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The Journal of Chemical Physics

Properties of the path-integral quantum hard-sphere fluid in k space

Luis M. Sesé

Published May 15, 2002• 18 citations

Abstract

The properties of quantum fluids in Fourier space, as the system response functions to weak external fields, are analyzed taking the quantum hard-sphere fluid as a probe. This serves to clarify the physical meaning of the different radial correlation functions that can be defined in a path-integral quantum fluid, since these functions are the r-space counterparts of the response functions. The basic feature of the external field relevant to this discussion is connected with its localizing/nonlocalizing effect on the quantum particles composing the fluid (i.e., a localizing field causes the collapse of the particle thermal packet). Fields that localize the quantum particles reveal the so-called instantaneous quantities (e.g., the conventional static structure factor), which are related with the diagonal elements of the density matrix. Fields that do not localize the quantum particles show the so-called linear response quantities, which are related to the diagonal and the off-diagonal density matrix elements. To perform this study the path-integral formalism is considered from the functional analysis approach. Given that the Gaussian Feynman–Hibbs effective potential picture is known to represent well many structural features of the quantum hard-sphere fluid, the parallel study of the response functions within this picture is also presented. In particular, the latter picture provides an accurate Ornstein–Zernike scheme that can be used for numerical calculations of response functions over a wide range of conditions, and also gives fine estimates for quantities difficult to compute with the path integral. Results for the quantum hard-sphere fluid obtained within the latter scheme are reported, tests of consistency are given, and the possibility of approximating the instantaneous response function by means of the coherent part of the linear response function is assessed.

DOI: 10.1063/1.1468223

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The properties of quantum fluids in Fourier space, as the system response functions to weak external fields, are analyzed taking the quantum hard-sphere fluid as a probe. This serves to clarify the physical meaning of the different radial correlation functions that can be defined in a path-integral quantum fluid, since these functions are the r-space counterparts of the response functions. The basic feature of the external field relevant to this discussion is connected with its localizing/nonlocalizing effect on the quantum particles composing the fluid (i.e., a localizing field causes the collapse of the particle thermal packet). Fields that localize the quantum particles reveal the so-called instantaneous quantities (e.g., the conventional static structure factor), which are related with the diagonal elements of the density matrix. Fields that do not localize the quantum particles show the so-called linear response quantities, which are related to the diagonal and the off-diagonal density matrix elements. To perform this study the path-integral formalism is considered from the functional analysis approach. Given that the Gaussian Feynman–Hibbs effective potential picture is known to represent well many structural features of the quantum hard-sphere fluid, the parallel study of the response functions within this picture is also presented. In particular, the latter picture provides an accurate Ornstein–Zernike scheme that can be used for numerical calculations of response functions over a wide range of conditions, and also gives fine estimates for quantities difficult to compute with the path integral. Results for the quantum hard-sphere fluid obtained within the latter scheme are reported, tests of consistency are given, and the possibility of approximating the instantaneous response function by means of the coherent part of the linear response function is assessed.

The dynamic structure factor (DSF) is a mathematical function that contains information about inter-particle correlations and their time evolution. Mostly the classical molecular dynamics is used to calculate the DSF of the classical systems. On the contrary this article deals with quantum systems and the quantum dynamic structure factor. The Wigner formulation of quantum mechanics was used to derive the path integral representation of the DSF, which is based on the Wiener-Khinchin theorem showing relation of the the power spectrum of a random paths to their correlation function. The $3 {\rm D} $ quantum system of strongly correlated soft-sphere fermions was considered as an interesting physical example. The developed Wigner path integral Monte Carlo (WPIMC) approach has been developed to calculate the spin--resolved DSFs, the radial distribution functions (RDFs) and other thermodynamic functions in a wide range of density and temperatures. The physical meaning of the peaks arising on the RDFs and DSF have been analyzed and explained by the manifestation of the interference effects of the exchange and interparticle interactions and, as well as, the wave interference between multiple-scattering. This phenomenon in the system of the soft-sphere scatterers may to be the precursor effect of the Anderson localization, which finds its origin in the wave interference between multiple-scattering paths.

In the linear-response regime, a normal Bose or Fermi fluid can be described by an exact kinetic equation whose kernel is nonlocal in space and time. We derive a general expression for this kernel and evaluate it explicitly to second order in the interparticle potential. The result is a wave-vector- and frequency-dependent generalization of the linear Uehling-Uhlenbeck kernel with the Born-approximation cross section. Our theory is developed in terms of a second-quantized form of the Wigner representation. Convenient expressions are obtained for the commutators and anticommutators of the phase-space density operators, and the equilibrium averages of these operators are analyzed in terms of momentum-dependent generalizations of the classical pair distribution function

h(k)

and direct correlation function

c(k)

. The central quantity in this study is a two-particle equilibrium correlation function, the phase-space density-density anticommutator, whose Fourier transform $S(k\ensuremath{\omega}p{p}^{\ensuremath{'}})$ gives the symmetrized scattering function $S(k\ensuremath{\omega})$ by integration over the momenta. The kinetic equation is obtained by a formal closure of the quantum BBGKY hierarchy, with the nonlocal kernel expressed in terms of correlation functions involving two, three, and four particles. We show that our method for approximating the kernel and initial condition by a second-order expansion preserves all the sum rules of $S(k\ensuremath{\omega}p{p}^{\ensuremath{'}})$ to the same order and that the result satisfies the appropriate positivity and symmetry conditions.

The short-time or high-frequency expansions of several dynamical correlation functions in a quantum hard-sphere fluid of spinless bosons are computed. These expansions are shown to be of a different form than those obtained for either a fluid of quantum particles interacting through a smooth potential or for a classical fluid of hard spheres. Assuming that the same dynamical contributions that are important in the short-time expansions are also responsible for dominant final-state interactions in high-energy neutron scattering, we also approximately compute the shape of the dynamic structure factor near the free-particle frequency.

The Wigner formulation of quantum mechanics is used to derive a path-integral representation of the quantum density of states (DOS) of strongly correlated fermions in the canonical ensemble. A path-integral Monte Carlo approach for the simulation of DOS and other thermodynamic functions is suggested. The derived Wigner function in the phase space resembles the Maxwell-Boltzmann distribution but allows for quantum effects. We consider a three-dimensional quantum system of strongly correlated soft-sphere fermions at different densities and temperatures. The calculated properties include the DOS, momentum distribution functions, spin-resolved radial distribution functions, potentials of mean force, and related energy levels obtained from the Bohr-Sommerfeld condition. We observe sharp peaks on DOS and momentum distribution curves, which are explained by the appearance of fermionic bound states.

An important mathematical function that describes the interparticle correlations and their time evolution is the dynamic structure factor (DSF). According to Van Hove the dynamic structure factor (DSF) is the space and time Fourier transform of the density-density correlation function of the unperturbed system. In the case of classical systems, computations of the DSF typically employ classical molecular dynamics techniques. However, the present study focuses on quantum systems, in particular on the quantum dynamic structure factor. The main idea of this article is to employ the Wigner formulation of quantum mechanics to derive a new path integral description of the quantum DSF. The next important idea is to consider paths in this representation as stationary random processes and apply the Wiener–Khinchin theorem to express the DSF through the expected values of the power spectrum of trajectories in this representation. To calculate the DSF, spin-resolved radial distribution functions (RDFs) and other thermodynamic parameters of a 3D system of soft sphere fermions we developed a Wigner Path Integral Monte Carlo (WPIMC) method. The observed peaks on the RDFs and DSFs were attributed to wave interference resulting from a multiple-scattering and perturbation effects due to particle exchange and interaction. This phenomenon, observed in a soft sphere scattering system, may serve as a precursor to Anderson localisation, a mechanism in which wave interference across multiple scattering channels prevents wave propagation.

We study the temperature dependence of various spatial correlation functions of normal fluid and liquid

^{4}\text{H}\text{e}

along an isochoric line of particle number density $\ensuremath{\varrho}=0.02185\text{ }{\text{\AA{}}}^{\ensuremath{-}3}$. The formal and numerical investigation employs correlated density-matrix theory as currently developed. The formalism permits a clear analysis of particle exchange (statistical) effects and so-called direct-direct quantum effects that appear only in a many-body system of interacting particles. At supercritical temperatures $T\ensuremath{\ge}12\text{ }\text{K}$ the helium fluid can be characterized as a perfect quantum Boltzmann system. In such a fluid, quantum statistical correlations are absent and the particle exchange follows classical Boltzmann statistics despite direct-direct quantum effects being present. At temperatures below 12 K, statistical quantum correlations begin to appear and are largest at the Bose-Einstein transition temperature

{T}_{\text{BE}}=2.17\text{ }\text{K}

. However, these correlations are considerably smaller than the statistical exchange correlations existing in a free Bose gas of bosons with helium mass. We present analytical and numerical results on the radial distribution function, the exchange correlation function, the phase-phase correlation function, the one-body reduced density matrix, and gross thermal quantities such as specific heat and total exchange energy. The role of particle exchange in determining their quantum properties is analyzed and discussed in detail.