Critical behavior of sliding charge-density waves in 4-ε dimensions

Consider two interacting sub-systems with nonoverlapping charge distributions. How can the properties of the combined system be expressed in terms of properties of the individual sub-systems alone? This general problem appears in a wide variety of physical, chemical, and biological systems [Joester 2003, Stone 1996, Moore 2001]. In this chapter we provide a prescription for addressing this issue by the computation of (i) response functions and (ii) correlation functions of spontaneous fluctuations of relevant degrees of freedom in the individual sub-systems. The computation of response and correlation functions is greatly simplified by using the density matrix in Liouville space [Mukamel 1995]. Hilbert and Liouville spaces offer very different languages for the description of nonlinear response. Computing dynamical observables in terms of the wavefunction in Hilbert space requires both forward and backward propagations in time. In contrast, the density matrix calculated in Liouville space should only be propagated forward. The choice is between following only the ket, moving it both forward and backward, or following the joint forward dynamics of the ket and the bra. Artificial time variables (Keldysh loops) commonly used in many-body theory [Haug 1996] are connected with the wavefunction. The density matrix which uses the real laboratory time throughout the calculation offers a more intuitive picture. Wavefunction-based theories calculate transition amplitudes, which by themselves are not observable. The density matrix, on the other hand directly calculates physical observables. Moreover, dephasing processes (damping of off-diagonal elements of the density matrix caused by phase fluctuations) which are naturally included into the Liouville space formulation may not be described in terms of the wavefunction. The “causality paradox” of TDDFT [Gross 1996] can be clearly resolved in Liouville space [Mukamel 2005]. In this chapter we present a method for expressing the joint response of two interacting sub-systems in terms of their correlation and response functions. This factorization appears quite naturally in Liouville space. The p order response of the individual systems is a linear combination of 2 distinct (p + 1)-point correlation functions known as Liouville space pathways [Mukamel 1995], which differ by whether the interaction at each time

The dynamics of liquids is discussed with the help of time-dependent correlation functions. They are related to response functions by the fluctuation-dissipation theorem. This theorem enables to relate experimentally measured inelastic scattering data to the Fourier-transformed correlation and response functions. The Laplace transform of the correlation functions can be represented as a continuous fraction with suitable residual terms (memory functions). The projection formalism of Mori and Zwanzig is introduced.KeywordsCorrelation FunctionMemory FunctionRelaxation FunctionLiouville OperatorGeneralize Langevin EquationThese keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

Atmospheric optical turbulence means refractive index random fluctuation of atmosphere. In this article, according to the concept of correlation function, the measurement principle, measurement schemes, and data processing method of spatial correlation function are given based on a high-quality fiber optical turbulence sensing array. Determining the statistical time and the calculation principle of the spatial correlation is the main point of current research. Emphasis is put on demonstrating the kinds of structural forms and analyzing the impact elements of spatial correlation function in turbulence as clear as possible. Using the sensing array, experimental measurement is promoted in the near ground layer and many forms of correlation functions are revealed. Results show that there are two main structural forms of the spatial correlation function:the first one shows an isotropy-model form, which tends to decrease with the increase of spatial displacement, and then tends to zero after outer scale, the coincidence rate is about 58.7%. The other one tends to oscillate around zero, and the coincidence rate is about 37.9%. By analyzing the probability and impact elements, it is not difficult to know that the spatial correlation of an optical turbulence mainly depends on the intensity and development degree of the optical turbulence; and the coherent structure is an important factor of oscillation in the correlation functions. On the one hand, the value of correlation coefficient is mainly determined by the intensity of the optical turbulence; and on a certain scale, the stronger the turbulence, the bigger the value of correlation coefficient becomes. On the other hand, the variation tendency of correlation function is not only determined by the intensity of turbulence, but also by the development degree of the optical turbulence. When the atmosphere is in advection or anisotropy, its spatial correlation coefficient will oscillate around zero and be unrelated to the spatial displacement. The spatial correlation function obtained by the sensor array set is not only the foundation of analyzing the spatial structure, but also the beginning of giving non-Kolmogorov model of turbulence.

We investigate the Kondo effect of the Fermi arcs in a time-reversal-invariant Weyl semimetal with the variational method. To show the consequence brought out by the nontrivial spin texture, we calculate the spatial spin-spin correlation functions. The correlation functions exhibit high anisotropy. The diagonal correlation functions are dominated by the antiferromagnetic correlation while the off-diagonal part has more complicated pattern. The correlation functions obey the same symmetry as the spin texture. Tuning chemical potential changes the pattern of the correlation functions and the correlation length. The correlation functions of the Weyl semimetal Fermi arcs and that from a Dirac semimetal show discrepancy.

We study the mode-coupling approximation for the Kardar-Parisi-Zhang equation in the strong-coupling regime. By constructing an ansatz consistent with the asymptotic forms of the correlation and response functions we determine the upper critical dimension d(c) = 4 and the expansion z = 2-(d-4)/4+O((4-d)2) around dc. We find the exact z = 3/2 value in d = 1, and estimate the values z approximately 1.62, z approximately 1.78 in d = 2, 3. The result dc = 4 and the expansion around dc are very robust and can be derived just from a mild assumption on the relative scale on which the response and correlation functions vary as z approaches 2.

We study the non-linear evolution of baryon acoustic oscillations in the matter power spectrum and correlation function from the improved perturbation theory (PT). Based on the framework of renormalized PT, we apply the {\it closure approximation} that truncates the infinite series of loop contributions at one-loop order, and obtain a closed set of integral equations for power spectrum and non-linear propagator. The resultant integral expressions keep important non-perturbative properties which can dramatically improve the prediction of non-linear power spectrum. Employing the Born approximation, we then derive the analytic expressions for non-linear power spectrum and the predictions are made for non-linear evolution of baryon acoustic oscillations in power spectrum and correlation function. A detailed comparison between improved PT results and N-body simulations shows that a percent-level agreement is achieved in a certain range in power spectrum and in a rather wider range in correlation function. Combining a model of non-linear redshift-space distortion, we also evaluate the power spectrum and correlation function in correlation function. In contrast to the results in real space, the agreement between N-body simulations and improved PT predictions tends to be worse, and a more elaborate modeling for redshift-space distortion needs to be developed. Nevertheless, with currently existing model, we find that the prediction of correlation function has a sufficient accuracy compared with the cosmic-variance errors for future galaxy surveys with volume of a few (Gpc/h)^3 at z>=0.5.

Exact three spin correlation function relations for the square and the honeycomb Ising lattices

Bayesian model comparison and selection of spatial correlation functions for soil parameters

This paper investigates the asymptotic behavior of the current correlation function at large distances by means of the expansion constructed by the authors and Isergin; the expansion was performed earlier. The correlation length is calculated for arbitrary coupling constant. In the framework of perturbation theory, the correlation functions of a model were considered in which a perturbation theory was constructed on the basis of successive integration, first over the fast and then over the slow fields, and this theoery was used to investigate the behavior of the correlation functions at large distances.

A spatial correlation function has been previously proposed as a means of quantifying chaotic characteristics of quantum wavefunctions. In this paper we show that the classical analogue of the spatial correlation of a quantum wavefunction is identical (over short ranges) to its quantum counterpart, for the majority of high lying eigenstates of the classically chaotic stadium billiard problem. This behaviour is identified with quantum ergodicity. Furthermore the classical correlation function is used to derive an energy‐scaling law for the correlation lengths. The spatial correlation approach is then generalized by considering cross‐correlation functions and the time evolution of the correlation lengths for non‐stationary quantum states. This quantity is shown to be a near constant of the motion for wavepackets comprised of chaotic states and may relate to mixing properties of the dynamics.

A fermion system with a simple attractive interaction is discussed with the aid of time-dependent correlation functions. Although perturbation theory is inapplicable, a sequence of correlation approximations described in the first paper of this series can be employed. The lowest approximation in the sequence expresses the two-particle correlation function in terms of single-particle functions and leads to the Hartree approximation; the second expresses three-particle correlation function in terms of one- and two-particle correlation functions and leads to the time-dependent correlation functions that characterize the super-conducting model of Bardeen, Cooper, and Schrieffer. In the second section of this paper these correlation functions are determined and the thermodynamic properties of the superconductor are calculated from them.In the third section of the paper, the electromagnetic effects of the superconductor predicted by the Bardeen-Cooper-Schrieffer time-dependent correlation functions are considered. Their unsatisfactory description of current conservation is indicated and overcome in the fourth section by a more accurate solution valid at nonvanishing temperature. This solution predicts different diffusive properties but the same Meissner effect and superconductive behavior, since the longitudinal current correlation function is modified while the transverse current correlation functions is not.The fifth section of the paper is devoted to the properties of a pure superconductor which depend on the lifetimes of the single-particle excitations. The effect of these lifetimes on the static electrical conductivity is determined, and it is shown that they do not destroy supercurrents although they eliminate a gap in the single-particle excitation spectrum. Their effect on the thermal conductivity is also calculated using heat current correlation functions. It is shown that a model which treats the lifetime of the single-particle excitation due to lattice interactions as constant yields results in agreement with observed thermal conductivities.

Contact terms and duality symmetry in the critical dissipative Hofstadter model

Nonstationary response statistics of structures with hysteretic damping to evolutionary stochastic excitation

The statistics of systems with good chaotic properties obey a formal fluctuation-response relation which gives the average linear response of a dynamical system to an external perturbation in terms of two-time correlation functions. Unfortunately, except for particularly simple cases, the appropriate form of correlation function is unknown because an analytic expression for the invariant density is lacking. The simplest situation is that in which the probability distribution is Gaussian. In that case, the fluctuation-response relation is a linear relation between the response matrix and the two-time two-point correlation matrix. Some numerical computations have been carried out in low-dimensional models of hydrodynamic systems. The results show that fluctuation-response relation for Gaussian distributions is not a useful approximation. Nevertheless, these calculations show that, even for non-Gaussian statistics, the response function and the two-time correlations can have similar qualitative features, which may be attributed to the existence of the more general fluctuation-response relation.

In a simple nonequilibrium statistical mechanics model (two coupled nonlinear stochastic reaction-diffusion equations), self-sustained oscillations appear as a symmetry-breaking effect for the associated probability density. Correlation and response functions are approximately calculated above and below the transition point. Renormalization-group theory yields the leading-order scaling of order parameter, relaxation time, correction to the collective frequency, and correlation length.