Two-body Density Research Articles

Counter-Ions Near a Charged Wall: Exact Results for Disc and Planar Geometries

Macromolecules, when immersed in a polar solvent like water, become charged by a fixed surface charge density which is compensated by ``counter-ions'' moving out of the surface. Such classical particle systems exhibit poor screening properties at any temperature and the trivial bulk regime (far away from the charged surface) with no particles, so the validity of standard Coulomb sum rules is questionable. In the present paper, we concentrate on the two-dimensional version of the model with the logarithmic interaction potential. We go from the finite disc to the semi-infinite planar geometry. The system is exactly solvable for two values of the coupling constant $\Gamma$: in the Poisson-Boltzmann mean-field limit $\Gamma\to 0$ and at the free-fermion point $\Gamma=2$. We show that the finite-size expansion of the free energy does not contain universal term as is usual for Coulomb fluids. For the coupling constant being an arbitrary positive even integer, using an anticommuting representation of the partition function and many-body densities we derive a sequence of sum rules. As a result, the contact density of counter-ions at the wall is available for the disc. The amplitude function, which characterizes the asymptotic inverse-power law behavior of the two-body density along the wall, is found to be related to the particle density profile. The dielectric susceptibility tensor, calculated exactly for an arbitrary coupling and the particle number, exhibits the anticipated disc value in the thermodynamic limit, in spite of zero contribution from the bulk region. Some of the results obtained in the Poisson-Boltzmann limit are generalized to an arbitrary Euclidean dimension.

Variance as a sensitive probe of correlations

Bose-Einstein condensates made of ultracold trapped bosonic atoms have become a central venue in which interacting many-body quantum systems are studied. The ground state of a trapped Bose-Einstein condensate has been proven to be 100% condensed in the limit of infinite particle number and constant interaction parameter [Lieb and Seiringer, Phys. Rev. Lett. {\bf 88}, 170409 (2002)]. The meaning of this result is that properties of the condensate, noticeably its energy and density, converge to those obtained by minimizing the Gross-Pitaevskii energy functional. This naturally raises the question whether correlations are of any importance in this limit. Here, we demonstrate both analytically and numerically that even in the infinite particle limit many-body correlations can lead to a substantial modification of the \textit{variance} of any operator compared to that expected from the Gross-Pitaevskii result. The strong deviation of the variance stems from its explicit dependence on terms of the reduced two-body density matrix which otherwise do not contribute to the energy and density in this limit. This makes the variance a sensitive probe of many-body correlations even when the energy and density of the system have already converged to the Gross-Pitaevskii result. We use the center-of-mass position operator to exemplify this persistence of correlations. Implications of this many-body effect are discussed.

Assessing the role of static length scales behind glassy dynamics in polydisperse hard disks

The possible role of growing static order in the dynamical slowing down toward the glass transition has recently attracted considerable attention. On the basis of random first-order transition theory, a new method to measure the static correlation length of amorphous order, called "point-to-set" (PTS) length, has been proposed and used to show that the dynamic length grows much faster than the static length. Here, we study the nature of the PTS length, using a polydisperse hard-disk system, which is a model that is known to exhibit a growing hexatic order upon densification. We show that the PTS correlation length is decoupled from the steeper increase of the correlation length of hexatic order and dynamic heterogeneity, while closely mirroring the decay length of two-body density correlations. Our results thus provide a clear example that other forms of order can play an important role in the slowing down of the dynamics, casting a serious doubt on the order-agnostic nature of the PTS length and its relevance to slow dynamics, provided that a polydisperse hard-disk system is a typical glass former.

Quantum decoherence in the entanglement entropy of a composite particle and its relationship to coarse graining in the Husimi function

I investigate quantum decoherence in a one-body density matrix of a composite particle consisting of two correlated particles. Because of a two-body correlation in the composite particle, quantum decoherence occurs in the one-body density matrix that has been reduced from the two-body density matrix. As the delocalization of the distribution of the composite particle grows, the entanglement entropy increases, and the system can be well-described by a semi-classical approximation, wherein the center position of the composite particle can be regarded as a classical coordinate. I connect the quantum decoherence in the one-body density matrix of a composite particle to the coarse graining in a phase space distribution function of a single particle and associate it with the Husimi function.

Extension of time-dependent Hartree-Fock-Bogoliubov equations

An extension of the time-dependent Hartree-Fock-Bogoliubov theory (ETDHFB) which includes higher-order effects such as screening of the pairing correlation is proposed. ETDHFB is applied to a pairing model and a fermion system trapped in a harmonic potential to test its feasibility by comparison with exact solutions. With the use of perturbative expressions for the pairing tensor and the two-body density matrix derived from ETDHFB, the screening effect is investigated for atomic fermion systems and isotopes of tin nuclei. It is found that the screening effect on the pairing correlation is not significant.

New truncation scheme for a time-dependent density-matrix approach applied to the ground state ofO16

The ground state of $^{16}\mathrm{O}$ is calculated by using a time-dependent density-matrix approach derived from a new truncation scheme of the Bogoliubov--Born--Green--Kirkwood--Yvon hierarchy for reduced density matrices, where a three-body density matrix is approximated by an antisymmetrized product of two-body density matrices. The new scheme is compared with a simpler truncation scheme previously used for the calculation of the ground state of $^{16}\mathrm{O}$ where the three-body density matrix is neglected and only two-particle--two-hole elements of the two-body density matrix are considered. It is shown that the results obtained from the two truncation schemes agree well with the exact solution.

Single-particle versus pair superfluidity in a bilayer system of dipolar bosons

We consider the ground state of a bilayer system of dipolar bosons, where dipoles are oriented by an external field in the direction perpendicular to the parallel planes. Quantum Monte Carlo methods are used to calculate the ground-state energy, the one-body and two-body density matrix, and the superfluid response as a function of the separation between layers. We find that by decreasing the interlayer distance for fixed value of the strength of the dipolar interaction, the system undergoes a quantum phase transition from a single-particle to a pair superfluid. The single-particle superfluid is characterized by a finite value of both the atomic condensate and the super-counterfluid density. The pair superfluid phase is found to be stable against formation of many-body cluster states and features a gap in the spectrum of elementary excitations.

Application of extended random-phase approximation with ground-state correlations to collective excitations of 16O

The ground-state correlation effects on the octupole excitation of 16O are studied using the extended random phase approximation (ERPA) derived from the time-dependent density-matrix theory. ERPA includes the effects of ground-state correlations through the fractional occupation of single-particle states and a correlated two-body density matrix. The coupling to two particle – two hole configurations is also included in ERPA. It is found that the first 3− state calculated in the random phase approximation (RPA) is shifted upward when the self-energy contributions due to ground-state correlations are included in particle – hole pairs and that the coupling to two particle – two hole states plays a role in shifting the first 3− state down to the right position.

Interaction cross sections for neon isotopes in the Glauber model and the halo structure ofNe31

In this work we calculate the interaction cross sections of neon isotopes (${}^{17--32}$Ne) on ${}^{12}\mathrm{C}$ at 240 and 950 MeV/nucleon within the framework of the Coulomb-modified correlation expansion for the Glauber model $S$ matrix. The calculations involve (i) up to the two-body density term in the correlation expansion, (ii) the relativistic mean-field (RMF) densities for the colliding nuclei, and (iii) the single Gaussian approximation for the nucleon-nucleon (NN) amplitude. To test the halo structure, the extended neutron density distribution for $^{31}\mathrm{Ne}$ is also used. Our results, using RMF densities, are in close agreement with the available experimental data at 240 MeV/nucleon except for $^{31}\mathrm{Ne}$, for which the extended neutron density distribution is found to play an important role in explaining the data, thereby supporting a halo structure in $^{31}\mathrm{Ne}$. The corresponding results at 950 MeV/nucleon are, however, not as satisfactory as those obtained at 240 MeV/nucleon. The results also demonstrate the importance of the two-body density term in the present calculations.

Truncation scheme of time-dependent density-matrix approach

A truncation scheme of the Bogoliubov-Born-Green-Kirkwood-Yvon hierarchy for reduced density matrices, where a three-body density matrix is approximated by the antisymmetrized products of two-body density matrices, is proposed. This truncation scheme is tested for three model Hamiltonians. It is shown that the obtained results are in good agreement with the exact solutions.

Correlation matrix renormalization approximation for total-energy calculations of correlated electron systems

We generalized the commonly used Gutzwiller approximation for calculating the electronic structure and total energy of strongly correlated electron systems. In our method, the evaluation of one-body and two-body density matrix elements of the Hamiltonian is simplified using a renormalization approximation to achieve better scaling of the computational effort as a function of system size. To achieve a clear presentation of the concept and methodology, we describe the detailed formalism for a finite hydrogen system with minimal basis set. We applied the correlation matrix renormalization approximation approach to a H${}_{2}$ dimer and H${}_{8}$ cubic fragment with minimal basis sets, as well as a H${}_{2}$ molecule with a large basis set. The results compare favorably with sophisticated quantum chemical calculations. We believe our approach can serve as an alternative way to build up the exchange-correlation energy functional for an improved density functional theory description of systems with strong electron correlations.

Condensate fraction of asymmetric three-component Fermi gas

In this paper, we investigate the condensate fraction (CF) of fermionic pairs in the BCS—BEC crossover for three-component Fermi gas with both asymmetric interactions and unequal chemical potentials in two-dimensional free space. By using the functional-path-integral method, we have analytically derived the number densities and bound-state energy, from which the off-diagonal long-range order is analyzed in terms of the asymptotic behavior of the two-body density matrix. The explicit formula of CF is obtained as a function of the bound-state energy and population imbalance. It is demonstrated that the CF spectrum with respect to the bound-state energy can be used to characterize the quantum phase transition between the two kinds of Sarma phases as well as the transition from three-component to two-component superfluid. Moreover we obtain the same analytic formula of CF in the BCS superfluid phase as that of homogeneous Fermi gas with equal chemical potentials.

CHAOTIC PARAMETER λ IN HANBURY-BROWN–TWISS INTERFEROMETRY IN AN ANISOTROPIC BOSON GAS MODEL

Using one- and two-body density matrices, we calculate the spatial and momentum distributions, two-particle Hanbury-Brown–Twiss (HBT) correlation functions, and the chaotic parameter λ in HBT interferometry for the systems of boson gas within the harmonic oscillator potentials with anisotropic frequencies in transverse and longitudinal directions. The HBT chaotic parameter, which can be obtained by measuring the correlation functions at zero relative momentum of the particle pair, is related to the degree of Bose–Einstein condensation and thus the system environment. We investigate the effects of system temperature, particle number and the average momentum of the particle pair on the chaotic parameter. The value of λ decreases with the condensed fraction, f0. It is one for f0 = 0 and zero for f0 = 1. For a certain f0 between 0 and 1, we find that λ increases with the average momentum of the particle pair and decreases with the particle number of system. The results of λ are sensitive to the ratio, ν = ωz/ωρ, of the frequencies in longitudinal and transverse directions. They are smaller for larger ν when ωρ is fixed. In the heavy-ion collisions at the Large Hadron Collider (LHC) energy the large identical pion multiplicity may possibly lead to a considerable Bose–Einstein condensation. Its effect on the chaotic parameter in two-pion interferometry is worth considering in earnest.

Inelastic electron scattering from light nuclei

The inelastic longitudinal form factors F(q) ’s, an expression for the transition charge density were studied where the deformation in nuclear collective modes besides the shell model transition density are taken into consideration. In this work, the core polarization transition density was evaluated by adopting the shape of Tassie model together with the form of the ground state two-body charge density distributions and the effect of two body short range correlation function. It was noticed that the core polarization effects, which represent the collective modes are essential in obtaining a good agreement between the calculated inelastic longitudinal F(q) ’s and those of the experimental data for 4He, 12C, 16O, 28Si, 32S and 40Ca nuclei. J.Natn.Sci.Foundation Sri Lanka 2013 41 (3): 209-217 DOI: http://dx.doi.org/10.4038/jnsfsr.v41i3.6053

Extended evolution equations for neutrino propagation in astrophysical and cosmological environments

We derive the evolution equations for a system of neutrinos interacting among themselves and with a matter background, based upon the Bogoliubov-Born-Green-Kirkwood-Yvon (BBGKY) hierarchy. This theoretical framework gives an (unclosed) set of first-order coupled integro-differential equations governing the evolution of the reduced density matrices. By employing the hierarchy, we first rederive the mean-field evolution equations for the neutrino one-body density matrix associated with a system of neutrinos and antineutrinos interacting with matter and with an anisotropic neutrino background. Then, we derive extended evolution equations to determine neutrino flavor conversion beyond the commonly used mean-field approximation. To this aim we include neutrino-antineutrino pairing correlations to the two-body density matrix. The inclusion of these new contributions leads to an extended evolution equation for the normal neutrino density and to an equation for the abnormal one involving the pairing mean-field. We discuss the possible impact of neutrino-antineutrino correlations on neutrino flavor conversion in the astrophysical and cosmological environments, and possibly upon the supernova dynamics. Our results can be easily generalized to an arbitrary number of neutrino families.

Importance of many-body correlations in glass transition: An example from polydisperse hard spheres

Most of the liquid-state theories, including glass-transition theories, are constructed on the basis of two-body density correlations. However, we have recently shown that many-body correlations, in particular, bond orientational correlations, play a key role in both the glass transition and the crystallization transition. Here we show, with numerical simulations of supercooled polydisperse hard spheres systems, that the length-scale associated with any two-point spatial correlation function does not increase toward the glass transition. A growing length-scale is instead revealed by considering many-body correlation functions, such as correlators of orientational order, which follows the length-scale of the dynamic heterogeneities. Despite the growing of crystal-like bond orientational order, we reveal that the stability against crystallization with increasing polydispersity is due to an increasing population of icosahedral arrangements of particles. Our results suggest that, for this type of systems, many-body correlations are a manifestation of the link between the vitrification and the crystallization phenomena. Whether a system is vitrified or crystallized can be controlled by the degree of frustration against crystallization, polydispersity in this case.

An effective energy gradient expression for divide-and-conquer second-order Møller–Plesset perturbation theory

We recently proposed a linear-scaling evaluation scheme for the second-order Mo̸ller-Plesset perturbation (MP2) energy based on the divide-and-conquer (DC) method [M. Kobayashi, Y. Imamura, and H. Nakai, J. Chem. Phys. 127, 074103 (2007)]. In this paper, we propose an approximate but effective expression for the first derivative of the DC-MP2 energy. The present scheme evaluates the one- and two-body density matrices, which appear in the MP2 gradient formula, in the DC manner; that is, the entire matrix is obtained as the sum of subsystem matrices masked by the partition matrix. Therefore, the method requires solving only the local Z-vector equations. Illustrative applications to three types of systems, peptides, Si surface model, and delocalized polyenes, reveal the effectiveness of the present method.

Importance of many-body orientational correlations in the physical description of liquids

Liquids are often assumed to be homogeneous and isotropic at any lengthscale and translationally invariant. The standard liquid-state theory is constructed on the basis of this picture and thus basically described in terms of the two-body density correlation. This picture is certainly valid at rather high temperatures, where a liquid is in a highly disordered state. However, it may not necessarily be valid at low temperatures or for a system which has strong directional bonding. Indeed, there remain fundamental unsolved problems in liquid science, which are difficult to explain by such a theory. They include water's thermodynamic and kinetic anomalies, liquid-liquid transitions, liquid-glass transitions, and liquid-solid transitions. We argue that for the physical description of these phenomena it is crucial to take into account many-body (orientational) correlations, which have been overlooked in the conventional liquid-state theory. It is essential to recognise that a liquid can lower its free energy by local or mesoscopic ordering without breaking global symmetry. Since such ordering must involve at least a central particle and its neighbours, which are more than two particles, it is intrinsically a consequence of many-body correlations. Particularly important ordering is associated with local breakdown of rotational symmetry, i.e., bond orientational ordering. We emphasize that translational ordering is global whereas orientational ordering can be local. Because of the strong first-order nature of translational ordering, its growth in a liquid state is modest. Thus any structural ordering in a liquid should be associated primarily with orientational ordering and not with translational ordering. We show that bond orientational ordering indeed plays a significant role in all the above-mentioned phenomena at least for (quasi-)single-component liquids. In this Introductory Lecture, we discuss how these phenomena can be explained by such local or mesoscopic ordering in a unified manner.

Quasiclassical molecular dynamics for the dilute Fermi gas at unitarity

We study the dilute Fermi gas at unitarity using molecular dynamics with an effective quantum potential constructed to reproduce the quantum two-body density matrix at unitarity. Results for the equation of state, the pair-correlation function, and the shear viscosity are presented. These quantities are well understood in the dilute, high-temperature limit. Using molecular dynamics we determine higher-order corrections in the diluteness parameter $n{\ensuremath{\lambda}}^{3}$, where $n$ is the density and $\ensuremath{\lambda}$ is the thermal de Broglie wavelength. In the case of the contact density, which parameterizes the short distance behavior of the correlation function, we find that the results of molecular dynamics interpolate between the truncated second- and third-order virial expansion and are in excellent agreement with existing $T$-matrix calculations. For the shear viscosity we reproduce the expected scaling behavior at high temperature, $\ensuremath{\eta}\ensuremath{\sim}1/{\ensuremath{\lambda}}^{3}$, and we determine the leading density-dependent correction to this result.

Molecular dynamics simulations and generalized Lenard-Balescu calculations of electron-ion temperature equilibration in plasmas

We study the problem of electron-ion temperature equilibration in plasmas. We consider pure H at various densities and temperatures and Ar-doped H at temperatures high enough so that the Ar is fully ionized. Two theoretical approaches are used: classical molecular dynamics (MD) with statistical two-body potentials and a generalized Lenard-Balescu (GLB) theory capable of treating multicomponent weakly coupled plasmas. The GLB is used in two modes: (1) with the quantum dielectric response in the random-phase approximation (RPA) together with the pure Coulomb interaction and (2) with the classical (ℏ→0) dielectric response (both with and without local-field corrections) together with the statistical potentials. We find that the MD results are described very well by classical GLB including the statistical potentials and without local-field corrections (RPA only); worse agreement is found when static local-field effects are included, in contradiction to the classical pure-Coulomb case with like charges. The results of the various approaches are all in excellent agreement with pure-Coulomb quantum GLB when the temperature is high enough. In addition, we show that classical calculations with statistical potentials derived from the exact quantum two-body density matrix produce results in far better agreement with pure-Coulomb quantum GLB than classical calculations performed with older existing statistical potentials.