非各态历经噪声的产生及其应用

用n阶时间导数噪声作为内部热噪声驱动自由粒子运动时,若该噪声所对应的非各态历经强度b?=0,且系统初始处于热平衡,则此时系统的长时稳态速度可以作为非各态历经噪声使用.非各态历经噪声具有谱密度在零频处发散的特点,一维半无界耦合简谐振子链对与之相连的气体分子的作用具有非各态历经噪声的特点,是非各态历经噪声的实例.最后将非各态历经噪声作为外噪声驱动倾斜周期势中的粒子运动,系统出现扩散指数α2的超弹道扩散现象.同时发现系统的速度分布将呈现出迁移态和锁定态两种不同状态,并且处于迁移态的粒子的速度分布为双峰.迁移态中双峰的出现是一种比较新奇的现象.     

Understanding synchronization induced by “common noise”

Noise-induced synchronization refers to the phenomenon where two uncoupled, independent nonlinear oscillators can achieve synchronization through a “common” noisy forcing. Here, “common” means identical. However, “common noise” is a construct which does not exist in practice. Noise by nature is unique and two noise signals cannot be exactly the same. How to justify and understand this central concept in noise-induced synchronization? What is the relation between noise-induced synchronization and the usual chaotic synchronization? Here we argue and demonstrate that noise-induced synchronization is closely related to generalized synchronization as characterized by the emergence of a functional relation between distinct dynamical systems through mutual interaction. We show that the same mechanism applies to the phenomenon of noise-induced (or chaos-induced) phase synchronization.     

Cosmography: Cosmology without the Einstein equations

How much of modern cosmology is really cosmography? How much of modern cosmology is independent of the Einstein equations? (Independent of the Friedmann equations?) These questions are becoming increasingly germane—as the models cosmologists use for the stress-energy content of the universe become increasingly baroque, it behaves us to step back a little and carefully disentangle cosmological kinematics from cosmological dynamics. The use of basic symmetry principles (such as the cosmological principle) permits us to do a considerable amount, without ever having to address the vexatious issues of just how much “dark energy”, “dark matter”, “quintessence”, and/or “phantom matter” is needed in order to satisfy the Einstein equations. This is the sub-sector of cosmology that Weinberg refers to as “cosmography”, and in this article I will explore the extent to which cosmography is sufficient for analyzing the Hubble law and so describing many of the features of the universe around us.     

How Gibbs Distributions May Naturally Arise from Synaptic Adaptation Mechanisms. A Model-Based Argumentation

This paper addresses two questions in the context of neuronal networks dynamics, using methods from dynamical systems theory and statistical physics: (i) How to characterize the statistical properties of sequences of action potentials (“spike trains”) produced by neuronal networks? and; (ii) what are the effects of synaptic plasticity on these statistics? We introduce a framework in which spike trains are associated to a coding of membrane potential trajectories, and actually, constitute a symbolic coding in important explicit examples (the so-called gIF models). On this basis, we use the thermodynamic formalism from ergodic theory to show how Gibbs distributions are natural probability measures to describe the statistics of spike trains, given the empirical averages of prescribed quantities. As a second result, we show that Gibbs distributions naturally arise when considering “slow” synaptic plasticity rules where the characteristic time for synapse adaptation is quite longer than the characteristic time for neurons dynamics.     

黑洞与奇点

黑洞可以说是引力最极端的体现,其视界内是个连光也逃不出去的时空区域。近来黑洞在天 文观测方面取得令人惊讶的发展,这其中包括：黑洞碰撞的引力波探测以及M87 星系的超大质量 黑洞的所谓第一张黑洞照片。但是在理论的层面上,黑洞物理尚有许多未解之谜。其中,信息遗失 的悖论是最有名的。但是,有另一个问题至少和信息的丢失一样{甚至更加{令人费解的,就是黑洞 内部的奇点性质。时空奇点是广义相对论本身无法描述的,在那里究竟发生什么事？黑洞内部的奇 点和宇宙大爆炸时的奇点有何不同？奇点是否会裸露在黑洞外面？所谓“宇宙监督猜想”的假设目 前有何进展？我们在这篇半科普的文章中简单的介绍这些课题,希望本文章对物理和数学的本科生 有所帮助。     

Physics and pseudogroups

The purpose of this paper is to give a new interpretation of the famous “Equivalence Problem” of E. Cartan, within the framework of differential invariants and bundles of geometric objects. The first example is the well-known theorem of G. Darboux for closed 2-forms. A second striking example is related to the so-called “Helmholtz postulate” in classical thermodynamics. Finally, we work out a counter-example of V. Guillemin and S. Sternberg, exhibiting geometric object involved and giving a new insight into the deformation theory of such structures.     

Diffusion in different models of active Brownian motion

Active Brownian particles (ABP) have served as phenomenological models of self-propelled motion in biology. We study the effective diffusion coefficient of two one-dimensional ABP models (simplified depot model and Rayleigh-Helmholtz model) differing in their nonlinear friction functions. Depending on the choice of the friction function the diffusion coefficient does or does not attain a minimum as a function of noise intensity. We furthermore discuss the case of an additional bias breaking the left-right symmetry of the system. We show that this bias induces a drift and that it generally reduces the diffusion coefficient. For a finite range of values of the bias, both models can exhibit a maximum in the diffusion coefficient vs. noise intensity.     

Thermal mechanics: A quantum mechanical analogue of nonequilibrium statistical thermodynamics

A formal but not conventional equivalence between stochastic processes in nonequilibrium statistical thermodynamics and Schrödinger dynamics in quantum mechanics is shown. It is found, for each stochastic process described by a stochastic differential equation of Itô type, there exists a Schrödinger-like dynamics in which the absolute square of a wavefunction gives us the same probability distribution as the original stochastic process. In utilizing this equivalence between them, that is, rewriting the stochastic differential equation by an equivalent Schrödinger equation, it is possible to obtain the notion of deterministic limit of the stochastic process as a semi-classical limit of the “Schrödinger” equation. The deterministic limit thus obtained improves the conventional deterministic approximation in the sense of Onsager-Machlup. The present approach is valid for a general class of stochastic equations where local drifts and diffusion coefficients depend on the position. Two concrete examples are given. It should be noticed that the approach in the present form has nothing to do with the conventional one where only a formal similarity between the Fokker-Planck equation and the Schrödinger equation is considered.     

Fundamental Problems in the Unification of Physics

We discuss the following problems, plaguing the present search for the “final theory”: (1) How to find a mathematical structure rich enough to be suitably approximated by the mathematical structures of general relativity and quantum mechanics? (2) How to reconcile nonlocal phenomena of quantum mechanics with time honored causality and reality postulates? (3) Does the collapse of the wave function contain some hints concerning the future quantum gravity theory? (4) It seems that the final theory cannot avoid the problem of dynamics, and consequently the problem of time. What kind of time, if this theory is supposed to be background free? (5) Will the dynamics of the “final theory” be probabilistic? Quantum probability exhibits some essential differences as compared with classical probability; are they but variations of some more general probabilistic measure theory? (6) Do we need a radically new interpretation of quantum mechanics, or rather an entirely new theory of which the present quantum mechanics is an approximation? (7) If the final theory is to be background free, it should provide a mechanism of space-time generation. Should we try to explain not only the generation of space-time, but also the generation of its material content? (8) As far as the existence of the initial singularity is concerned, one usually expects either “yes” or “not” answers from the final theory. However, if the mathematical structure of the future theory is supposed to be truly more general that the mathematical structures of the present general relativity and quantum mechanics, is a “third answer“ possible? Could this third answer be related to the probabilistic character of the final theory? We discuss these questions in the framework of a working model unifying gravity and quanta. The analysis reveals unexpected aspects of these rather wildly discussed issues.     

The diffusion equation of random genetic drift – biology’s analogue of the Schrödinger equation?

This work is a self-contained introduction to some basic aspects of the dynamics that occurs in biological populations. It focusses on the proportion (or frequency) of a population that carries a particular gene. We make use of the notion of a force, in the context of genetics and evolution, to describe the dynamics of the frequency in an effectively infinite population. We then show how randomness enters into the dynamics of populations with a finite size, a randomness known as random genetic drift. We derive an equation, involving random numbers, which describes how the frequency behaves in a population of finite size. It is shown that in some situations this equation exhibits irreversible absorption phenomena. These phenomena are associated with the extinction (or loss) of the gene, or the complete takeover by the gene (termed fixation), where 100% of the population carries the gene. Taking the theory further, we show how an approximation leads to a stochastic differential equation for the frequency, where random genetic drift takes the form of an additional contribution to the force, that randomly fluctuates. The stochastic differential equation is, in turn, related to a diffusion equation, which encompasses many fundamental phenomena. Because of this, the diffusion equation plausibly has a similar status in biology to the Schrödinger equation in physics. It is notable that both the Schrödinger equation and the diffusion equation have a somewhat similar mathematical structure: they both involve first order derivatives of time and second order derivatives of space (or the analogue of space). There are, however, some significant mathematical differences. In contrast to the Schrödinger equation, the diffusion equation can routinely have solutions which are singular, in that the solutions contain Dirac delta functions. The delta functions are not, however, problematic, and have an explicit biological significance. We illustrate results with some basic calculations and computer simulations.     

Transport and Dissipation in Quantum Pumps

This paper is about adiabatic transport in quantum pumps. The notion of “energy shift,” a self-adjoint operator dual to the Wigner time delay, plays a role in our approach: It determines the current, the dissipation, the noise and the entropy currents in quantum pumps. We discuss the geometric and topological content of adiabatic transport and show that the mechanism of Thouless and Niu for quantized transport via Chern numbers cannot be realized in quantum pumps where Chern numbers necessarily vanish.     

Topology of the gauge condition and new confinement phases in non-abelian gauge theories

The gauge-fixing constraint in a gauge field theory is crucial for understanding both short-distance and long-distance behavior of non-abelian gauge field theories. We define what we call “non-propagating” gauge conditions such as the unitary gauge and “approximately non-propagating” or renormalizable gauge conditions, and study their topological properties. By first fixing the non-abelian part of the gauge ambiguity we find that SU(N) gauge theories can be written in the form of abelian gauge theories with N ? 1 fold multiplicity enriched with magnetic monopoles with certain magnetic charge combinations. Their electric chargesare governed by the instanton angle θ.If θ is continuously varied from 0 to 2π and a confinement mode is assumed for some θ, then at least one phase-transition must occur. We speculate on the possibility of new phases: e.g., “oblique confinement,” where $\text{θ ? π}$, and explain some peculiar features of this mode. In principle there may be infinitely many such modes, all separated by phase transition boundaries.     

Is the electric charge conserved?

A possible nonconservation of the electric charge is considered in the framework of spontaneously broken gauge models. Assuming the validity of perturbation theory, such models are shown to be inconsistent with experiment unless there exists a scalar particle with the “pseudocharge” ?ρ where ??10^?.     

后掠叶片锯齿尾缘宽频噪声实验研究

本文采用线性传声器阵列分别对具有常规尾缘及锯齿形尾缘的后掠叶片的尾缘噪声进行了实验测量;运用CLEAN-SC数据处理方法精确地识别出叶片尾缘噪声的声学参数.并且基于多组实验结果的对比,深入研究了不同的尾缘锯齿长度、周期、几何比例对后掠叶片尾缘噪声降噪效果的影响.实验结果表明:在低湍流度、自由来流情况下,在总声压级降噪方...     

Effective long-time phase dynamics of limit-cycle oscillators driven by weak colored noise

An effective white-noise Langevin equation is derived that describes long-time phase dynamics of a limit-cycle oscillator driven by weak stationary colored noise. Effective drift and diffusion coefficients are given in terms of the phase sensitivity of the oscillator and the correlation function of the noise, and are explicitly calculated for oscillators with sinusoidal phase sensitivity functions driven by two typical colored Gaussian processes. The results are verified by numerical simulations using several types of stochastic or chaotic noise. The drift and diffusion coefficients of oscillators driven by chaotic noise exhibit anomalous dependence on the oscillator frequency, reflecting the peculiar power spectrum of the chaotic noise.     

Quantum statistical treatment of multimode self-pulsing in optical systems

We elaborate a formalism which is appropriate to describe the effects of quantum noise in multimode optical instabilities. The multimode Fokker-Planck equation is reformulated in terms of suitable “dressed mode” variables, which diagonalize the linearized part of the drift matrix. We work out explicitly the relations of our formalism with the quantum theory of multiwave mixing.     

An adaptive way for improving noise reduction using local geometric projection

We propose an adaptive way to improve noise reduction by local geometric projection. From the neighborhood of each candidate point in phase space, we identify the best subspace that the point will be orthogonally projected to. The signal subspace is formed by the most significant eigendirections of the neighborhood, while the less significant ones define the noise subspace. We provide a simple criterion to separate the most significant eigendirections from the less significant ones. This criterion is based on the maximum logarithmic difference between the neighborhood eigendirection lengths, and the assumption that there is at least one eigendirection that corresponds to the noise subspace. In this way, we take into account the special characteristics of each neighborhood and introduce a more successful noise reduction technique. Results are presented for a chaotic time series of the Henon map and Ikeda map, as well as on the Nasdaq Composite index.     

Are global terrorist attacks time-correlated?

Is there any kind of “memory” in the sequence of terror attacks worldwide? Are the terrorist attacks non-randomly time distributed? Our analysis suggests that they are correlated, which means that a terror event is not independent from the time elapsed since the previous event. But, if we consider terror attacks with a large severity index (number of fatalities or injured), the phenomenon is unpredictable, since it approaches a Poisson process (random, independent and uncorrelated).     

Block Spins for Partial Differential Equations

We investigate the use of renormalization group methods to solve partial differential equations (PDEs) numerically. Our approach focuses on coarse-graining the underlying continuum process as opposed to the conventional numerical analysis method of sampling it. We calculate exactly the coarse grained or “perfect” Laplacian operator and investigate the numerical effectiveness of the technique on a series of 1 + 1-dimensional PDEs with varying levels of smoothness in the dynamics: the diffusion equation, the time-dependent Ginzburg–Landau equation, the Swift–Hohenberg equation, and the damped Kuramoto–Sivashinsky equation. We find that the renormalization group is superior to conventional sampling-based discretizations in representing faithfully the dynamics with a large grid spacing, introducing no detectable lattice artifacts as long as there is a natural ultraviolet cutoff in the problem. We discuss limitations and open problems of this approach.