigorous proof that the difference between exact and asymptotic solutions is of order ?², uniformly in time and for arbitrary initial data is given.     

数学物理方程中的通解问题

通过例证说明,通解法对于偏微分方程而言,其适用范围要比通常认为的大.并表明,只要适当变通,行波法就可用于有界情况.     

Classical solutions of a model of quark confinement

We find the classical solutions of a model of quark confinement defined by the vanishing of colour currents. Both plane-wave type of solutions extending all over space as well as string-type of solutions confined to restricted regions of space are found.     

Analytic solutions of simple flows and analysis of nonslip boundary conditions for the lattice Boltzmann BGK model

In this paper we analytically solve the velocity of the lattice Boltzmann BGK equation (LBGK) for several simple flows. The analysis provides a framework to theoretically analyze various boundary conditions. In particular, the analysis is used to derive the slip velocities generated by various schemes for the nonslip boundary condition. We find that the slip velocity is zero as long as fe=0 at boundaries, no matter what combination of distributions is chosen. The schemes proposed by Nobleet al. and by Inamuroet al. yield the correct zeroslip velocity, while some other schemes, such as the bounce-back scheme and the equilibrium distribution scheme, would inevitably generate a nonzero slip velocity. The bounce-back scheme with the wall located halfway between a flow node and a bounce-back node is also studied for the simple flows considered and is shown to produce results of second-order accuracy. The momentum exchange at boundaries seems to be highly related to the slip velocity at boundaries. To be specific, the slip velocity is zero only when the momentum dissipated by boundaries is equal to the stress provided by fluids.     

Non-Markovian dynamics of a qubit in a reservoir: different solutions of non-Markovian master equation

We present a non-Markovian master equation for a qubit interacting with a general reservoir, which is derived according to the Nakajima-Zwanzig and the time convolutionless projection operator technique. The non-Markovian solutions and Markovian solution of dynamical decay of a qubit are compared. The results indicate the validity of non-Markovian approach in different coupling regimes and also show that the Markovian master equation may not precisely describe the dynamics of an open quantum system in some situation. The non-Markovian solutions may be effective for many qubits independently interacting with the heated reservoirs.     

Chapman-Enskog and Burnett solutions for a simple, rigid-sphere gas: Numerical solutions using a subtraction technique

Chapman-Enskog and Burnett solutions are obtained for a simple, rigid-sphere gas through application of a subtraction technique to the linearized Boltzmann equation. The numerical results obtained converge and compare well with results reported previously using other techniques. This method is straightforward and may prove quite useful in addressing several other similar problems in the kinetic theory of gases.     

On the decay of infinite energy solutions to the Navier-Stokes equations in the plane

Infinite energy solutions to the Navier-Stokes equations in R² may be constructed by decomposing the initial data into a finite energy piece and an infinite energy piece, which are then treated separately. We prove that the finite energy part of such solutions is bounded for all time and decays algebraically in time when the same can be said of heat energy starting from the same data. As a consequence, we describe the asymptotic behavior of the infinite energy solutions. Specifically, we consider the solutions of Gallagher and Planchon (2002) [2] as well as solutions constructed from a “radial energy decomposition”. Our proof uses the Fourier Splitting technique of M.E. Schonbek.     

Analytic study of a model of diffusion on random comb-like structures

Summary An analytic study of a model of diffusion on random comb-like structures in which a bias field may or not exist along the backbone is presented. First, when no bias is present, the method allows to compute in an exact manner, for any given disordered structure, the asymptotic behaviour at large time of the probability of presence of the particle at its initial site and on the backbone, and of the particle position and dispersion. The expressions of these quantities are shown to coincide asymptotically with those derived in simple ?mean-field? treatments. The results for any given sample do not depend on the particular configuration (self-averaging property). When a bias field is present along the backbone, one can also compute directly in an exact manner the asymptotic behaviour at large time of the disorder average of the probability of presence of the particle at its initial site. As for the particle position and dispersion, they can be computed in a periodized system of arbitrary periodN. The corresponding quantities for the random system can then be obtained by taking the limitN→∞. As a result, in both cases the behaviours strongly depend on the distribution of the lengths of the branches. With an exponential distribution transport is normal while anomalous drift and diffusion may take place for a power law distribution (when long branches are present with sufficiently high weights). Paper presented at the I International Conference on Scaling Concepts and Complex Fluids, Copanello, Italy, July 4–8, 1994. Laboratoire associé au C.N.R.S. (U.A. no. 17) et aux Universités Paris VII et Paris VI.     

Multiplication for solutions of the equation

Linear first-order systems of partial differential equations (PDEs) of the form f=Mg, where M is a constant matrix, are studied on vector spaces over the fields of real and complex numbers. The Cauchy–Riemann equations belong to this class. We introduce on the solution space a bilinear *-multiplication, playing the role of a nonlinear superposition principle, that allows for algebraic construction of new solutions from known solutions. The gradient equation f=Mg is a simple special case of a large class of systems of PDEs, admitting a *-multiplication of solutions. We prove that any gradient equation has the exceptional property that the general analytic solution can be expressed as *-power series of certain simple solutions.