Strong Law of Large Numbers for Betti Numbers in the Thermodynamic Regime

Journal of Statistical Physics - We establish the strong law of large numbers for Betti numbers of random ?ech complexes built on $${\mathbb {R}}^N$$ -valued binomial point processes and...     

A Law of Large Numbers for the Zeroes of Heine–Stieltjes Polynomials

We determine the limiting density of the zeroes of Heine–Stieltjes polynomials (or of any set of points satisfying the conclusion of Heine–Stieltjes Theorem) in the thermodynamic limit and use this to prove a strong law of large numbers for the zeroes.     

Large Numbers and the Time Variation of Physical Constants

We consider a cosmological model consistent withobservation which not only explains the well-knownlarge-number coincidences, but also deduces the valuesof the mass, radius, and age of the universe, the Hubble constant and the cosmological constant,a relation between the pion mass and the Hubble constantknown so far only as a mysterious empirical coincidence,and other features. This model predicts an ever-expanding universe, as indeed latestastrophysical data indicate.     

Lyapunov Exponents for Particles Advected in Compressible Random Velocity Fields at Small and Large Kubo Numbers

We calculate the Lyapunov exponents describing spatial clustering of particles advected in one- and two-dimensional random velocity fields at finite Kubo numbers $\operatorname {Ku}$ (a dimensionless parameter characterising the correlation time of the velocity field). In one dimension we obtain accurate results up to $\operatorname {Ku}\sim 1$ by resummation of a perturbation expansion in $\operatorname {Ku}$ . At large Kubo numbers we compute the Lyapunov exponent by taking into account the fact that the particles follow the minima of the potential function corresponding to the velocity field. The Lyapunov exponent is always negative. In two spatial dimensions the sign of the maximal Lyapunov exponent λ ₁ may change, depending upon the degree of compressibility of the flow and the Kubo number. For small Kubo numbers we compute the first four non-vanishing terms in the small- $\operatorname {Ku}$ expansion of the Lyapunov exponents. By resumming these expansions we obtain a precise estimate of the location of the path-coalescence transition (where λ ₁ changes sign) for Kubo numbers up to approximately $\operatorname{Ku} = 0.5$ . For large Kubo numbers we estimate the Lyapunov exponents for a partially compressible velocity field by assuming that the particles sample those stagnation points of the velocity field that have a negative real part of the maximal eigenvalue of the matrix of flow-velocity gradients.     

一种模拟大密度比多相流的混合算法

在扩散界面法(diffuse interface method,DIM)的基础上提出一种能够处理大密度比(large density ratio)的多相流混合算法.流场信息通过格子波尔兹曼方法(lattice Boltzmann method,LBM)获得;相界面通过直接求解Cahn-Hilliard(C-H)方程确定.为保证在大密度比情况下求解界面方程的稳定性,采用二阶迎风格式来离散方程的对流项.通过对Rayleigh-Taylor(R-T)不稳定、液体中的气泡上升及液滴撞击干燥壁面的数值模拟,验证了方法的可行性.     

A Blow-Up Result for Dyadic Models of the Euler Equations

We partially answer a question raised by Kiselev and Zlatos (Int Math Res Not 2005(38):2315–2339, 2005); in the generalized dyadic model of the Euler equation, a blow-up of \({H^{1/3+\delta}}\)-norm occurs. We recover a few previous blow-up results for various related dyadic models as corollaries.     

颗粒溶液颗粒尺寸及浓度的差分偏振弹性散射光谱在线分析方法

偏振弹性散射光谱技术的基本原理为在偏振光入射条件下,根据出射光的偏振特性不同可以筛选出浅表层组织的单次散射光信息和深层组织的漫散射光信息。该研究的创新点在于将这种方法应用于颗粒溶液检测,目的是在颗粒溶液原始状态下实现对颗粒尺寸及浓度的同时检测。设计了一个共轴笼式光学系统,测量了聚苯乙烯微球颗粒溶液某一角度的背向散射信号,通过控制入射端和收集端偏振片的偏振方向获得了颗粒溶液的偏振平行光谱与偏振垂直光谱,两者之差即偏振差分光谱对应颗粒的单次散射信息,将该单次散射信息与Mie散射数据库进行比对获得颗粒的尺寸,然后在颗粒尺寸作为已知的条件下进一步分析偏振垂直光谱,将该垂直光谱对应的颗粒溶液的漫散射信息代入光漫散射下的近似表达式拟合得到颗粒的浓度信息。将实验结果与样品提供值进行了比对,并进一步分析了在获取颗粒数浓度时,颗粒直径的方差分布对结果的影响,最终验证了该实验方法的可行性。该方法的潜在应用包括对标准颗粒制造厂商的产品在线检测以及对牛奶制品中脂肪和蛋白质的浓度检测研究。     

Sedimentation Image Analysis (SIA) for the Simultaneous Determination of Particle Mass Density and Particle Size

A new method for the simultaneous determination of the distribution of particle mass density and the distribution of particle size with a technique with only a single measurement is presented. The basic idea of the new optical method is the analysis of gravitational particle settling by a digital image acquisition system. Individual particles illuminated by a laser light sheet are tracked by a continuously operating CCD camera. The projected area, shape factors and the centre of gravity are detected during the sedimentation process from a series of images with a constant time spread. As the algorithm is based on single particle tracking, the heterogeneity of the sample can be taken into account. From these measured particle characteristics, particle size and settling rate are calculated. Thus particle mass density is obtained taking into account also the influence of particle shape on the settling process. This method, which we name sedimentation image analysis (SIA), is particularly suitable for the characterization of heterogeneous material, e.g. soil, in the micrometer range.     

A Mechanical Model of Brownian Motion for One Massive Particle Including Slow Light Particles

We provide a connection between Brownian motion and a classical mechanical system. Precisely, we consider a system of one massive particle interacting with an ideal gas, evolved according to non-random mechanical principles, via interaction potentials, without any assumption requiring that the initial velocities of the environmental particles should be restricted to be “fast enough”. We prove the convergence of the (position, velocity)-process of the massive particle under a certain scaling limit, such that the mass of the environmental particles converges to 0 while the density and the velocities of them go to infinity, and give the precise expression of the limiting process, a diffusion process.     

A Thermodynamic Formalism for Continuous Time Markov Chains with Values on the Bernoulli Space: Entropy, Pressure and Large Deviations

Through this paper we analyze the ergodic properties of continuous time Markov chains with values on the one-dimensional spin lattice $\{1,\dots,d\}^{{\mathbb{N}}}$ (also known as the Bernoulli space). Initially, we consider as the infinitesimal generator the operator , where is a discrete time Ruelle operator (transfer operator), and $A:\{1,\dots,d\}^{{\mathbb{N}}}\to\mathbb{R}$ is a given fixed Lipschitz function. The associated continuous time stationary Markov chain will define the a priori probability. Given a Lipschitz interaction $V:\{1,\dots,d\}^{{\mathbb{N}}}\to\mathbb{R}$ , we are interested in Gibbs (equilibrium) state for such V. This will be another continuous time stationary Markov chain. In order to analyze this problem we will use a continuous time Ruelle operator (transfer operator) naturally associated to V. Among other things we will show that a continuous time Perron-Frobenius Theorem is true in the case V is a Lipschitz function. We also introduce an entropy, which is negative (see also Lopes et al. in Entropy and Variational Principle for one-dimensional Lattice Systems with a general a-priori probability: positive and zero temperature. Arxiv, 2012), and we consider a variational principle of pressure. Finally, we analyze large deviations properties for the empirical measure in the continuous time setting using results by Y. Kifer (Tamsui Oxf. J. Manag. Sci. 321(2):505–524, 1990). In the last appendix of the paper we explain why the techniques we develop here have the capability to be applied to the analysis of convergence of a certain version of the Metropolis algorithm.     

Probability Density and Statistical Properties for a Three-State Markovian Noise and Escape of Particles for a System Driven by This Noise

A three-state Markovian noise is investigated. Its probability density and statistical properties are obtained. Escape of particles for a system with potential barrier only driven by this noise is investigated. It is shown that, in some circumstances, this noise can make the particles escape over the potential barrier; but in other circumstances, it cannot. Resonant activation phenomenon appears for the system considered by us.     

Static and Time Dependent Density Functional Theory with Internal Degrees of Freedom: Merits and Limitations Demonstrated for the Potts Model

We present an extension of the density-functional theory (DFT) formalism for lattice gases to systems with internal degrees of freedom. In order to test approximations commonly used in DFT approaches, we investigate the statics and dynamics of occupation (density) profiles in the one-dimensional Potts model. In particular, by taking the exact functional for this model we can directly evaluate the quality of the local equilibrium approximation used in time-dependent density-functional theory (TDFT). Excellent agreement is found in comparison with Monte Carlo simulations. Finally, principal limitations of TDFT are demonstrated.     

A Diagnostic Method for Time Resolved Spatial Profile Measurements of Proton and Impurity Density and Temperature

A new diagnostic technique, based on elastic scattering of neutral atoms off the plasma ions, is proposed and analyzed. Space and time resolved measurements of the number densities and temperatures of the various components of the plasma are possible by energy analysis of the scattered neutral atoms. The ability to resolve ions of different masses is limited by ion thermal motion and the energy and angular dependence of the scattering cross sections. In hydrogenic plasmas with impurities, the scattering by the impurities is easily resolvable from the scattering of protons, even when the individual impurity contributions cannot be separately resolved. Detected particle count rates are calculated for a conceputal system for a tokamak plasma (n ~ 1013 cm-3, Ti ~ 300 eV). Time resolution of ~ 10 ms with spatial resolution of a few cm are predicted.     

A Limit Result for a System of Particles in Random Environment

We consider an infinite system of particles in one dimension, each particle performs independent Sinai’s random walk in random environment. Considering an instant t, large enough, we prove a result in probability showing that the particles are trapped in the neighborhood of well defined points of the lattice depending on the random environment, t and the starting points of the particles. Supported by GREFI-MEFI and Departimento di Mathematica, Universita di Roma II “Tor Vergata”, Italy.     

Time of Relaxation to a Stationary Nonequilibrium Distribution of Brownian Particles in a System with a Source and a Sink

We generalize a method for determination of time scales of nonstationary diffusion in a one-dimensional medium. The time of relaxation to a stationary non-equilibrium density of substance in a system with a source, a sink, and an arbitrary potential profile are obtained and studied in detail. A few specific examples are considered.     

Optimality of the Triangular Lattice for a Particle System with Wasserstein Interaction

We prove strong crystallization results in two dimensions for an energy that arises in the theory of block copolymers. The energy is defined on sets of points and their weights, or equivalently on the set of atomic measures. It consists of two terms; the first term is the sum of the square root of the weights, and the second is the quadratic optimal transport cost between the atomic measure and the Lebesgue measure. We prove that this system admits crystallization in several different ways: (1) the energy is bounded from below by the energy of a triangular lattice (called ${\mathcal{T}}$ ); (2) if the energy equals that of ${\mathcal{T}}$ , then the measure is a rotated and translated copy of ${\mathcal{T}}$ ; (3) if the energy is close to that of ${\mathcal{T}}$ , then locally the measure is close to a rotated and translated copy of ${\mathcal{T}}$ . These three results require the domain to be a polygon with at most six sides. A fourth result states that the energy of ${\mathcal{T}}$ can be achieved in the limit of large domains, for domains with arbitrary boundaries. The proofs make use of three ingredients. First, the optimal transport cost associates to each point a polygonal cell; the energy can be bounded from below by a sum over all cells of a function that depends only on the cell. Second, this function has a convex lower bound that is sharp at ${\mathcal{T}}$ . Third, Euler’s polytope formula limits the average number of sides of the polygonal cells to six, where six is the number corresponding to the triangular lattice.     

Flux Large Deviations of Independent and Reacting Particle Systems,with Implications for Macroscopic Fluctuation Theory

We consider a system of independent particles and a system of reacting particles on a discrete state space. For the independent case, we rigorously prove a dynamic large-deviation principle for the empirical measure-integrated flux pair, taking the one-way fluxes rather than net fluxes into account. We prove the large deviations under deterministic initial conditions, and under random initial conditions satisfying a large-deviation principle. We formally calculate similar one-way flux large deviations for the more general reacting particles case. Next, we use these results to derive a number of principles from Macroscopic Fluctuation Theory like time-reversal principles and generalised gradient structures.     

Bell's Inequality for a System Composed of Particles with Different Spins

For two particles with different spins, we derive the Bell's inequality. The inequality is investigated for two systems combining spin-1 and spin-1/2; spin-1/2 and spin-3/2. We show that for these states Bell's inequality is violated.