Four-body Correlation Functions Research Articles

Simulating two-dimensional correlation spectroscopies with third-order infrared and fifth-order infrared-Raman processes of liquid water.

To investigate the possibility of measuring the intermolecular and intramolecular anharmonic coupling of balk water, we calculate third-order two-dimensional (2D) infrared spectra and fifth-order 2D IR-IR-Raman-Raman spectra expressed in terms of four-body correlation functions of optical observables. For this purpose, a multimode Brownian oscillator model of four interacting anharmonic oscillators strongly coupled to their respective heat baths is employed. The nonlinearity of system-bath interactions is considered to describe thermal relaxation and vibrational dephasing. The linear and nonlinear spectra are then computed in a non-Markovian and nonperturbative regime in a rigorous manner using discretized hierarchical equations of motion in mixed Liouville-Wigner space. The calculated 2D spectra for stretching-bending, bending-librational, stretching-librational, and stretching-translational modes consist of various positive and negative peaks exhibiting essential details of intermolecular and intramolecular mode-mode interactions under thermal relaxation and dephasing at finite temperature.

Method of difference-differential equations for some Bethe-ansatz-solvable models

In studies of one-dimensional Bethe ansatz solvable models, a Fredholm integral equation of the second kind with a difference kernel on a finite interval often appears. This equation does not generally admit a closed-form solution and hence its analysis is quite complicated. Here we study a family of such equations, concentrating on their moments. We find exact relations between the moments in the form of difference-differential equations. The latter results significantly advance the analysis, enabling one to practically determine all the moments from the explicit knowledge of the lowest one. As applications, several examples are considered. First, we study the moments of the quasimomentum distribution in the Lieb-Liniger model and find explicit analytical results. The latter moments determine several basic quantities, e.g., the $N$-body local correlation functions. We prove the equivalence between different expressions found in the literature for the three-body local correlation functions and find an exact result for the four-body local correlation function in terms of the moments of the quasimomentum distributions. We eventually find the analytical results for the three- and four-body correlation functions in the form of asymptotic series in the regimes of weak and strong interactions. Next, we study the exact form of the low-energy spectrum of a magnon (a polaron) excitation in the two-component Bose gas described by the Yang-Gaudin model. We find its explicit form, which depends on the moments of the quasimomentum distributions of the Lieb-Liniger model. Then, we address a seemingly unrelated problem of capacitance of a circular capacitor and express the exact result for the capacitance in the parametric form. In the most interesting case of short plate separations, the parametric form has a single logarithmic term. This should be contrasted with the explicit result that has a complicated structure of logarithms.

Relation between dynamic heterogeneities observed in scattering experiments and four-body correlations

Dynamic heterogeneity is expected to be a key concept for understanding the origin of slow dynamics near the glass transition. In previous studies, quantitative evaluations of dynamic heterogeneity have been attempted using two different routes, i.e., the speckle patterns in scattering experiments or the four-body correlation functions of microscopic configuration data obtained from molecular dynamics simulations or real-space observations using confocal microscopy. However, the physical relationship between these dynamic heterogeneities obtained using different methods has not been clarified. This study proposes a connection between dynamic heterogeneities characterized based on speckle patterns and those obtained from four-body correlations. The validity of the relationship is also clarified.

Nonlinear Density Response and Higher Order Correlation Functions in Warm Dense Matter

In a recent letter [Phys. Rev. Lett. 125, 085001 (2020)], Dornheim et al. have presented the first ab initio path integral Monte Carlo (PIMC) results for the nonlinear electronic density response at warm dense matter (WDM) conditions. In the present work, we extend these considerations by exploring the relation between the nonlinear response and three-/four-body correlation functions from many-body theory. In particular, this connection directly implies a comparably increased sensitivity of the nonlinear response to electronic exchange–correlation (XC) effects, which is indeed confirmed by our analysis over the entire relevant range of densities (\(r_{s} = 0.5, \ldots ,10\)) and temperatures (\(\theta = 0.01, \ldots ,4\)). Finally, our work suggests the possibility of deliberately probing the nonlinear regime to experimentally probe three- and potentially even four-body correlation functions in WDM.

Local Number Fluctuations in Hyperuniform and Nonhyperuniform Systems: Higher-Order Moments and Distribution Functions

The local number variance associated with a spherical sampling window of radius $R$ enables a classification of many-particle systems in $d$-dimensional Euclidean space according to the degree to which large-scale density fluctuations are suppressed, resulting in a demarcation between hyperuniform and nonhyperuniform phyla. To better characterize density fluctuations, we carry out an extensive study of higher-order moments, including the skewness $\gamma_1(R)$, excess kurtosis $\gamma_2(R)$ and the corresponding probability distribution function $P[N(R)]$ of a large family of models across the first three space dimensions, including both hyperuniform and nonhyperuniform models. Specifically, we derive explicit integral expressions for $\gamma_1(R)$ and $\gamma_2(R)$ involving up to three- and four-body correlation functions, respectively. We also derive rigorous bounds on $\gamma_1(R)$, $\gamma_2(R)$ and $P[N(R)]$. High-quality simulation data for these quantities are generated for each model. We also ascertain the proximity of $P[N(R)]$ to the normal distribution via a novel Gaussian distance metric $l_2(R)$. Among all models, the convergence to a central limit theorem (CLT) is generally fastest for the disordered hyperuniform processes. The convergence to a CLT is slower for standard nonhyperuniform models, and slowest for the antihyperuniform model studied here. We prove that one-dimensional hyperuniform systems of class I or any $d$-dimensional lattice cannot obey a CLT. Remarkably, we discovered that the gamma distribution provides a good approximation to $P[N(R)]$ for all models that obey a CLT, enabling us to estimate the large-$R$ scalings of $\gamma_1(R)$, $\gamma_2(R)$ and $l_2(R)$. For any $d$-dimensional model that "decorrelates" or "correlates" with $d$, we elucidate why $P[N(R)]$ increasingly moves toward or away from Gaussian-like behavior, respectively.

The Importance of Averaging to Interpret Electron Correlographs of Disordered Materials

The development of effective new tools for structural characterization of disordered materials and systems is becoming increasingly important as such tools provide the key to understanding, and ultimately controlling, their properties. The relatively novel technique of correlograph analysis (i.e., the approach of calculating angular autocorrelations within diffraction patterns) promises unique advantages for probing the local symmetries of disordered structures. Because correlograph analysis examines a component of the high-order four-body correlation function, it is more sensitive to medium-range ordering than conventional diffraction methods. As a follow-up of our previous publication, where we studied thin samples of sputtered amorphous silicon, we describe here the practical experimental method and common systematic errors of electron correlograph analysis. Using both experimental data and numerical simulations, we demonstrate that reliable structural information about the sample can only be extracted from the mean correlograph averaged over a sufficient number of individual results.

Theoretical study on the sound absorption of electrolytic solutions. I. Theoretical formulation

A theory is formulated that describes the sound absorption of electrolytic solutions due to the relative motion of ions, including the formation of ion pairs. The theory is based on the Kubo-Green formula for the bulk viscosity. The time correlation function of the pressure is projected onto the bilinear product of the density modes of ions. The time development of the product of density modes is described by the diffusive limit of the generalized Langevin equation, and approximate expressions for the three- and four-body correlation functions required are given with the hypernetted-chain integral equation theory. Calculations on the aqueous solutions of model electrolytes are performed. It is demonstrated that the theory describes both the activated barrier crossing between contact and solvent-separated ion pairs and the Coulombic correlation between ions.

Molecular-dynamics analysis of the nucleation and crystallization process of Si

Molecular-dynamics (MD) simulations of excimer-laser annealed Si using the Tersoff potential have been performed and the processes of the nucleation and crystallization of Si are monitored by the order parameter for each atom. The order parameter which identifies each atom belonging to the crystalline phase in amorphous and liquid-like environment is presented by a four-body correlation function. The simulation results show that the atomistic processes of the nucleation and crystallization of Si on crystalline Si (c-Si) substrates can be divided into several stages. Twins with low formation energy disturb the further crystallization of nucleus along the [ 1 1 1 ] direction.

Many-body correlations versus mode-coupling effects in the dynamics of dense gases

Time dependent contributions due to the triplet and quadruplet correlation functions have been combined with microscopically determined contribution due to the pair correlation function for the transverse stress correlation function. Comparison of the results with simulation data shows that contributions due to two- and three-body static correlation functions are sufficient to understand the viscous dynamics of dense gases. However, close to the triple point, it becomes necessary to include the contribution due to the four-body correlation function. The contribution due to the mode coupling effect has been contrasted with the contribution due to three- and four-body correlation functions. It is noted that the mode coupling contribution plays a similar role as that due to many-body correlation functions.

Fluctuation microscopy – a tool for examining medium-range order in noncrystalline systems

Fluctuation microscopy examines the spatial variations in coherent microdiffraction from an ensemble of small volumes in noncrystalline systems. The variance of scattered intensity into regions of reciprocal space can be used to give insight into three-body and four-body correlation functions, which are sensitive to medium-range order. The technique was originally developed for transmission electron microscopy and was successfully used to understand medium-range order in amorphous silicon and germanium. Applying this method to X-rays, we have developed a new approach: fluctuation X-ray microscopy (FXM). The approach offers quantitative insight into medium-range correlations in materials at nanometer and larger length scales. The FXM technique can be used to explore medium-range order and subtle structural changes in a wide range of disordered materials from soft matter to nanocomposites, nanowire and quantum dot arrays. We have demonstrated this new technique by studying films of polystyrene latex spheres.

Multidimensional vibrational spectroscopy for tunneling processes in a dissipative environment

Simulating tunneling processes as well as their observation are challenging problems for many areas. In this study, we consider a double-well potential system coupled to a heat bath with a linear-linear (LL) and square-linear (SL) system-bath interactions. The LL interaction leads to longitudinal (T1) and transversal (T2) homogeneous relaxations, whereas the SL interaction leads to the inhomogeneous dephasing (T2*) relaxation in the white noise limit with a rotating wave approximation. We discuss the dynamics of the double-well system under infrared (IR) laser excitations from a Gaussian-Markovian quantum Fokker-Planck equation approach, which was developed by generalizing Kubo's stochastic Liouville equation. Analytical expression of the Green function is obtained for a case of two-state-jump modulation by performing the Fourier-Laplace transformation. We then calculate a two-dimensional infrared signal, which is defined by the four-body correlation function of optical dipole, for various noise correlation time, system-bath coupling parameters, and temperatures. It is shown that the bath-induced vibrational excitation and relaxation dynamics between the tunneling splitting levels can be detected as the isolated off-diagonal peaks in the third-order two-dimensional infrared (2D-IR) spectroscopy for a specific phase matching condition. Furthermore, this spectroscopy also allows us to directly evaluate the rate constants for tunneling reactions, which relates to the coherence between the splitting levels; it can be regarded as a novel technique for measuring chemical reaction rates. We depict the change of reaction rates as a function of system-bath coupling strength and a temperature through the 2D-IR signal.

Stokes–Einstein relations for a square-well fluid

A Stokes-Einstein relation, relating the shear viscosity eta to the self-diffusion coefficient D, is constructed for a classical fluid subject to an effective two-body intermolecular force, derived from a square-well potential, undergoing dynamics as described by a Smoluchowski equation for pair diffusion. The time correlation functions for eta and 1D are separated into contributions from delta function, hard-sphere forces, and from delta function, square-well soft forces. Furthermore, D is separated into its two- and three-body time correlation functions, and eta into its two- to four-body terms. D shows activated diffusion, as in Arrhenius behavior, and on the level of two-body dynamics, the Deta product adheres to the Stokes-Einstein relation, subject to a small correction for potential softness. Three-body time correlation functions increase D, whereas three- and four-body correlation functions in eta are partially offsetting. The deviation of Deta product from the Stokes-Einstein law arises from the three-body time correlations functions in D.

Integrable Magnetic Model of Two Chains Coupled by Four-Body Interactions

An exact solution for an XXZ chain with four-body interactions is obtained and its phase diagram is determined. The model can be reduced to two chains coupled by four-body interactions, and it is shown that the ground state of the two-chain model is magnetized in part. Furthermore, a twisted four-body correlation function of the anti-ferromagnetic Heisenberg chain is obtained.

Correlation dynamics of Green functions

We generalize the methods used in the theory of correlation dynamics and establish a set of equations of motion for many-body correlation Green functions in the nonrelativistic case. These nonlinear and coupled equations of motion describe the dynamical evolution of correlation Green functions of different order and transparently show how many-body correlations are generated by the different interaction terms in a genuine nonperturbative framework. The nonperturbative results of the conventional Green function theory are included in the present formalism as two limiting cases (the so-called ladder-diagram summation and ring-diagram summation) as well as the familiar correlation dynamics of density matrices in the equal-time limit. We present explicit expressions for three- and four-body correlation functions that can be used to dynamically restore the trace relations for spin-symmetric Fermi systems and study numerically the relative importance of two-, three- and four-body correlations for nuclear configurations close to the ground state.

THREE- AND FOUR-BODY CORRELATION FUNCTIONS FOR THE BLUME-CAPEL MODEL

The Cluster Variation Method (CVM), in its recent formulation using the Moebius Inversion, is applied to study the critical properties of the face-centered-cubic Blume-Capel model in the triangle and tetrahedron approximations. We develop a procedure which preserves the characteristics of Kikuchi’s Natural Iteration Method, moreover reducing considerably the number of involved variables. The resulting phase diagram is compared with that obtained from lower orders of CVM approximation (single-site and pair) and from other methods (series expansions, Monte Carlo, …). The tetrahedron approximation shows a good agreement with these high-precision methods, in particular for the location of the tricritical point. The behavior of order parameters and multisite (up to four-body) correlation functions is determined for all significant regions of the phase space.

Theoretical investigation of the dip in the far infrared absorption spectrum of dense rare gas mixtures

The far infrared absorption spectrum of dense rare gas mixtures is analyzed in terms of two-, three-, and four-body contributions. It is argued that pair, triplet, and quadruplet terms must be accurately known at short times, in order to describe the overall spectral shape, since the long time region, where diffusion processes can occur, is drastically reduced by the range of the interaction and the cancellation effect. A model calculation is presented where each piece of the total correlation function C(t) is expressed as a Gram–Charlier series. The parameters are then estimated with the help of a lattice gas model which includes a correction to the Kirkwood superposition approximation in the case of three-body correlations. The dip in the reduced spectral density is interpreted as a dynamical cancellation effect between two-, three-, and four-body correlation functions, the role played by the cage effect at liquid density being exacerbated by the cancellation. The width and the depth of the dip are found to be strongly dependent on the mass ratio, density, and concentrations, and some comparisons with experimental data are made for Ne–Ar and Kr–Ar liquid mixtures. Finally, the phenomenological interpretation of the dip given by Van Kranendonk and Lewis in the 1970’s (intercollisional interference effect at low density) is discussed in detail and the possible relationship with the present theory is examined.

Hypernetted-chain approximation for three- and four-body correlations in simple fluids.

New approximate integral equations for the three- and four-body correlation functions and structure functions are derived. These approximations are based on the hypernetted-chain resummation. The equation for the three-body correlation function satisfies a self-consistency condition, the Born-Green-Yvon equation, and has the Kirkwood superposition approximation as the leading or nonhomogeneous term. Similarly, the new approximation for the third-order structure function has the convolution approximation as its leading term. In addition to the formalism, numeric solutions are presented for a Lennard-Jones system corresponding to argon at its triple point and at room temperature (with its triple-point density). At this high density, the integral equations give results significantly different from their leading terms.