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Journal of Mathematical Physics

Correlations near phase transition in a simple fluid

G. V. Ramanathan

Published Oct 1, 1977• 2 citations
Publisher

Abstract

Correlations in a system with a weak, long-ranged attractive potential are studied. Using a natural small parameter, the asymptotic orders of all correlations are established. Explicit leading order solutions are obtained for all correlations when the system is away from phase change As it approaches transition, It is shown that the many-body correlations, even though still small, develop very long range and that they are shown extremely slowly varying functions over space. As a results all the correlations contribut equally significantly to the computation of the pair correlation. In an asymptotic ragion near transition, a hierarchy is derived in which each corelation is expressed in terms of its immediate predecessor, its immediate successor and the pair correlation. The pair potential does not appear in these equations. Instead the correlations are expressed in terms of a Yukawa potential whose strength and range are related to the density and other parameters but not to the detailed form of the potential. In the leading approximation, the pair correlation is proportional to this Yukawa function.

DOI
10.1063/1.523166

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Correlations in a system with a weak, long-ranged attractive potential are studied. Using a natural small parameter, the asymptotic orders of all correlations are established. Explicit leading order solutions are obtained for all correlations when the system is away from phase change As it approaches transition, It is shown that the many-body correlations, even though still small, develop very long range and that they are shown extremely slowly varying functions over space. As a results all the correlations contribut equally significantly to the computation of the pair correlation. In an asymptotic ragion near transition, a hierarchy is derived in which each corelation is expressed in terms of its immediate predecessor, its immediate successor and the pair correlation. The pair potential does not appear in these equations. Instead the correlations are expressed in terms of a Yukawa potential whose strength and range are related to the density and other parameters but not to the detailed form of the potential. In the leading approximation, the pair correlation is proportional to this Yukawa function.
Ground-state properties of a two-dimensional fluid of bosons with repulsive dipole-dipole interactions are studied by means of the Euler-Lagrange hypernetted-chain approximation. We present a self-consistent semianalytical theory of the pair distribution function g(r)g(r)g(r) and ground-state energy of this system. Our approach is based on the solution of a zero-energy scattering Schr\"odinger equation for the ``pair amplitude'' g(r)\sqrt{g(r)}g(r)​ with an effective potential from Jastrow-Feenberg correlations. We find excellent agreement with quantum Monte Carlo results over a wide range of coupling strength, nearly up to the critical coupling for the liquid-to-crystal quantum phase transition. We also calculate the one-body density matrix and related quantities, such as the momentum distribution function and the condensate fraction.
We study correlation functions of the Calogero-Sutherland model in the whole range of the interaction parameter. Using the replica method we obtain analytical expressions for the long-distance asymptotics of the one-body density matrix in addition to the previously derived asymptotics of the pair-distribution function [D.M. Gangardt and A. Kamenev, Nucl. Phys. B 610, 578 (2001)]. The leading analytic and nonanalytic terms in the short-distance expansion of the one-body density matrix are discussed. Numerical results for these correlation functions are obtained using Monte Carlo techniques for all distances. The momentum distribution and static structure factor are calculated. The potential and kinetic energies are obtained using the Hellmann-Feynman theorem. Perfect agreement is found between the analytical expressions and numerical data. These results allow for the description of physical regimes of the Calogero-Sutherland model. The zero temperature phase diagram is found to be of a crossover type and includes quasicondensation, quasicrystallization and quasisupersolid regimes.
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)h(k)h(k) and direct correlation function c(k)c(k)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.
We have investigated the correlation functions of interacting bosons at the generic superfluid-insulator transition, a prototypical quantum phase transition, in two dimensions in the spherical limit. Unexpectedly the spatial correlation functions show non-power-law behavior consisting of two parts: short-range correlation due to the particle-hole pair excitations and long-range off-diagonal order due to the single-particle condensation. The temporal correlation functions, on the other hand, show power-law behavior.
The low-density expansions for the energy, chemical potential, and condensate depletion of the homogeneous dilute dipolar Bose gas are obtained by regularizing the dipole-dipole interaction at long distances. It is shown that the leading term, proportional to the density, allows a simple physical interpretation and consistently describes the thermodynamic stability of the system. The long-range asymptotics are obtained analytically for the normal and anomalous one-particle correlation functions and the pair distribution function. We discuss the properties of the two-body scattering with zero relative momentum for the dipole-dipole interaction; in particular, we derive the asymptotics of the wave function and a correction to the scattering length for small values of the dipolar range. We show how the density expansions can be derived within the Bogoliubov model of weakly interacting particles without any divergence from the assumption of universality of the expansions at low densities.