Statistical Theory of Equilibrium Fluctuations

An analysis of the BBGKY hierarchy shows that the fluctuation theory for hydrodynamic variables (i.e., the fluid density, velocity, and temperature) must be constructed based on the hydrodynamic equations. We show that fluctuations in the hydrodynamic variables uniquely determine the fluctuations in all other thermodynamic parameters of a fluid (such as the pressure, entropy, and intrinsic energy). We find the spectral structure of fluctuation waves of the hydrodynamic variables, calculate their amplitudes, and derive the dispersion equation establishing the relation between the phase velocity of the fluctuation wave propagation and its frequency. We explain the effects of the flicker noise generation and of the stability loss on the first-order phase equilibrium lines. We also analyze the fluctuation decay process and show that this process gives rise to the effect of the Mandelstam–Brillouin scattering. We compare the obtained results with the existing fluctuation theories.     

Generalized Informational Entropy and Noncanonical Distribution in Equilibrium Statistical Mechanics

Based on the Jaynes principle of maximum for informational entropy, we find a generalized probability distribution and construct a generalized equilibrium statistical mechanics (ESM) for a wide class of objects to which the usual (canonical) ESM cannot be applied. We consistently consider the case of a continuous, not discrete, random variable characterizing the state of the object. For large values of the argument, the resulting distribution is characterized by a power-law, not exponential, asymptotic behavior, and the corresponding power asymptotic expression agrees with the empirical laws established for these objects. The -deformed Boltzmann–Gibbs–Shannon functional satisfying the requirements of the entropy axiomatics and leading to the canonical ESM for =0 is used as the original entropy functional. We also consider nonlinear transformations of this functional. We show that depending on how the averages of the dynamical characteristics of the object are defined, the different (Tsallis, Renyi, and Hardy–Littlewood–Pólya) versions of the generalized ESM can be used, and we give their comparative analysis. We find conditions under which the Gibbs–Helmholtz thermodynamic relations hold and the Legendre transformation can be applied to the generalized entropy and the Massieu–Planck function. We consider the Tsallis and Renyi ESM versions in detail for the case of a one-dimensional probabilistic object with a single dynamical characteristic whose role is played by a generalized positive energy with a monotonic power growth. We obtain constraints on the Renyi index under which the equilibrium distribution relates to a definite class of stable Gaussian or Levy–Khinchin distributions.     

Coherent States and Quantum Asymptotic Features by Weak KAM Theory

We propose a sketch for a proof of an interesting theorem on the evolution of coherent states, whose statement has been first presented in Paul (Séminaire: Équations aux Dérivées Partielles. 2007–2008, pages Exp. No. IV, 21. École Polytech., Palaiseau, 2009), and give some further insights on the asymptotic behavior, involving weak KAM theory.     

New Identities for Weak KAM Theory

This paper records for the Hamiltonian H =1/2│P│2 + W(x) some old and new identities relevant for the PDE/variational approach to weak KAM theory.     

KAM Theory for Partial Differential Equations

In the last years much progress has been achieved in KAM theory concerning bifurcation of quasi-periodic solutions of Hamiltonian or reversible partial differential equations.We provide an overview of the state of the art in this field.     

Metric Properties of Bogoliubov Trajectories in Statistical Equilibrium Theory

We investigate some properties of the Bogoliubov measure that appear in statistical equilibrium theory for quantum systems and establish the nondifferentiability of the Bogoliubov trajectories in the corresponding function space. We prove a theorem on the quadratic variation of trajectories and study the properties implied by this theorem for the scale transformations. We construct some examples of semigroups related to the Bogoliubov measure. Independent increments are found for this measure. We consider the relation between the Bogoliubov measure and parabolic partial differential equations.     

Geometric Models of the Statistical Theory of Fragmentation

We propose an approach to describing a medium fragmentation process based on studying the stochastic geometry of the medium states. This approach allows accounting for the interrelation of the produced fragments relative to their positions and, in particular, allows taking the size of the fragmenting object into account. We use this approach to analyze a one-dimensional model—a stochastic process with discrete time and a phase space consisting of partitions into fragments of the real axis. We derive the driving equation for the partition function with respect to sizes and prove the existence of a limit distribution.     

Weak KAM Theory topics in the stationary ergodic setting

We perform a qualitative analysis of the critical equation associated with a stationary ergodic Hamiltonian through a stochastic version of the metric method, where the notion of closed random stationary set, issued from stochastic geometry, plays a major role. Our purpose is to give an appropriate notion of random Aubry set, to single out characterizing conditions for the existence of exact or approximate correctors, and write down representation formulae for them. For the last task, we make use of a Lax-type formula, adapted to the stochastic environment. This material can be regarded as a first step of a long-term project to develop a random analog of Weak KAM Theory, generalizing what done in the periodic case or, more generally, when the underlying space is a compact manifold.     

Local product structure for Equilibrium States

The usual way to study the local structure of Equilibrium State of an Axiom-A diffeomorphism or flow is to use the symbolic dynamic and to push results on the manifold. A new geometrical method is given. It consists in proving that Equilibrium States for Hölder-continuous functions are related to other Equilibrium States of some special sub-systems satisfying a sort of expansiveness. Using different kinds of extensions the local product structure of Gibbs-measure is proven.

     

单指标模型的统计推断

本文对单指标模型的统计推断方法进行了系统阐述,其中包括联系函数和指标系数的估计,经验似然,模型检验和变量选择等。本文的取材来自近二十年来的最新研究成果。     

单指标模型的统计推断

本文对单指标模型的统计推断方法进行了系统阐述,其中包括联系函数和指标系数的估计,经验似然,模型检验和变量选择等。本文的取材来自近二十年来的最新研究成果。     

一维可压缩Euler方程组的两个模型

作者考察了一维可压缩Euler方程组的两个模型.利用特征分解和Gronwall不等式,首先得到具有几何结构且绝热指数γ=3的一维可压缩Euler方程组L~∞模的一致有界性.进一步,考虑当绝热指数γ=-1时,一维非等熵可压缩Euler方程组的Cauchy问题.在适当的假设下,得到该系统的整体经典解.     

Coherence and Uniqueness Theorems for Averaging Processes in Statistical Mechanics

Let S be the set of scalings {n ^–1:n=1,2,3,...} and let L _z =z Z ², zS, be the corresponding set of scaled lattices in R ². In this paper averaging operators are defined for plaquette functions on L _z to plaquette functions on L _z for all z, zS, z=dz, d{2,3,4,...}, and their coherence is proved. This generalizes the averaging operators introduced by Balaban and Federbush. There are such coherent families of averaging operators for any dimension D=1,2,3,... and not only for D=2. Finally there are uniqueness theorems saying that in a sense, besides a form of straightforward averaging, the weights used are the only ones that give coherent families of averaging operators.     

Analyticity for Some Operator Functions from Statistical Quantum Mechanics

For rather general thermodynamic equilibrium distribution functions the density of a statistical ensemble of quantum mechanical particles depends analytically on the potential in the Schr?dinger operator describing the quantum system. A key to the proof is that the resolvent to a power less than one of an elliptic operator with non-smooth coefficients, and mixed Dirichlet/Neumann boundary conditions on a bounded up to three-dimensional Lipschitz domain boundedly maps the space of square integrable functions to the space of essentially bounded functions. Dedicated to Günter Albinus Submitted: November 21, 2008. Accepted: March 31, 2009.     

On the Coupling of 3D-1D Muscle Models for Lumbar Spine Mechanics

The aim of this research is to represent, within one modelling framework, selected parts of the musculoskeletal system using principles of continuum mechanics, while other parts are modelled using lumped-parameter models and principles of Multi-Body Dynamics. The most challenging part within such a framework will be to properly model the transition from 3D to 1D models for skeletal muscles as many of the skeletal muscles extend beyond the selected part. Hence, this paper focuses on an interface condition for the 3D-1D transition within a skeletal muscle. (© 2011 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

New Identities for Weak KAM Theory(For Haim Brezis,in continuing admiration)

This paper records for the Hamiltonian H =1/2|p|~2+ W(x) some old and new identities relevant for the PDE/variational approach to weak KAM theory.     

Computational Statistical Methods for Social Network Models

We review the broad range of recent statistical work in social network models, with emphasis on computational aspects of these methods. Particular focus is applied to exponential-family random graph models (ERGM) and latent variable models for data on complete networks observed at a single time point, though we also briefly review many methods for incompletely observed networks and networks observed at multiple time points. Although we mention far more modeling techniques than we can possibly cover in depth, we provide numerous citations to current literature. We illustrate several of the methods on a small, well-known network dataset, Sampson's monks, providing code where possible so that these analyses may be duplicated.     

尖峰厚尾保险损失数据的统计建模

保险损失数据的一个重要特点是尖峰厚尾性,即既有大量的小额损失,又有少量的高额损失,使得通常的损失分布模型很难拟合此类数据,从而出现了对各种损失分布模型进行改进的尝试.改进后的模型一方面要有较高的峰度,另一方面又要有较厚的尾部.最近几年文献中出现的改进模型主要是组合模型,即把一个具有非零众数的模型(如对数正态分布或威布尔分布)与一个厚尾分布模型(如帕累托分布或广义帕累托分布)进行组合.讨论了这些组合模型的性质和特点,并与偏t正态分布和偏t分布进行了比较分析,最后应用MCMC方法估计模型参数,并通过一个实际损失数据的拟合分析,表明偏t分布对尖峰厚尾损失数据的拟合要优于目前已经提出的各种组合模型.     

On the entropy minimization problem in Statistical Mechanics

In many works on Statistical Mechanics and Statistical Physics, when deriving the distribution of particles of ideal gases, one uses the method of Lagrange multipliers in a formal way. In this paper we treat rigorously this problem for Bose–Einstein, Fermi–Dirac and Maxwell–Boltzmann entropies and present a complete study in the case of the Maxwell–Boltzmann entropy. Our approach is based on recent results on series of convex functions.