Three-body Correlation Function Research Articles

Spatial distribution of reduced density of hard spheres near a hard-sphere dimer: Results from three-dimensional Ornstein–Zernike equations coupled with several different closures and from grand canonical Monte Carlo simulation

We calculated the spatial distribution function, which is the one-body reduced density distribution of solvent particles around a nonspherical solute, using the three-dimensional Ornstein–Zernike equations coupled with closures. The solvent was a hard-sphere fluid, and the contact dimer of the solvent particles was the nonspherical solute. Two traditional closures, the Percus–Yevick and hypernetted-chain approximations, and three closures with bridge functions (BFs) were examined. These spatial distribution functions were compared with the results obtained by grand canonical Monte Carlo (GCMC) simulations. Three closures with BFs, such as the modified hypernetted-chain (MHNC) closure using a bridge function proposed by Kinoshita, gave precise spatial distribution functions. These results were much more accurate than those obtained by the traditional closures. The deviations from the spatial distribution function obtained by the GCMC simulation appeared only near the concave surface of the solute dimer. However, the deviations were minor compared with those between the simulation and the predictions of the two traditional closures. By contrast, the three-body correlation functions for the closures with BFs were not more accurate than those obtained by the traditional closures.

Crossover from attractive to repulsive induced interactions and bound states of two distinguishable Bose polarons

We study the impact of induced correlations and quasiparticle properties by immersing two distinguishable impurities in a harmonically trapped bosonic medium. It is found that when the impurities couple both either repulsively or attractively to their host, the latter mediates a two-body correlated behavior between them. In the reverse case, namely the impurities interact oppositely with the host, they feature anti-bunching. Monitoring the impurities relative distance and constructing an effective two-body model to be compared with the full many-body calculations, we are able to associate the induced (anti-) correlated behavior of the impurities with the presence of attractive (repulsive) induced interactions. Furthermore, we capture the formation of a bipolaron and a trimer state in the strongly attractive regime. The trimer refers to the correlated behavior of two impurities and a representative atom of the bosonic medium and it is characterized by an ellipsoidal shape of the three-body correlation function. Our results open the way for controlling polaron induced correlations and creating relevant bound states.

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.

A method to remove lower order contributions in multi-particle femtoscopic correlation functions

In recent years the femtoscopy technique has been used by the ALICE Collaboration in small colliding systems at the LHC to investigate the strong-interaction of hadron pairs in the low-energy regime. The extension of this technique to the study of many-body correlations aims to deliver in the next years the first experimental measurements of the genuine many-hadron interactions, provided that the contributions due to the lower order terms are properly accounted for. In this paper we present a method that allows to determine the residual lower order contributions to the three-body correlation functions, based on the cumulant decomposition approach and on kinematic transformations. A procedure to simulate genuine three-body correlations in three-baryon correlation functions is also developed. A qualitative study of the produced correlation signal is performed by varying the strength of the adopted three-body interaction model and comparisons with the expectations for the lower order contributions to the correlation function are shown. The method can be also applied to evaluate the combinatorial background in the two-body correlation functions, providing an improved statistical accuracy with respect to the standard techniques. The example of the contribution by the pK^+K^- channel to the recently measured pupphi correlation is discussed.

Nonlinear Fermi liquid transport through a quantum dot in asymmetric tunnel junctions

We study the nonlinear conductance through a quantum dot, specifically its dependence on the asymmetries in the tunnel couplings and bias voltages $V$, at low energies. Extending the microscopic Fermi liquid theory for the Anderson impurity model, we obtain an exact formula for the steady current $I$ up to terms of order ${V}^{3}$ in the presence of these asymmetries. The coefficients for the nonlinear terms are described in terms of a set of the Fermi liquid parameters: the phase shift, static susceptibilities, and three-body correlation functions of electrons in the quantum dots, defined with respect to the equilibrium ground state. We calculate these correlation functions, using the numerical renormalization group approach (NRG), over a wide range of impurity-electron filling that can be controlled by a gate voltage in real systems. The NRG results show that the order ${V}^{2}$ nonlinear current is enhanced significantly in the valence fluctuation regime. It is caused by the order $V$ energy shift of the impurity level, induced in the presence of the tunneling or bias asymmetry. Furthermore, in the valence fluctuation regime, we also find that the order ${V}^{3}$ nonlinear current exhibits a shoulder structure, for which the three-body correlations that evolve for large asymmetries play an essential role.

Observation of Three-Body Correlations for Photons Coupled to a Rydberg Superatom.

We report on the experimental observation of nontrivial three-photon correlations imprinted onto initially uncorrelated photons through an interaction with a single Rydberg superatom. Exploiting the Rydberg blockade mechanism, we turn a cold atomic cloud into a single effective emitter with collectively enhanced coupling to a focused photonic mode which gives rise to clear signatures in the connected part of the three-body correlation function of the outgoing photons. We show that our results are in good agreement with a quantitative model for a single, strongly coupled Rydberg superatom. Furthermore, we present an idealized but exactly solvable model of a single two-level system coupled to a photonic mode, which allows for an interpretation of our experimental observations in terms of bound states and scattering states.

DeePCG: Constructing coarse-grained models via deep neural networks.

We introduce a general framework for constructing coarse-grained potential models without ad hoc approximations such as limiting the potential to two- and/or three-body contributions. The scheme, called the Deep Coarse-Grained Potential (abbreviated DeePCG), exploits a carefully crafted neural network to construct a many-body coarse-grained potential. The network is trained with full atomistic data in a way that preserves the natural symmetries of the system. The resulting model is very accurate and can be used to sample the configurations of the coarse-grained variables in a much faster way than with the original atomistic model. As an application, we consider liquid water and use the oxygen coordinates as the coarse-grained variables, starting from a full atomistic simulation of this system at the ab initio molecular dynamics level. We find that the two-body, three-body, and higher-order oxygen correlation functions produced by the coarse-grained and full atomistic models agree very well with each other, illustrating the effectiveness of the DeePCG model on a rather challenging task.

The Three-Body Direct Correlation Function of Hard Sphere and Hard Ellipsoid Fluids

In previous work, the direct correlation functions of hard spheres and hard ellipsoidal fluids were calculated. In the present work, using Marko variational method and Pynn–Wulf approximation, the three-body direct correlation function of hard ellipsoid fluid is calculated. In these calculations, the triplet direct correlation function of hard spheres is required. The new expressions for the direct correlation function of hard spheres and hard ellipsoids and triplet correlation of hard spheres, introduced by Muller and coworker, are used. The Fourier transform of new triplet direct correlation of hard sphere fluid is compared with the simulation data. As mentioned in previous work, the expansion coefficients of two-body correlations of hard ellipsoids (components of three-body correlations) are fairly in agreement with the Monte Carlo simulation and the other theories. The Fourier transform of new triplet direct correlation of hard spheres and hard ellipsoid fluids is calculated. These Fourier transforms can be used for calculation of triplet structure factor of hard spheres and hard ellipsoid fluids. The Fourier transforms of triplet correlation of hard ellipsoids with elongations $$x = 1.50,\,\,1.20,\,\,1.10$$ as limited cases are compared with exact result of hard spheres of unit diameter. There is qualitative agreement between hard ellipsoids and hard spheres results of triplet correlations.

Structural properties of transition-metal clusters via force-biased Monte Carlo and ab initio calculations: A comparative study

We present a force-biased Monte Carlo (FMC) method for structural modeling of the transition-metal clusters of Fe, Ni, and Cu with sizes of 13, 30, and 55 atoms. By employing the Finnis-Sinclair potential for Fe and the Sutton-Chen potential for Ni and Cu, the total energy of the clusters is minimized using the local gradient of the potentials in Monte Carlo simulations. The structural configurations of the clusters, obtained from the biased Monte Carlo approach, are analyzed and compared with the same configurations from the Cambridge Cluster Database (CCD) upon relaxation of the clusters using the first-principles density-functional code nwchem. The results show that the total-energy value and the structure of the FMC clusters are essentially identical to the corresponding value and the structure of the CCD clusters. A comparison of the nwchem-relax FMC and CCD structures is presented by computing the pair-correlation function, the bond-angle distribution, the coordination number of the first-coordination shell, and the Steinhardt bond-orientational order parameter, which provide information about the two- and three-body correlation functions, the local bonding environment of the atoms, and the geometry of the clusters. An atom-by-atom comparison of the FMC and CCD clusters is also provided by superposing one set of clusters onto another, and the electronic properties of the clusters are addressed by computing the density of electronic states.

Structure of transition metal clusters: A force-biased Monte Carlo approach

We present a force-biased Monte Carlo (FMC) method for structural modeling of transition metal clusters of Fe, Ni, and Cu with 5 to 60 atoms. By employing the Finnis-Sinclair potential for Fe and the Sutton-Chen potential for Ni and Cu, the total energy of the clusters is minimized using a method that utilizes atomic forces in Monte Carlo simulations. The structural configurations of the clusters obtained from this biased Monte Carlo approach are analyzed and compared with the same from the Cambridge Cluster Database (CCD). The results show that the total-energy of the FMC clusters is very close to the corresponding value of the CCD clusters as listed in the Cambridge Cluster Database. A comparison of the FMC and CCD clusters is presented by computing the pair-correlation function, the bond-angle distribution, and the distribution of atomic-coordination numbers in the first-coordination shell, which provide information about the two-body and three-body correlation functions, the local atomic structure, and the bonding environment of the atoms in the clusters.

Connection between nonlocal one-body and local three-body correlations of the Lieb-Liniger model

We derive a connection between the fourth coefficient of the short-distance Taylor expansion of the one-body correlation function, and the local three-body correlation function of the Lieb-Liniger model of $\delta$-interacting spinless bosons in one dimension. This connection, valid at arbitrary interaction strength, involves the fourth moment of the density of quasi-momenta. Generalizing recent conjectures, we propose approximate analytical expressions for the fourth coefficient covering the whole range of repulsive interactions, validated by comparison with accurate numerics. In particular, we find that the fourth coefficient changes sign at interaction strength $\gamma_c\simeq 3.816$, while the first three coefficients of the Taylor expansion of the one-body correlation function retain the same sign throughout the whole range of interaction strengths.

The local structure factor near an interface; beyond extended capillary-wave models

We investigate the local structure factor S (z;q) at a free liquid–gas interface in systems with short-ranged intermolecular forces and determine the corrections to the leading-order, capillary-wave-like, Goldstone mode divergence of S (z;q) known to occur for parallel (i.e. measured along the interface) wavevectors . We show from explicit solution of the inhomogeneous Ornstein–Zernike equation that for distances z far from the interface, where the profile decays exponentially, S (z;q) splits unambiguously into bulk and interfacial contributions. On each side of the interface, the interfacial contributions can be characterised by distinct liquid and gas wavevector dependent surface tensions, and , which are determined solely by the bulk two-body and three-body direct correlation functions. At high temperatures, the wavevector dependence simplifies and is determined almost entirely by the appropriate bulk structure factor, leading to positive rigidity coefficients. Our predictions are confirmed by explicit calculation of S (z;q) within square-gradient theory and the Sullivan model. The results for the latter predict a striking temperature dependence for and , and have implications for fluctuation effects. Our results account quantitatively for the findings of a recent very extensive simulation study by Höfling and Dietrich of the total structure factor in the interfacial region, in a system with a cut-off Lennard-Jones potential, in sharp contrast to extended capillary-wave models which failed completely to describe the simulation results.

SIMULATION ANALYSIS OF THE TOTALLY ASYMETRIC SIMPLE EXCLUSION PROCESS TO DETERMINE THE PROFIL OF THE ONE-BODY CORRELATION FUNCTION, TWO-BODY CORRELATION FUNCTION, AND THREE-BODY CORRELATION FUNCTION AROUND THE ENDS OF THE LATTICE WITH LATTICE NUMBER VARIATION

This study aimed to determine the behavior of one-body, two-body, and three-body correlation functions of the model dynamics TASEP with sequential updating rules and open boundary conditions on vehicular traffic around the end of the traffic light. The study began with the determination of algorithm to model the dynamics of TASEP and coding, with the variation of the input rate (α) , the output rate (β), and the number of lattice sites (N). Then the program run with specific time limit (t) and number of systems (M). The value of the one-body correlation function determines the average occupancy of particles in lattice site-i at time t. Two-body correlation function determines the average occupancy of particle at site-i when there is another particle occupying the nearest neighbor lattice, i+1, at time t. Three-body correlation function determines the average occupancy of particles to occupy lattice site-i when there are other particles occupying the nearest and next nearest neighbor lattice sites, i+1 and i+2, at time t. The value of the one-body correlation function turns out to be larger than the value of the two-body correlation function. The value of the two-body correlation function is larger than the value of the three-body correlation function for all phases. The correlation between a vehicle to another vehicle will be even greater. Keywords: TASEP, sequential updating, n-body correlation function

A generalized-Yvon-Born-Green method for coarse-grained modeling

The Yvon-Born-Green (YBG) integral equation is a basic result of liquid state theory that relates the pair potential of a simple fluid to the resulting equilibrium two- and three-body correlation functions. Quite recently, we derived a more general form that can be applied to complex molecular systems. This generalized-YBG (g-YBG) theory provides not only an exact relation between a given potential and the resulting equilibrium correlation functions, but also a remarkably powerful framework for directly solving the statistical mechanics inverse problem of determining potentials from equilibrium structure ensembles. In the context of coarse-grained (CG) modeling, the g-YBG theory determines a variationally optimal approximation to the many-body potential of mean force directly (i.e., noniteratively) from structural correlation functions and, in particular, allows “force-matching” without forces. While our initial efforts numerically validated the g-YBG theory with relatively simple systems, our more recent efforts have considered increasingly complex systems, such as peptides and polymers. This minireview summarizes this progress and the resulting insight, as well as discusses the outstanding challenges and future directions for the g-YBG theory.

Phase separation in one-dimensional hard-core boson system with two- and three-body interactions

We investigate the ground state phase diagram of hard-core boson system with repulsive two-body and attractive three-body interactions in one-dimensional optic lattice. When these two interactions are comparable and increasing the hopping rate, physically intuitive analysis indicates that there exists an exotic phase separation regime between the solid phase with charge density wave order and superfluid phase. We identify these phases and phase transitions by numerically analyzing the density distribution, structure factor of density-density correlation function, three-body correlation function and von Neumann entropy estimator obtained by density matrix renormalization group method. These exotic phases and phase transitions are expected to be observed in the ultra-cold polar molecule experiments by properly tuning interaction parameters, which is constructive to understand the physics of ubiquitous insulating-superconducting phase transitions in condensed matter systems.

Triplet correlations and bridge functions in classical density functional theory for liquid water

Density functional theory for molecular fluids developed by Donley et al. (J. Chem. Phys. 101 (1994) 3205) is extended to include the effects of orientation-dependent bridge functions associated with the inter-particle, triplet correlations. Resultant integral equations for the pair and direct correlation functions are solved for water, where the three-body direct correlation functions are approximated in terms of two-body functions. A test calculation employing a simple Gaussian form for the two-body function between the oxygen sites then provides a promising result to improve the description of the oxygen–oxygen correlations in liquid water at room temperature.

Three-Body Correlation Functions and Recombination Rates for Bosons in Three Dimensions and One Dimension

We investigate local three-body correlations for bosonic particles in three dimensions and one dimension as a function of the interaction strength. The three-body correlation function g(3) is determined by measuring the three-body recombination rate in an ultracold gas of Cs atoms. In three dimensions, we measure the dependence of g(3) on the gas parameter in a BEC, finding good agreement with the theoretical prediction accounting for beyond-mean-field effects. In one dimension, we observe a reduction of g(3) by several orders of magnitude upon increasing interactions from the weakly interacting BEC to the strongly interacting Tonks-Girardeau regime, in good agreement with predictions from the Lieb-Liniger model for all strengths of interaction.

Thermalization in a quasi-one-dimensional ultracold bosonic gas

We study the collisional processes that can lead to thermalization in one-dimensional (1D) systems. For two-body collisions, excitations of transverse modes are the prerequisite for energy exchange and thermalization. At very low temperatures, excitations of transverse modes are exponentially suppressed, thermalization by two-body collisions stops and the system should become integrable. In quantum mechanics, virtual excitations of higher radial modes are possible. These virtually excited radial modes give rise to effective three-body velocity-changing collisions, which lead to thermalization. We show that these three-body elastic interactions are suppressed by pairwise quantum correlations when approaching the strongly correlated regime. If the relative momentum k is small compared with the two-body coupling constant c, the three-particle scattering state is suppressed by a factor of (k/c)12, which is proportional to γ−12, that is, to the square of the three-body correlation function at zero distance in the limit of the Lieb–Liniger parameter γ≫1. This demonstrates that in 1D quantum systems, it is not the freeze-out of two-body collisions but the strong quantum correlations that ensure absence of thermalization on experimentally relevant time scales.

The behavior of ions near a charged wall-dependence on ion size, concentration, and surface charge.

A renormalization of the 3D-RISM-HNC integral equation is used to study the solvent and ion distributions at neutral and negatively charged planar atomistic surfaces. The charge density of the surfaces ranged from 0.0 to 0.4116 C/m(2), and the modeled electrolyte solutions consist of the salts NaCl, KCl, and CsCl at concentrations of 0.1, 0.25, and 1.0 M in SPC/E water. The results are qualitatively compared to the results from other integral equation methods and simulations for similar models. We find that the 3D-IEs predict an electric multilayer screening behavior in the solvent and ion distributions in contrast to the double layer anticipated from Poisson-Boltzmann theory. It is observed that the cation size has a significant effect on the distributions near the surface up to three solvation layers beyond which the behavior is the same among the different cations. The response of the distributions to the charged surface is described as an increase in ion and solvent density near the wall. The higher concentration solutions screen the electrostatic source more strongly at the wall as expected. The importance of ion-solvent and ion-ion correlations near the surface is shown through three-body correlation functions which are obtainable from the 3D-IEs in this study.

Inversion of radial distribution functions to pair forces by solving the Yvon–Born–Green equation iteratively

We develop a new method to invert the target profiles of radial distribution functions (RDFs) to the pair forces between particles. The target profiles of RDFs can be obtained from all-atom molecular dynamics (MD) simulations or experiments and the inverted pair forces can be used in molecular simulations at a coarse-grained (CG) scale. Our method is based on a variational principle that determines the mean forces between CG sites after integrating out the unwanted degrees of freedom. The solution of this variational principle has been shown to correspond to the Yvon-Born-Green (YBG) equation [Noid et al., J. Phys. Chem. B 111, 4116 (2007)]. To invert RDFs, we solve the YBG equation iteratively by running a CG MD simulation at each step of iteration. A novelty of the iterative-YBG method is that during iteration, CG forces are updated according to the YBG equation without imposing any approximation as is required by other methods. As a result, only three to ten iterations are required to achieve convergence for all cases tested in this work. Furthermore, we show that not only are the target RDFs reproduced by the iterative solution; the profiles of the three-body correlation function in the YBG equation computed from all-atom and CG simulations also have a better agreement. The iterative-YBG method is applied to compute the CG forces of four molecular liquids to illustrate its efficiency and robustness: water, ethane, ethanol, and a water/methanol mixture. Using the resulting CG forces, all of the target RDFs observed in all-atom MD simulations are reproduced. We also show that the iterative-YBG method can be applied with a virial constraint to expand the representability of a CG force field. The iterative-YBG method thus provides a general and robust framework for computing CG forces from RDFs and could be systematically generalized to go beyond pairwise forces and to include higher-body interactions in a CG force field by applying the aforementioned variational principle to derive the corresponding YBG equation for iterative solution.