A formal density expansion of transport coefficients expressed in terms of momentum autocorrelation functions is derived for a degenerate quantum gas (Bose-Einstein or Fermi-Dirac statistics) at any frequency and for noncentral as well as central forces. The result to lowest order in density is reduced to the solution of a well-defined quantum two-body problem, and the first density correction is reduced to the solution of a well-defined quantum three-body problem. It is confirmed that at zero frequency one only needs the asymptotic forms of the collision operator to calculate quantum transport coefficients. It is further pointed out that the coefficients of the third- and higher order terms of the density expansion diverge at zero frequency, in analogy with the classical case. A "renormalization" is suggested which, it is believed, leads to a nonanalytic density dependence for quantum gases distinct from the nonanalytic behavior associated with degeneracy statistics.
Within the framework of the adiabatic theory of pressure broadening, an expansion of the spectrum in powers of the density is obtained by making use of a technique of local averaging in the time domain. The first term of the expansion, ${I}_{c}(\ensuremath{\omega})$, is essentially the impact distribution; the second term is $n{J}_{w}(\ensuremath{\omega})$ folded into ${I}_{c}(\ensuremath{\omega})$, where ${J}_{w}(\ensuremath{\omega})$ is the wing of the one-perturber distribution, that is, the intensity distribution obtained if there is only one perturbing atom interacting with the radiating atom. Higher-order terms are successive convolution powers of $n{J}_{w}(\ensuremath{\omega})$, all folded into ${I}_{c}(\ensuremath{\omega})$. The expansion should be useful mostly for understanding and predicting qualitative features of the spectrum. In particular, the results imply that structure in the wings, such as satellite bands, is broadened and shifted in the same way as the line center under increasing pressure, in agreement with experimental observations.
Modified moments provide coefficients in expansions for time-autocorrelation functions (TAF's) and for their spectral densities. The expansions for TAF's are very generally convergent; the expansions for their spectral densities converge when these densities satisfy sufficient continuity or smoothness conditions. Whenever a partial sum to the density is non-negative, the corresponding partial sum to the TAF necessarily lies between the rigorous bounds recently obtained by Platz and Gordon. The accuracy of these expansions is illustrated by an application to the harmonic solid model considered by Platz and Gordon. The expansion for the TAF is consistently one or two orders of magnitude more accurate than the bounds obtained from the same number of moments and accurately represents the TAF to substantially longer times than do the bounds.
We present a general scheme based on nonlinear response theory to calculate the expansion of correlation functions such as the pair-correlation function or the exchange-correlation hole of an inhomogeneous many-particle system in terms of density derivatives of arbitrary order. We further derive a consistency condition that is necessary for the existence of the gradient expansion. This condition is used to carry out an infinite summation of terms involving response functions up to infinite order from which it follows that the coefficient functions of the gradient expansion can be expressed in terms of the local density profile rather than the background density around which the expansion is carried out. We apply the method to the calculation of the gradient expansion of the one-particle density matrix to second order in the density gradients and recover in an alternative manner the result of Gross and Dreizler [Gross and Dreizler, Z. Phys. A 302, 103 (1981)], which was derived using the Kirzhnits method. The nonlinear response method is more general and avoids the turning point problem of the Kirzhnits expansion. We further give a description of the exchange hole in momentum space and confirm the wave vector analysis of Langreth and Perdew [Langreth and Perdew, Phys. Rev. B 21, 5469 (1980)] for this case. This is used to derive that the second-order gradient expansion of the system averaged exchange hole satisfies the hole sum rule and to calculate the gradient coefficient of the exchange energy without the need to regularize divergent integrals.
For a system of point charges that interact through the three-dimensional electrostatic Coulomb potential (without any regularization) and obey the laws of nonrelativistic quantum mechanics with Bose or Fermi statistics, the static correlations are expressed in terms of Mayer-like diagrams. The exchange effects are taken into account systematically and the long-range Coulomb divergencies are exactly resummed in order to get finite diagrams. For this purpose, in the framework of the grand canonical ensemble, the matrix elements of the imaginary-time evolution operator are represented by the Feynman-Kac functional integral according to Ginibre's idea [J. Math. Phys. 6, 238 (1965); 6, 252 (1965), 6, 1432 (1965)] and we exhibit a correspondence between the correlations in the quantum system of point particles and the distribution functions in a classical fluid of "exchange" loops with Brownian random shapes. The size of a loop, which corresponds to the number of particles involved in the corresponding cyclic permutation, and the shape of a loop, which describes the quantum fluctuations, play the part of internal degrees of freedom that must be integrated over when calculating the distributions relative to the quantum point charges. The loops interact through a two-body potential that is different from the electrostatic interaction between two charged curves: each line element of a loop interacts only with a discrete number of line elements in every other loop. The linear response theory to an infinitesimal external charge distribution can be written in this formalism by a formula analogous to that of classical statistics, and the loop-fugacity and loop-density expansions of the loop-distribution functions are derived by the usual techniques of Mayer diagrams generalized to extended objects. At large distances
r, the loop potential behaves as
r1 and, as in the classical case, every Mayer diagram diverges. However, this nonintegrable tail is independent of the shape of the loops and it is explicitly and exactly resummed by a generalization of the method developed by Meeron [J. Chem. Phys. 28, 630 (1958); Plasma Physics (McGraw-Hill, New York, 1961)], for classical fluids of point entities. Auxiliary
r1 bonds are introduced and subdiagrams involving chains of
r1 bonds are integrated over first in a systematic way. The new diagrams contain bonds between loops that decay either exponentially or algebraically, with a
r31 leading term, and the new diagrams are at least conditionally integrable. The part of the quantum particle-particle correlation arising directly from exchange, which is derived from the density of loop, decays faster than any inverse power law, whereas, as shown in the following paper [Phys. Rev. E 53, 4595 (1996)], the whole quantum particle-particle correlation, which also involves the correlations between loops, decays only as
r61.