Generalized optical theorems for the reconstruction of Green's function of an inhomogeneous elastic medium

This paper investigates the reconstruction of elastic Green's function from the cross-correlation of waves excited by random noise in the context of scattering theory. Using a general operator equation-the resolvent formula-Green's function reconstruction is established when the noise sources satisfy an equipartition condition. In an inhomogeneous medium, the operator formalism leads to generalized forms of optical theorem involving the off-shell T-matrix of elastic waves, which describes scattering in the near-field. The role of temporal absorption in the formulation of the theorem is discussed. Previously established symmetry and reciprocity relations involving the on-shell T-matrix are recovered in the usual far-field and infinitesimal absorption limits. The theory is applied to a point scattering model for elastic waves. The T-matrix of the point scatterer incorporating all recurrent scattering loops is obtained by a regularization procedure. The physical significance of the point scatterer is discussed. In particular this model satisfies the off-shell version of the generalized optical theorem. The link between equipartition and Green's function reconstruction in a scattering medium is discussed.     

Retrieval of Green's function and generalized optical theorem for the scattering of complete dyadic fields

Green's function retrieval has been widely used in different research fields due to the fact that the Green's function can be extracted by cross-correlating the records at two receivers. In this paper, the retrieval of the dyadic Green's function is studied by investigating the representation theorem. The generalized optical theorem for the dyadic fields is derived based on the elastic dynamic interferometric equation. By addressing the cross-correlation recorded at two receivers, the important role of the generalized optical theorem and energy equipartition in retrieving the exact Green's function is shown. The presented derivation also shows the Newton-Marchenko equation holdsif the condition of equipartition is not satisfied.     

热振动在纳米力学测量中的应用

基于能量均分定理和悬臂梁理论,分析讨论了微悬臂梁进行热振动的物理模型。通过三个实例,介绍了热振动在纳米力学测量中的应用,其中重点阐述了单根纳米晶须的杨氏模量的热振动定量测量方法。     

Harmonic lattice behavior of two-dimensional colloidal crystals

Using positional data from videomicroscopy and applying the equipartition theorem for harmonic Hamiltonians, we determine the wave-vector-dependent normal mode spring constants of a two-dimensional colloidal model crystal and compare the measured band structure to predictions of the harmonic lattice theory. We find good agreement for both the transversal and the longitudinal modes. For q-->0, the measured spring constants are consistent with the elastic moduli of the crystal.     

Nonhomogeneous Cooling, Entropic Gravity and MOND Theory

In this paper, by using the holographic principle, a modified equipartition theorem where we assume that below a critical temperature the energy is not equally divided on all bits, and the Unruh temperature, we derive MOND theory and a modified Friedmann equation compatible with MOND theory. Furthermore, we rederive a modified Newton’s law of gravitation by employing an adequate redefinition of the numbers of bits.     

The equipartition for quantum systems

For KMS-States an inequality between the first and second moment of the time-auto-correlation function is derived and applied to Boson systems to prove that the classical equipartition theorem is strictly non valid.     

用推广的Maxwell速度分布律证明转动、振动方式能均分定理

将描述分子平动速度分布的Maxwell速度分布律推广到转动和振动方式,并用推广的Maxwell分布律成功地证明了转动、振动方式的能均分定理．     

A hybrid formalism combining fluctuating hydrodynamics and generalized Langevin dynamics for the simulation of nanoparticle thermal motion in an incompressible fluid medium

A novel hybrid scheme based on Markovian fluctuating hydrodynamics of the fluid and a non-Markovian Langevin dynamics with the Ornstein-Uhlenbeck noise perturbing the translational and rotational equations of motion of the nanoparticle is employed to study the thermal motion of a nanoparticle in an incompressible Newtonian fluid medium. A direct numerical simulation adopting an arbitrary Lagrangian-Eulerian (ALE) based finite element method (FEM) is employed in simulating the thermal motion of a particle suspended in the fluid confined in a cylindrical vessel. The results for thermal equilibrium between the particle and the fluid are validated by comparing the numerically predicted temperature of the nanoparticle with that obtained from the equipartition theorem. The nature of the hydrodynamic interactions is verified by comparing the velocity autocorrelation function (VACF) and mean squared displacement (MSD) with well-known analytical results. For nanoparticle motion in an incompressible fluid, the fluctuating hydrodynamics approach resolves the hydrodynamics correctly but does not impose the correct equipartition of energy based on the nanoparticle mass because of the added mass of the displaced fluid. In contrast, the Langevin approach with an appropriate memory is able to show the correct equipartition of energy, but not the correct short- and long-time hydrodynamic correlations. Using our hybrid approach presented here, we show for the first time, that we can simultaneously satisfy the equipartition theorem and the (short- and long-time) hydrodynamic correlations. In effect, this results in a thermostat that also simultaneously preserves the true hydrodynamic correlations. The significance of this result is that our new algorithm provides a robust computational approach to explore nanoparticle motion in arbitrary geometries and flow fields, while simultaneously enabling us to study carrier adhesion mediated by biological reactions (receptor-ligand interactions) at the vessel wall at a specified finite temperature.     

The anti-FPU problem

We present a detailed analysis of the modulational instability of the zone-boundary mode for one and higher-dimensional Fermi-Pasta-Ulam (FPU) lattices. Following this instability, a process of relaxation to equipartition takes place, which we have called the Anti-FPU problem because the energy is initially fed into the highest frequency part of the spectrum, at variance with the original FPU problem (low frequency excitations of the lattice). This process leads to the formation of chaotic breathers in both one and two dimensions. Finally, the system relaxes to energy equipartition on time scales which increase as the energy density is decreased. We show that breathers formed when cooling the lattice at the edges, starting from a random initial state, bear strong qualitative similarities with chaotic breathers.     

Deriving a differential equation for the spin-spin correlation in Ising systems

Summary The equipartition theorem and the high-temperature expansion are used to derive a differential equation for the spin-spin correlation in Ising systems. Some applications of this equation including improved upper bound on the three-dimensional transition temperature and correlation length are discussed. Due to the relevance of its scientific content, this paper has been given priority by the Journal Direction.     

Time scale for energy equipartition in a two-dimensional FPU model

The FPU problem, i.e., the problem of energy equipartition among normal modes in a weakly nonlinear lattice, is here studied in dimension two, more precisely in a model with triangular cell and nearest-neighbors Lennard-Jones interaction. The number n of degrees of freedom ranges from 182 to 6338. Energy is initially equidistributed among a small number n(0) of low frequency modes, with n(0) proportional to n. We study numerically the time evolution of the so-called spectral entropy and the related "effective number" n(eff) of degrees of freedom involved in the dynamics; in this (rather typical) way we can estimate, for each n and each specific energy (energy per degree of freedom) epsilon, the time scale T(n)(epsilon) for energy equipartition. Numerical results indicate that in the thermodynamic limit the equipartition times are short: more precisely, for large n at fixed epsilon we find a limit curve T(infinity)(epsilon), and T(infinity) grows only as epsilon(-1) for small epsilon. Larger equipartition times are obtained by lowering epsilon, at fixed n, below a crossover value epsilon(c)(n). However, epsilon(c) appears to vanish by increasing n (faster than 1n), and the total energy E=nepsilon, rather than epsilon, appears to be the relevant variable when n is large and epsilon相似文献   

A Novel Method for the Detection of Trace Gases Based on Photo-Acoustic Spectroscopy

We study quartz tuning fork (QTF) characteristics using a 532 nm semiconductor laser with a power of 39 mW and calculate QTF vibrations caused by thermal noise and disturbance of the air using the equipartition theorem; the vibration value is about 1.152 Pm. The signal-to-noise ratio and QTF resonance amplitude acquired experimentally are 104.56 and 214.75 Pm, respectively. In addition, we develop a new photo-acoustic spectroscopic system for detection of trace acetylene using a CW diode laser source with distributed feedback operating near 1,532 nm, measure the absorption spectrum of acetylene employing this system, and show that the method elaborated is more sensitive than photoelectric detectors that provides new directions for research in photo-acoustic spectroscopy.     

Noncentral forces in crystals of charged colloids

The elastic properties of fcc crystals consisting of charge stabilized colloidal particles are determined from real space imaging experiments using confocal microscopy. The normal modes and the force constants of the crystal are obtained from the fluctuations of the particles around their lattice sites using the equipartition theorem. We show that the Cauchy relation is not fulfilled and that only noncentral many-body forces can account for the elastic properties.     

等离子体中的颗粒成长模型

研究等离子体环境下的带电微小颗粒的成长过程．在收集离子和颗粒间凝合两种成长机制中,后者起主要作用,使生成的颗粒粒度接近．基于能量均分定理提出一个简化的模型,推导出颗粒粒度分布函数的动力方程．初步的数值模拟结果证实了上述成长机制的设想. 关键词：     

DNA分子力学性质的测量

利用单分子实验技术测量了DNA分子的力学性质.通过生化手段将单根DNA分子连在顺磁小球与玻璃侧壁之间,利用磁铁对顺磁球施加力进而拉伸DNA.图像分析得到小球运动的轨迹以及DNA的长度,利用均分定理可以求得施加的外力.测量发现在外力作用下DNA分子的力学反应与高分子物理学的蠕虫状链模型符合良好.     

Central Limit Theorem for Exponentially Quasi-local Statistics of Spin Models on Cayley Graphs

Central limit theorems for linear statistics of lattice random fields (including spin models) are usually proven under suitable mixing conditions or quasi-associativity. Many interesting examples of spin models do not satisfy mixing conditions, and on the other hand, it does not seem easy to show central limit theorem for local statistics via quasi-associativity. In this work, we prove general central limit theorems for local statistics and exponentially quasi-local statistics of spin models on discrete Cayley graphs with polynomial growth. Further, we supplement these results by proving similar central limit theorems for random fields on discrete Cayley graphs taking values in a countable space, but under the stronger assumptions of \(\alpha \)-mixing (for local statistics) and exponential \(\alpha \)-mixing (for exponentially quasi-local statistics). All our central limit theorems assume a suitable variance lower bound like many others in the literature. We illustrate our general central limit theorem with specific examples of lattice spin models and statistics arising in computational topology, statistical physics and random networks. Examples of clustering spin models include quasi-associated spin models with fast decaying covariances like the off-critical Ising model, level sets of Gaussian random fields with fast decaying covariances like the massive Gaussian free field and determinantal point processes with fast decaying kernels. Examples of local statistics include intrinsic volumes, face counts, component counts of random cubical complexes while exponentially quasi-local statistics include nearest neighbour distances in spin models and Betti numbers of sub-critical random cubical complexes.     

Thermodynamics with generalized ensembles: The class of dual orthodes

We address the problem of the foundation of generalized ensembles in statistical physics. The approach is based on Boltzmann's concept of orthodes. These are the statistical ensembles that satisfy the heat theorem, according to which the heat exchanged divided by the temperature is an exact differential. This approach can be seen as a mechanical approach alternative to the well established information-theoretic one based on the maximization of generalized information entropy. Our starting point are the Tsallis escort ensembles which have been previously proved to be orthodes, and have been proved to interpolate between canonical and microcanonical ensembles. Here we shall see that the Tsallis escort ensembles belong to a wider class of orthodes that include the most diverse types of ensembles. All such ensembles admit both a microcanonical-like parametrization (via the energy), and a canonical-like one (via the parameter β). For this reason we name them “dual”. One central result used to build the theory is a generalized equipartition theorem. The theory is illustrated with a few examples and the equivalence of all the dual orthodes is discussed.     

Neutrino flavor conversion in random magnetic fields

If massive neutrinos possess magnetic moments, a magnetic field can cause a spin flip. In the case of Dirac neutrinos the spin flip converts an active neutrino into a sterile one and vice versa. By constrast, if neutrinos are Majorana particles, a spin flip converts them to a neutrino of a different flavor. We examine the behavior of neutrinos in a random magnetic field as it occurs, for instance, in certain astronomical objects, such as an active galactic nucleus. Both Dirac and Majorana neutrinos behave ergodically: independently of their initial density matrix, they tend towards an equipartition of the helicity states. As a result, about half of the Dirac neutrinos produced becomes sterile. For Majorana neutrinos, there will be an approximate equipartition of flavors, independently of the production mechanism.     

Holographic equipartition from first order action

Recently, the idea that gravity is emergent has attract many people’s attention. The “Emergent Gravity Paradigm” is a program that develop this idea from the thermodynamical point of view. It expresses the Einstein equation in the language of thermodynamics. A key equation in this paradigm is the holographic equipartition which says that, in all static spacetimes, the degrees of freedom on the boundary equal those in the bulk. And the time evolution of spacetime is drove by the departure from the holographic equipartition. In this paper, we get the holographic equipartition and its generalization from the first order formalism, that is, the connection and its conjugate momentum are considered to be the canonical variables. The final results have similar structure as those from the metric formalism. It gives another proof of holographic equipartition.     

Quantum Brownian Motion in a Simple Model System

We consider a quantum particle coupled (with strength λ) to a spatial array of independent non-interacting reservoirs in thermal states (heat baths). Under the assumption that the reservoir correlations decay exponentially in time, we prove that the motion of the particle is diffusive at large times for small, but finite λ. Our proof relies on an expansion around the kinetic scaling limit ( l\searrow 0{\lambda \searrow 0}, while time and space scale as λ⁻²) in which the particle satisfies a Boltzmann equation. We also show an equipartition theorem: the distribution of the kinetic energy of the particle tends to a Maxwell-Boltzmann distribution, up to a correction of O(λ²).