Expression For Correlation Function Research Articles

Solution of the Dynamics of Liquids in the Large-Dimensional Limit.

We obtain analytic expressions for the time correlation functions of a liquid of spherical particles, exact in the limit of high dimensions d. The derivation is long but straightforward: a dynamic virial expansion for which only the first two terms survive, followed by a change to generalized spherical coordinates in the dynamic variables leading to saddle-point evaluation of integrals for large d. The problem is, thus, mapped onto a one-dimensional diffusion in a perturbed harmonic potential with colored noise. At high density, an ergodicity-breaking glass transition is found. In this regime, our results agree with thermodynamics, consistently with the general random first order transition scenario. The glass transition density is higher than the best known lower bound for hard sphere packings in large d. Because our calculation is, if not rigorous, elementary, an improvement in the bound for sphere packings in large dimensions is at hand.

Deriving the exact nonadiabatic quantum propagator in the mapping variable representation.

We derive an exact quantum propagator for nonadiabatic dynamics in multi-state systems using the mapping variable representation, where classical-like Cartesian variables are used to represent both continuous nuclear degrees of freedom and discrete electronic states. The resulting Liouvillian is a Moyal series that, when suitably approximated, can allow for the use of classical dynamics to efficiently model large systems. We demonstrate that different truncations of the exact Liouvillian lead to existing approximate semiclassical and mixed quantum-classical methods and we derive an associated error term for each method. Furthermore, by combining the imaginary-time path-integral representation of the Boltzmann operator with the exact Liouvillian, we obtain an analytic expression for thermal quantum real-time correlation functions. These results provide a rigorous theoretical foundation for the development of accurate and efficient classical-like dynamics to compute observables such as electron transfer reaction rates in complex quantized systems.

Four-point correlation function of stress-energy tensors in N = 4 $$ \mathcal{N}=4 $$ superconformal theories

We derive the explicit expression for the four-point correlation function of stress-energy tensors in four-dimensional N=4 superconformal theory. We show that it has a remarkably simple and suggestive form allowing us to predict a large class of four-point correlation functions involving the stress-energy tensor and other conserved currents. We then apply the obtained results on the correlation functions to computing the energy-energy correlations, which measure the flow of energy in the final states created from the vacuum by a source. We demonstrate that they are given by a universal function independent of the choice of the source. Our analysis relies only on N=4 superconformal symmetry and does not use the dynamics of the theory.

Generalized hydrodynamic correlations and fractional memory functions

Abstract A fractional generalized hydrodynamic (GH) model of the longitudinal velocity fluctuations correlation, and its associated memory function, for a complex fluid is analyzed. The adiabatic elimination of fast variables introduces memory effects in the transport equations, and the dynamic of the fluctuations is described by a generalized Langevin equation with long-range noise correlations. These features motivate the introduction of Caputo time fractional derivatives and allows us to calculate analytic expressions for the fractional longitudinal velocity correlation function and its associated memory function. Our analysis eliminates a spurious constant term in the non-fractional memory function found in the non-fractional description. It also produces a significantly slower power-law decay of the memory function in the GH regime that reduces to the well-known exponential decay in the non-fractional Navier–Stokes limit.

Non-classical light from superconducting resonators coupled to voltage-biased Josephson junctions

The interplay of the tunneling transfer of charges and the emission and absorption of light can be investigated in a set-up, where a voltage-biased Josephson junction (JJ) is placed in series to a microwave cavity. Here, we concentrate on the emission process and show that due to the inherent nonlinearity of the JJ tunneling Cooper-pairs can create a variety of non-classical states of light. Depending on experimental parameters and tuning the device can be described by effective Hamiltonians, indicating specific photon creation mechanisms which lead to strongly bunched or anti-bunched light emission. We derive explicit analytical expressions for the photon correlation function for these different processes and investigate their full crossover numerically. Photon distribution functions show squeezing and other interesting non-Poissonian behavior.

Flexocoupling impact on the generalized susceptibility and soft phonon modes in the ordered phase of ferroics

Flexoelectric effect impact on the generalized susceptibility and soft phonons dispersion was not studied in the long-range ordered phases of ferroics. The gap in the knowledge motivated us to establish the impact of the flexocoupling on the correlation function of the long-range order parameter fluctuations in ferroelectric phase of ferroics with local disordering sources. Within Landau-Ginzburg-Devonshire approach we obtained analytical expressions for the generalized susceptibility and phonon dispersion relations in the ferroelectric phase. Unexpectedly, the joint action of static and dynamic flexoelectric effect induces non-diagonal components of generalized susceptibility, which amplitude is proportional to the convolution of the spontaneous polarization with flexocoupling constants. The flexocoupling essentially broaden the k-spectrum of generalized susceptibility and so decreases the correlation radii, as well as leads to the additional "pushing away" of the optical and acoustic soft mode phonon branches. The contribution of spontaneous polarization via ferroelectric nonlinearity and electrostriction mechanisms can lead to both broadening and narrowing on the susceptibility k-spectrum. Due to the joint action of flexoelectric coupling and spontaneous polarization the degeneration of the transverse optic and acoustic modes disappears in the soft phonon spectrum in a ferroelectric phase in comparison with the spectra in a paraelectric phase. These effects have been studied quantitatively for ferroelectric lead zirconate titanate using realistic values of static and dynamic flexocouplings and cubic symmetry approximation for the elastic properties. Also we derived general expressions for correlation function for arbitrary symmetry, elastic and electrostrictive anisotropy.

Correlation functions from a unified variational principle: Trial Lie groups

Time-dependent expectation values and correlation functions for many-body quantum systems are evaluated by means of a unified variational principle. It optimizes a generating functional depending on sources associated with the observables of interest. It is built by imposing through Lagrange multipliers constraints that account for the initial state (at equilibrium or off equilibrium) and for the backward Heisenberg evolution of the observables. The trial objects are respectively akin to a density operator and to an operator involving the observables of interest and the sources. We work out here the case where trial spaces constitute Lie groups. This choice reduces the original degrees of freedom to those of the underlying Lie algebra, consisting of simple observables; the resulting objects are labeled by the indices of a basis of this algebra. Explicit results are obtained by expanding in powers of the sources. Zeroth and first orders provide thermodynamic quantities and expectation values in the form of mean-field approximations, with dynamical equations having a classical Lie–Poisson structure. At second order, the variational expression for two-time correlation functions separates–as does its exact counterpart–the approximate dynamics of the observables from the approximate correlations in the initial state. Two building blocks are involved: (i) a commutation matrix which stems from the structure constants of the Lie algebra; and (ii) the second-derivative matrix of a free-energy function. The diagonalization of both matrices, required for practical calculations, is worked out, in a way analogous to the standard RPA. The ensuing structure of the variational formulae is the same as for a system of non-interacting bosons (or of harmonic oscillators) plus, at non-zero temperature, classical Gaussian variables. This property is explained by mapping the original Lie algebra onto a simpler Lie algebra. The results, valid for any trial Lie group, fulfill consistency properties and encompass several special cases: linear responses, static and time-dependent fluctuations, zero- and high-temperature limits, static and dynamic stability of small deviations.

Anomalous processes with general waiting times: functionals and multipoint structure.

Many transport processes in nature exhibit anomalous diffusive properties with nontrivial scaling of the mean square displacement, e.g., diffusion of cells or of biomolecules inside the cell nucleus, where typically a crossover between different scaling regimes appears over time. Here, we investigate a class of anomalous diffusion processes that is able to capture such complex dynamics by virtue of a general waiting time distribution. We obtain a complete characterization of such generalized anomalous processes, including their functionals and multipoint structure, using a representation in terms of a normal diffusive process plus a stochastic time change. In particular, we derive analytical closed form expressions for the two-point correlation functions, which can be readily compared with experimental data.

Ground states of stealthy hyperuniform potentials. II. Stacked-slider phases.

Stealthy potentials, a family of long-range isotropic pair potentials, produce infinitely degenerate disordered ground states at high densities and crystalline ground states at low densities in d-dimensional Euclidean space R^{d}. In the previous paper in this series, we numerically studied the entropically favored ground states in the canonical ensemble in the zero-temperature limit across the first three Euclidean space dimensions. In this paper, we investigate using both numerical and theoretical techniques metastable stacked-slider phases, which are part of the ground-state manifold of stealthy potentials at densities in which crystal ground states are favored entropically. Our numerical results enable us to devise analytical models of this phase in two, three, and higher dimensions. Utilizing this model, we estimated the size of the feasible region in configuration space of the stacked-slider phase, finding it to be smaller than that of crystal structures in the infinite-system-size limit, which is consistent with our recent previous work. In two dimensions, we also determine exact expressions for the pair correlation function and structure factor of the analytical model of stacked-slider phases and analyze the connectedness of the ground-state manifold of stealthy potentials in this density regime. We demonstrate that stacked-slider phases are distinguishable states of matter; they are nonperiodic, statistically anisotropic structures that possess long-range orientational order but have zero shear modulus. We outline some possible future avenues of research to elucidate our understanding of this unusual phase of matter.

Motional narrowing under diffusion of two spin sub-lattices in LiNaSO4

An expression for the dipolar correlation function characterizing the fluctuations of the Hamiltonian under uncorrelated diffusion of two cationic sub-lattices in a solid has been obtained. A corresponding formula, which can be used for fitting the two-step temperature dependencies of the NMR line width, has been deduced. We also considered the case of a distribution of correlation times and deduced a corresponding analytical fitting function. Elaborated expressions have been applied to the NMR line width analysis of LiNaSO4 in the temperature range 295–788K.

Infinite matrix product states, boundary conformal field theory, and the open Haldane-Shastry model

We show that infinite matrix product states (MPSs) constructed from conformal field theories can describe ground states of one-dimensional critical systems with open boundary conditions. To illustrate this, we consider a simple infinite MPS for a spin-1/2 chain and derive an inhomogeneous open Haldane-Shastry model. For the spin-1/2 open Haldane-Shastry model, we derive an exact expression for the two-point spin correlation function. We also provide an $\mathrm{SU}(n)$ generalization of the open Haldane-Shastry model and determine its twisted Yangian generators responsible for the highly degenerate multiplets in the energy spectrum.

Gas phase infrared spectra from quasi-classical Kubo time correlation functions

We generalise the recently developed phase integration method (PIM) to obtain a computable approximation of the Kubo expression for quantum time correlation functions. Our scheme combines exact sampling of the quantum thermal density with classical dynamics to provide a quasi-classical approximation for the correlation function. The method will be specialised to the evaluation of the momentum autocorrelation function, with the goal to compute infrared spectra of simple molecules in the gas phase. Application to two simple but interesting benchmark systems shows that the approach is accurate and stable over a broad range of temperatures.

A Generalized Spatial Correlation Model for 3D MIMO Channels Based on the Fourier Coefficients of Power Spectrums

Previous studies have confirmed the adverse impact of fading correlation on the mutual information (MI) of two-dimensional (2D) multiple-input multiple-output (MIMO) systems. More recently, the trend is to enhance the system performance by exploiting the channel's degrees of freedom in the elevation, which necessitates the derivation and characterization of three-dimensional (3D) channels in the presence of spatial correlation. In this paper, an exact closed-form expression for the Spatial Correlation Function (SCF) is derived for 3D MIMO channels. This novel SCF is developed for a uniform linear array of antennas with nonisotropic antenna patterns. The proposed method resorts to the spherical harmonic expansion (SHE) of plane waves and the trigonometric expansion of Legendre and associated Legendre polynomials. The resulting expression depends on the underlying arbitrary angular distributions and antenna patterns through the Fourier Series (FS) coefficients of power azimuth and elevation spectrums. The novelty of the proposed method lies in the SCF being valid for any 3D propagation environment. The developed SCF determines the covariance matrices at the transmitter and the receiver that form the Kronecker channel model. In order to quantify the effects of correlation on the system performance, the information-theoretic deterministic equivalents of the MI for the Kronecker model are utilized in both mono-user and multi-user cases. Numerical results validate the proposed analytical expressions and elucidate the dependence of the system performance on azimuth and elevation angular spreads and antenna patterns. Some useful insights into the behaviour of MI as a function of downtilt angles are provided. The derived model will help evaluate the performance of correlated 3D MIMO channels in the future.

Evaluation of reaction fluxes in stationary and oscillating far-from-equilibrium biological systems

The complex spatio-temporal structures that appear in chemical and biological systems require far-from-equilibrium conditions which may lead to the circulation of reaction fluxes. We investigate how time asymmetry of cross-correlation functions of concentration fluctuations may be exploited to determine reaction fluxes at the cellular level. Using simulations of the master equation as a reference, we show that, far from a bifurcation, the Langevin approach provides a reliable tool to compute analytical expressions for time correlation functions. Biochemical mechanisms associated with bistability and oscillations issued from a Hopf bifurcation or a saddle–node infinite period bifurcation are considered. We show that the blind use of the simple relation obtained when assuming a linear deterministic dynamics often leads to a poor estimation of the value of the reaction flux and even of its sign.

Signatures of pairing and spin-orbit coupling in correlation functions of Fermi gases

We derive expressions for spin and density correlation functions in the (greatly enhanced) pseudogap phase of spin-orbit coupled Fermi superfluids. Density-density correlation functions are found to be relatively insensitive to the presence of these Rashba effects. To arrive at spin-spin correlation functions we derive new $f$-sum rules, valid even in the absence of a spin conservation law. Our spin-spin correlation functions are shown to be fully consistent with these $f$-sum rules. Importantly, they provide a clear signature of the Rashba band-structure and separately help to establish the presence of a pseudogap.

An Exact Closed Formula of a Spin-spin Correlation Function of the Two-Dimensional Ising Model with Finite Size

Some expressions of spin-spin correlation functions of the two-dimensional Ising model in the thermodynamic limit have been obtained, however, so far, no one correlation function that can show long range order of the model with finite size has been obtained. The purpose of this paper is to fill this gap. We present an exact closed formula for the correlation functions of the farthest pairs of spins in a column of the two-dimensional rectangular Ising model with periodic-free boundary condition and finite size by the spinor analysis method. The concrete expression correct to \(\mathrm {e}^{-LC_{0} }\) order of magnitude of such correlation functions is presented, from which it is shown clearly how long range order emerges as the decrease of the temperature. Some properties are discussed.

The influence of phonon bath on the control of single photon**Project supported by the Fundamental Research Funds for the Central Universities of Central South University, China (Grant No. 2014zzts145).

The influence of vacuum fluctuation and phonon bath on the probability of single photon emission are both considered in the two-level system model theoretically; by using the master equations and generating function method we get the analytical expression of the second-order fluorescence correlation function, probability of single photon emission, and Mandel’s Q parameter. The results manifest that the coupling between the phonon bath and single photon source destroys the superposition state induced by the square laser pulse, the Rabi oscillation damped rapidly with the increasing of temperature. Theoretically, when the structure parameter of arsenide quantum dots α scaled to 0.1 times of the sample, the critical coherence-temperature will rise up to hundreds of Kelvin, which means a step forward to the realization of coherent control of single photon source at room temperature.

Shear-stress function approach of hydration layer based on the Green-Kubo formula.

We present the analytic expression of the stress correlation (SC) function for the ubiquitous hydration water layer (HWL) using the Green-Kubo equation and the shear modulus of HWL. The SC function is then experimentally obtained by measuring the viscoelastic properties of HWL using shear-mode dynamic force spectroscopy. Interestingly, the SC changes sign from positive to negative as the HWL thickness increases, where the shear stresses acting on the HWLs bound to two nearby surfaces are out of phase. We also suggest that the repulsive hydration force originates from the SC of HWL. Our results provide the first demonstration of the microscopic understanding of the HWL viscoelasticity and may allow a deeper insight on the HWL dynamics as well as the complex liquids.

The role of coherence on two-particle quantum walks

Abstract We investigate the dynamical properties of the two-boson quantum walk in systems with different degrees of coherence, and where the effect of the coherence on the two-boson quantum walk can be naturally introduced. A general analytical expression for the two-boson correlation function, for both pure states and mixed states, is given.We propose a possible two-photon quantum-walk scheme with a mixed initial state, and find that the twophoton correlation function and the average distance between two photons can be influenced by the initial photon distribution, the relative phase, or the degree of coherence. The propagation features of our numerical results can be explained by our analytical two-photon correlation function.

Seeing asymptotic freedom in an exact correlator of a large-Nmatrix field theory

Exact expressions for correlation functions are known for the large-$N$ (planar) limit of the $(1+1)$-dimensional ${\rm SU}(N)\times {\rm SU}(N)$ principal chiral sigma model. These were obtained with the form-factor bootstrap, an entirely nonperturbative method. The large-$N$ solution of this asymptotically-free model is far less trivial than that of O($N$) sigma model (or other isovector models). Here we study the Euclidean two-point correlation function $N^{-1}< {\rm Tr}\,\Phi(0)^{\dagger} \Phi(x)>$, where $\Phi(x)\sim Z^{-1/2}U(x)$ is the scaling field and $U(x)\in SU(N)$ is the bare field. We express the two-point function in terms of the spectrum of the operator $\sqrt{-d^{2}/du^{2}}$, where $u\in (-1,1)$. At short distances, this expression perfectly matches the result from the perturbative renormalization group.