Continuous Time Finite State Mean Field Games

In this paper we consider symmetric games where a large number of players can be in any one of d states. We derive a limiting mean field model and characterize its main properties. This mean field limit is a system of coupled ordinary differential equations with initial-terminal data. For this mean field problem we prove a trend to equilibrium theorem, that is convergence, in an appropriate limit, to stationary solutions. Then we study an N+1-player problem, which the mean field model attempts to approximate. Our main result is the convergence as N→∞ of the mean field model and an estimate of the rate of convergence. We end the paper with some further examples for potential mean field games.     

Continuum limit in the fermionic hierarchical model

We discuss the problem of rigorously constructing the continuum limit in the fermionic hierarchical model. The continuum limit constructed as the limit of fields on the refined hierarchical lattices is a field on a p-adic continuum. We investigate the problem of reconstructing the coupling constants of the continuum model from the coupling constants of the discretized model. Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 118, No. 1, pp. 40–50, January, 1999.     

The p-Spin Interaction Model with External Field

This paper is devoted to a detailed study of a p-spins interaction model with external field, including some sharp bounds on the speed of self averaging of the overlap as well as a central limit theorem for its fluctuations, the thermodynamical limit for the free energy and the definition of an Almeida–Thouless type line. Those results show that the external field dominates the tendency to disorder induced by the increasing level of interaction between spins, and our system will share many of its features with the SK model, which is certainly not the case when the external magnetic field vanishes.     

长程选举模型的平均场极限

本文研究长程选举模型的平均场极限，利用对偶关系和特征函数方法证得长程选举模型的平均场极限满足下列微分方程：     

Solutions of the variational problem in the Curie—Weiss—Potts model

The variational problem for the Curie—Weiss—Potts model is solved completely. The results extend those of Ellis and Wang (1990, 1992), in which we study limit theorems and parameter estimations for the model and consider only the case of zero external field. In contrast to the Curie—Weiss model, this model has phase transitions in non-zero external field. All the solutions of the variational problem are non-degenerate points, so all the results in Ellis and Wang (1990, 1992) can be easily extended to the case considered here. We will also point out that simultaneous parameter estimation is impossible.     

THREE-DIMENSIONAL NUMERICAL LOCALIZATION OF IMPERFECTIONS BASED ON A LIMIT MODEL IN ELECTRIC FIELD AND A LIMIT PERTURBATION MODEL

From a limit model in electric field obtained by letting the frequency vanish in the time-harmonic Maxwell equations, we consider a limit perturbation model in the tangential boundary trace of the curl of the electric field for localizing numerically certain small electromagnetic inhomogeneities, in a three-dimensional bounded domain. We introduce here two localization procedures resulting from the combination of this limit perturbation model with each of the following inversion processes： the Current Projection method and an Inverse Fourier method. Each localization procedure uses, as data, a finite number of boundary measurements, and is employed in the single inhomogeneity case; only the one based on an Inverse Fourier method is required in the multiple inhomogeneities case. Our localization approach is numerically suitable for the context of inhomogeneities that are not purely electric. We compare the numerical results obtained from the two localization procedures in the single inhomogeneity configuration, and describe, in various settings of multiple inhomogeneities, the results provided by the procedure based on an Inverse Fourier method.     

Long-Time Accuracy for Approximate Slow Manifolds in a Finite-Dimensional Model of Balance

We study the slow singular limit for planar anharmonic oscillatory motion of a charged particle under the influence of a perpendicular magnetic field when the mass of the particle goes to zero. This model has been used by the authors as a toy model for exploring variational high-order approximations to the slow dynamics in rotating fluids. In this paper, we address the long time validity of the slow limit equations in the simplest nontrivial case. We show that the first-order reduced model remains O(ε) accurate over a long 1/ε timescale. The proof is elementary, but involves subtle estimates on the nonautonomous linearized dynamics.     

The Stratonovich Interpretation of Quantum Stochastic Approximations

The Stratonovich version of non-commutative stochastic calculus is introduced and shown to be equivalent to the Itô version developed by Hudson and Parthasarathy [1]. The conversion from Stratonovich to Itô version is shown to be implemented by a stochastic form of Wick's theorem: that is, involving the normal ordering of time-dependent noise fields. It is shown for a model of a quantum mechanical system coupled to a Bosonic field in a Gaussian state that under suitable scaling limits, in particular the weak coupling limit (for linear interactions) and low density limit (for scattering interactions), the limit form of the dynamical equation of motion is most naturally described as a quantum stochastic differential equation of Stratonovich form. We then convert the limit dynamical equations from Stratonovich to Itô form. Thermal Stratonovich noises are also presented.     

Mean field limit of a dynamical model of polymer systems

This paper provides a mathematically rigorous foundation for self-consistent mean feld theory of the polymeric physics.We study a new model for dynamics of mono-polymer systems.Every polymer is regarded as a string of points which are moving randomly as Brownian motions and under elastic forces.Every two points on the same string or on two diferent strings also interact under a pairwise potential V.The dynamics of the system is described by a system of N coupled stochastic partial diferential equations(SPDEs).We show that the mean feld limit as N→∞of the system is a self-consistent McKean-Vlasov type equation,under suitable assumptions on the initial and boundary conditions and regularity of V.We also prove that both the SPDE system of the polymers and the mean feld limit equation are well-posed.     

The combined relaxation and vanishing Debye length limit in the hydrodynamic model for semiconductors

The combined relaxation and vanishing Debye length limit for the hydrodynamic model for semiconductors is considered in both the unipolar and the bipolar case. The resulting limit problems are non‐linear drift driven hyperbolic equations. We make use of non‐standard entropy functions and the related entropy productions in order to obtain uniform estimates. In the bipolar case additional time‐dependent L^∞‐type estimates, available from the existence theory, are needed in order to control the entropy production terms. Finally, strong convergence of the electric field allows the limit towards the limiting problem. Copyright © 2001 John Wiley & Sons, Ltd.     

极限应变法在圆形隧洞稳定分析中的应用

材料的屈服和破坏是不同的,屈服准则已有大量研究,但缺少严格的破坏准则.理想弹塑性模型用应力表述难以区别屈服与破坏,为此该文提出极限应变破坏判据,可用于判断材料的局部和整体破坏.给出了不同材料极限应变的确定方法,并作为破坏判据用于岩土类材料的稳定分析,称为极限应变法.将极限应变法应用于圆形隧洞,研究隧洞的破坏过程、围岩破坏深度及其安全系数,并与滑移线理论和实际模型试验的结果进行对比.研究表明:极限应变法能够判断圆形隧洞的破坏过程与极限状态,求得准确的安全系数,与滑移线场法和模型试验的结果一致,验证了极限应变法在隧洞中应用的可行性.极限应变判据具有明确的力学意义,能反映材料破坏的全过程,为岩土类材料极限分析提供了一种新的方法.     

The hydrodynamic limit for a system with interactions prescribed by Ginzburg-Landau type random Hamiltonian

Summary As a microscopic model we consider a system of interacting continuum like spin field overR ^d . Its evolution law is determined by the Ginzburg-Landau type random Hamiltonian and the total spin of the system is preserved by this evolution. We show that the spin field converges, under the hydrodynamic space-time scalling, to a deterministic limit which is a solution of a certain nonlinear diffusion equation. This equation describes the time evolution of the macroscopic field. The hydrodynamic scaling has an effect of the homogenization on the system at the same time.     

Isogeometric Analysis of a Phase Field Model for Darcy Flows with Discontinuous Data

The authors consider a phase field model for Darcy flows with discontinuous data in porous media; specifically,they adopt the Hele-Shaw-Cahn-Hillard equations of[Lee,Lowengrub,Goodman,Physics of Fluids,2002] to model flows in the Hele-Shaw cell through a phase field formulation which incorporates discontinuities of physical data,namely density and viscosity,across interfaces. For the spatial approximation of the problem,the authors use NURBS—based isogeometric analysis in the framework of the Galerkin method,a computational framework which is particularly advantageous for the solution of high order partial differential equations and phase field problems which exhibit sharp but smooth interfaces. In this paper,the authors verify through numerical tests the sharp interface limit of the phase field model which in fact leads to an internal discontinuity interface problem; finally,they show the efficiency of isogeometric analysis for the numerical approximation of the model by solving a benchmark problem,the so-called"rising bubble" problem.     

Radiative transport limit of Dirac equations with random electromagnetic field

Abstract

This paper concerns the kinetic limit of the Dirac equation with a random electromagnetic field. We give a detailed mathematical analysis of the radiative transport limit for the phase space energy density of solutions to the Dirac equation. Our derivation is based on a martingale method and a perturbed test function expansion. This requires the electromagnetic field to be a Markovian space-time random field. The main mathematical tool in the derivation of the kinetic limit is the matrix-valued Wigner transform of the vector-valued Dirac solution. The major novelty compared with the scalar (Schrödinger) case is the proof of the weak convergence of cross modes to zero. The propagating modes are shown to converge in an appropriate probabilistic sense to their deterministic limit.     

The energy density in the planar Ising model

We study the critical Ising model on the square lattice in bounded simply connected domains with + and free boundary conditions. We relate the energy density of the model to a discrete fermionic correlator and compute its scaling limit by discrete complex analysis methods. As a consequence, we obtain a simple exact formula for the scaling limit of the energy field one-point function in terms of the hyperbolic metric. This confirms the predictions originating in physics, but also provides a higher precision.     

White noise in atmospheric optics

We consider a stochastic bilinear system model for laser propagation in atmospheric turbulence. The model consists of a random Schrödinger equation in which the white noise input is multiplied by the state. We consider approximate product form solutions of the Trotter-Kato type, and use these product forms to relate the Hilbert space-valued white noise model and the Itô equation model. We also consider white noise as the limit of a sequence Ornstein-Uhlenbeck processes. Finally, we consider approximate solutions using the Feynman-Itô equation, and an approximate calculation of the mean field without using the Markov approximation.     

Conformal covariance of the Abelian sandpile height one field

We study the scaling limit for the height one field of the two-dimensional Abelian sandpile model. The scaling limit for the covariance having height one at two macroscopically distant sites, more generally the centred height one joint moment of a finite number of macroscopically distant sites, is identified and shown to be conformally covariant. The result is based on a representation of the height one joint intensities that is close to a block-determinantal structure.     

Quasistatic evolution of damage in an elastic body with random inputs

Abstract

A model for the quasistatic evolution of the motion of an elastic body which is subject to material damage is presented and analyzed. The model takes the form of an elliptic system for the displacements coupled with a parabolic inclusion for the damage field. In both the system and the inclusion the coefficient and input functions that are present are assumed to be stochastic processes dependent on a random variable. The existence of a weak solution to the model is established using a sequence of approximate problems and passage to a limit. Moreover, the weak solution is shown to be product measurable.