Time behavior of the correlation functions in a simple dissipative quantum model

The force and velocity correlation functions for a particle interacting with a bath are calculated within a model allowing for finite memory effects. The relevance of a Brownian picture is delineated in view of the respective behavior of these functions and appears fully inadequate below some cross-over temperature; then, the interplay between quantum and thermal fluctuations yields some long time tails on the same time scale for both correlation functions. The real space transient diffusion coefficient is found to exceed its asymptotic Einstein value for most times in that regime. The limiting case of an infinitely short memory time is also investigated and is seen to produce weak divergences on a time scale which is small as compared to the other characteristic times.     

Time reversal and symmetries of time correlation functions

ABSTRACT

The time reversal invariance of classical dynamics is reconsidered in this paper with specific focus on its consequences for time correlation functions and associated properties such as transport coefficients. We show that, under fairly common assumptions on the interparticle potential, an isolated Hamiltonian system obeys more than one time reversal symmetry and that this entails non trivial consequences. Under an isotropic and homogeneous potential, in particular, eight valid time reversal operations exist. The presence of external fields that reduce the symmetry of space decreases this number, but does not necessarily impair all time reversal symmetries. Thus, analytic predictions of symmetry properties of time correlation functions and, in some cases, even of their null value are still possible. The noteworthy case of a constant external magnetic field, usually assumed to destroy time reversal symmetry, is considered in some detail. We show that, in this case too, some of the new time reversal operations hold, and that this makes it possible to derive relevant properties of correlation functions without the uninteresting inversion of the direction of the magnetic field commonly enforced in the literature.     

A nonlinear correlation measure for multivariable data set

The paper states that the mutual information carried by the rank sequences that are obtained from the original two sequences is a good measure of nonlinear correlation. Based on that, the nonlinear correlation information entropy (NCIE) is proposed for measuring the general relationship of a multivariable data set. NCIE uses a number in the closed interval [0, 1] to indicate the nonlinear correlation degree of the concerned multivariable data set, with 0 and 1 denotes the weakest and the strongest relationship, respectively. Data sets generated from an autoregressive model, a nonlinear chaotic Lorenz system, and a logistic function as known correlation sequences are analyzed using the proposed NCIE. The results testify the suitableness and correctness of the proposed concepts as nonlinear correlation measure.     

Properties of then-body correlation functions near the liquid-gas critical point. Correlation inequalities

We investigate the behavior of the many-body correlation functions in the vicinity of the gas-liquid critical point. We use the framework of the liquid state theory and, accordingly, no reference to an effective Landau-Ginzburg Hamiltonian is made. The critical condition is introduced by means of the equation of state. From the Baxter equation relating the many-body correlation functionsh(n) andh(n+1), we find that the integrals of all theh(n) diverge at the critical point. Then we present strong arguments and this leads to GKS-like inequalities, under some limiting conditions: the interparticle distances must be large and the thermodynamic state of the system must be close to the critical point. In order to get these inequalities, an upper bound forh(n) is obtained. Particular attention must be paid to the fact that the usual asymptotic approximations of the liquid state theory are no longer valid.     

On the measurability of quantum correlation functions

The concept of correlation function is widely used in classical statistical mechanics to characterize how two or more variables depend on each other. In quantum mechanics, on the other hand, there are observables that cannot be measured at the same time; the so-called incompatible observables. This prospect imposes a limitation on the definition of a quantum analog for the correlation function in terms of a sequence of measurements. Here, based on the notion of sequential weak measurements, we circumvent this limitation by introducing a framework to measure general quantum correlation functions, in principle, independently of the state of the system and the operators involved. To illustrate, we propose an experimental configuration to obtain explicitly the quantum correlation function between two Pauli operators, in which the input state is an arbitrary mixed qubit state encoded on the polarization of photons.     

Evaluation of time correlation functions from a generalized Enskog equation

Numerical results for the density and current correlation functions in dense hard-sphere fluids are obtained from a kinetic equation which is the extension of the linearized Enskog equation to finite wavelengths in order to demonstrate the convergence of the method of solution. Comparison is made to a previously proposed approximate solution.This work was performed in part under the auspices of the U.S. Department of Energy by the Lawrence Livermore Laboratory under contract number W-7405-ENG-48 and in part supported by the National Science Foundation.     

Mutual information analysis reveals bigeminy patterns in Andersen-Tawil syndrome and in subjects with a history of sudden cardiac death

Herein we introduce the Mutual Information Function (MIF) as a mathematical method to analyze ventricular bigeminy in certain pathological conditions of the heart known to be associated with frequent ventricular arrhythmias. In particular, we show that the MIF is sensitive enough to detect the bigeminy pattern in symbolic series from patients with Andersen-Tawil syndrome as well as in a group of patients from the Sudden Cardiac Death Holter Databases. The results confirm that MIF is an adequate method to detect the autocorrelation between the appearance of sinus and ventricular premature beats resulting in a bigeminy pattern. It is also shown that MIF reflects the bigeminy patterns as a function of the percentage of ventricular premature beats present in the symbolic series and also as a function of the percentage of bigeminy. The MIF was also useful to establish a consistent difference in the bigeminy pattern related to the diurnal and nocturnal periods presumably associated to the circadian rhythm of the heart. Understanding of the ventricular bigeminy patterns throughout 24-hours could provide some insights into the pathogenesis of ventricular tachyarrhythmias in these pathological conditions.     

Billiards correlation functions

We discuss various experiments on the time decay of velocity autocorrelation functions in billiards. We perform new experiments and find results which are compatible with an exponential mixing hypothesis first put forward by Friedman and Martin (FM): they do not seem compatible with the stretched exponentials believed, in spite of FM and more recently of Chernov, to describe the mixing. The analysis leads to several byproducts: we obtain information about the normal diffusive nature of the motion and we consider the probability distribution of the number of collisions in timet _m (ast _m ), finding a strong dependence on some geometric characteristics of the locus of the billiard obstacles.This paper is dedicated to Philippe Choquard on his 65th birthday     

Equilibrium time correlation functions in the low-density limit

We consider a system of hard spheres in thermal equilibrium. Using Lanford's result about the convergence of the solutions of the BBGKY hierarchy to the solutions of the Boltzmann hierarchy, we show that in the low-density limit (Boltzmann-Grad limit): (i) the total time correlation function is governed by the linearized Boltzmann equation (proved to be valid for short times), (ii) the self time correlation function, equivalently the distribution of a tagged particle in an equilibrium fluid, is governed by the Rayleigh-Boltzmann equation (proved to be valid for all times). In the latter case the fluid (not including the tagged particle) is to zeroth order in thermal equilibrium and to first order its distribution is governed by a combination of the Rayleigh-Boltzmann equation and the linearized Boltzmann equation (proved to be valid for short times).Supported in part by NSF Grant PHY 78-22302.     

Time correlation functions of a one-dimensional infinite system

We investigate the time evolution of a simple one-dimensional system with an infinite number of particles. We calculate some time correlation functions and show that they behave asymptotically as 1/t.     

A renormalization group analysis of correlation functions for the dipole gas

We develop a renormalization group method for analyzing the generating functional for charge correlations of a dilute classical dipole gas. It is based on and extends the renormalization group analysis introduced by Brydges and Yau for the dipole gas partition function. Our method leads to systematic formulas for the large-distance behavior of correlation functions of all orders. We prove that in any dimensiond2, at any value>0 of the inverse temperature, and at sufficiently small activityz, the correlation functions exhibit at large distances the same behavior as for a vacuum (z=0), but with a new dielectric constant 1+ over which we have good control. The results proved here extend existing results on the two-point correlations to all higher correlations, and constitute a general confirmation of the fact that dipoles do not screen.     

A new quantum statistical evaluation method for time correlation functions

Considering a system ofN identical interacting particles, which obey Fermi-Dirac or Bose-Einstein statistics, we derive new formulas for correlation functions of the type (whereB _j is diagonal in the free-particle states) in the thermodynamic limit. Thereby we apply and extend a superoperator formalism, recently developed for the derivation of long-time tails in semiclassical systems. As an illustrative application, the Boltzmann equation value of the time-integrated correlation functionC(t) is derived in a straightforward manner. Due to exchange effects, the obtained t-matrix and the resulting scattering cross section, which occurs in the Boltzmann collision operator, are now functionals of the Fermi-Dirac or Bose-Einstein distribution.     

Molecular dynamics studies of thermal conductivity time correlation functions

The Green–Kubo time correlation function for the thermal conductivity in liquid argon is studied for a thermodynamic state close to the triple point by standard molecular dynamics simulations. The collective heat flux vector has been separated into contributions originated at the kinetic energy, the intermolecular potential and the pair virial function. Furthermore, the Green–Kubo time correlation functions have been broken down into partial n-body terms (n=1,2,3,4). The most important contribution to the thermal conductivity is represented by the auto correlation of the virial term. In contrast to other collective phenomena described by time correlation functions involving n-body terms, the partial Green–Kubo time correlation functions for the thermal conductivity are not affected by exponential long-time tails.     

开边界六顶角模型的边界关联函数

讨论了具有开边界六顶角模型的关联函数，计算了涉及边界自发极化的边界关联函数，得到了它们的行列式表示. 关键词： 六顶角模型 关联函数 开边界     

An investigation into the linear and nonlinear correlation of two music walk sequences

In order to investigate the features of a multiple-part musical score which enhance its appeal to the listener’s ear, this study performs a robust analysis of the correlation between two musical sequences. In the proposed approach, a series of notes are extracted from seven well-known classical pieces of music and are converted into one-variable “music walks”. The linear correlation between pairs of music walks is assessed using the conventional linear correlation coefficient, while the nonlinear correlation is examined using the mutual information concept. The results show that even though two musical time walks may exhibit virtually no linear correlation, they invariably have a certain degree of nonlinear correlation. In other words, to truly understand the correlation between two musical sequences, it is necessary to consider not only the linear correlation between them, but also the nonlinear correlation. In addition, it is shown that the normalized mutual information coefficient between musical sequences has a relatively low value and varies significantly over the course of the musical score. Thus, it can be inferred that the appeal of a musical score stems at least in part from significant variations in both the melody and the rhythm of the constituent parts such that the overall score has a rich and unpredictable property.     

Multivariate Gaussian Copula Mutual Information to Estimate Functional Connectivity with Less Random Architecture

Recognition of a brain region’s interaction is an important field in neuroscience. Most studies use the Pearson correlation to find the interaction between the regions. According to the experimental evidence, there is a nonlinear dependence between the activities of different brain regions that is ignored by Pearson correlation as a linear measure. Typically, the average activity of each region is used as input because it is a univariate measure. This dimensional reduction, i.e., averaging, leads to a loss of spatial information across voxels within the region. In this study, we propose using an information-theoretic measure, multivariate mutual information (mvMI), as a nonlinear dependence to find the interaction between regions. This measure, which has been recently proposed, simplifies the mutual information calculation complexity using the Gaussian copula. Using simulated data, we show that the using this measure overcomes the mentioned limitations. Additionally using the real resting-state fMRI data, we compare the level of significance and randomness of graphs constructed using different methods. Our results indicate that the proposed method estimates the functional connectivity more significantly and leads to a smaller number of random connections than the common measure, Pearson correlation. Moreover, we find that the similarity of the estimated functional networks of the individuals is higher when the proposed method is used.     

The poisson representation. II Two-time correlation functions

Basic formulas for the two-time correlation functions are derived using the Poisson representation method. The formulas for the chemical system in thermodynamic equilibrium are shown to relate directly to the fluctuationdissipation theorems, which may be derived from equilibrium statistical mechanical considerations. For nonequilibrium systems, the formulas are shown to be generalizations of these fluctuation-dissipation theorems, but containing an extra term which arises entirely from the nonequilibrium nature of the system. These formulas are applied to two representative examples of equilibrium reactions (without spatial diffusion) and to a nonequilibrium chemical reaction model (including the process of spatial diffusion) for which the first two terms in a systematic expansion for the two-time correlation functions are calculated. The relation between the Poisson representation method and Glauber-SudarshanP-representation used in quantum optics is discussed.     

Exact expressions for row correlation functions in the isotropicd=2 ising model

We present exact explicit expressions for the row spin-spin correlation functions _{00 n}0 in the isotropicd= 2 Ising model, in terms of elliptic integrals, forn 5. We also give a general structural formula for _{00 n}0.     

Stress-stress correlation functions in lattice gases beyond Boltzmann's approximation

The complete time dependence of the stress-stress correlation functions in lattice gas cellular automata is calculated from the ring kinetic theory using numerical and analytical methods. This provides corrections, typically of 10–20%, to the usual molecular chaos calculations, where correlation functions decay exponentially. The resulting correlation function crosses over from an initial exponential decay to the long-time behavior calculated from mode coupling theory. The present theory, applied to the viscosity, accounts for a substantial part of the observed difference between the Boltzmann theory and simulations.