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.
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 derive two expansions in powers of the density for the autocorrelation function of a spectral-line profile in a plasma. The expansions can be regarded as systematic generalizations of the well-known unified theory and of the scalar theory which appear in a natural way as the respective first-order terms. Under the additional assumption of completeness of collisions, but without assuming nonoverlapping, we obtain a simplified expansion. It is a generalization of the impact theory, and the latter results again from the first-order term. We present quick and direct derivations of the first- and second-order terms for all three density expansions. The Nth-order terms contain time-development operators and S matrices for N or fewer particles.
In this paper, four distinct ideas are combined, which under a wide range of circumstances can give very rapidly converging series expansions for functional integrals. (1) Expansion of the functional being integrated in functional Taylor series. In the familiar case arising in quantum statistical mechanics, that of the Wiener integral of exp[−∫V(x(t))dt],V(x) being the perturbing potential, this is equivalent to expanding the characteristic functional of the probability functional of V(x(t)) in central moments of V(x(t)). The lowest-order term of the series is the approximation obtained by Feynman and Hibbs through a variational method. (2) Transfer of the harmonic term of the potential, when the functional integral is the quantum-statistical density matrix (Green's function of the Bloch equation), to the weighting function. This transforms the functional integral from a Wiener to an Uhlenbeck-Ornstein integral. The formal expressions for the terms of the expansion are somewhat more complicated, but they can be worked out, and the result is a great improvement in the speed of convergence of the series with decreasing temperature and/or decreasing relative magnitude of the anharmonic part of the potential. (3) ``Reservation of variables'' in the integration. This amounts to breaking the averaging process down into an average over subsets of the distributed random function (conditional average), followed by an averaging of these averages. Any step of this kind (it may be repeated within the subsets, etc.) gives an improvement of accuracy. (4) When the quantity being evaluated through functional integration is the partition function, the device introduced by Feynman and Hibbs, of interchanging the functional integration with the integration of the Green's function over the equated initial and final configuration-space points, may be combined with the above techniques. This eliminates one integration in the terms of the expansion and seems to improve accuracy at the same time. The general series obtained is correlated with more conventional operator techniques of quantum-mechanical perturbation theory, in order to answer the perennial question, does the path-integral method bring with it anything that could not be derived by other methods? It is in some sense a Feynman-Dyson expansion of the Green's function, but one that is further modified mathematically in a way characteristic only of the path-integral point of view, and which, moreover, improves its accuracy. It thus appears unlikely that the result is merely one of standard type disguised as a functional integral result. Sample numerical calculations are given to assay the accuracy of the methods, which are shown to compare very favorably with the traditional approximation of finite subdivision of the time interval.
In this paper we have presented a general scheme whereby the method of Padé approximants may be applied to the power series expansion of time dependent correlation functions. It has been demonstrated by actual numerical calculations, using the computer simulated results of Berne and Harp [5], that even a few terms in the power series expansion are capable of reproducing the correlation curves to a fairly large extent. It is pointed out that the method can be extended to higher orders without the introduction of fitting parameters, and without violating the time parity requirements of the Liouville equation. In the final analysis the method is a truly microscopic method since it relies on the microscopic hamiltonian, and the underlying principles are very straightforward. The basic requirements are the spectral moments, but we do not need many of these as the Padé approximants are highly convergent.
Two representation of the Laplace transforms of time autocorrelation functions, namely the moment expansion and Mori's continued fraction representation, are studied from the point of view of their convergence, with the aim of obtaining new general properties of time autocorrelation functions. First, the relation between them is established by using mathematical techniques in the case of a general dynamical variable, and a direct method, useful for applications, in the case of Hermitian variables. The mathematical structure associated to Mori's generalized random forces is investigated and it is shown that these random forces can be obtained by a Schmidt orthogonalization of the sequence of initial time derivatives of the dynamical variable considered. Then, the convergence criteria for both representations are examined and an illustration is given with the exactly solvable case of an isotopic impurity in a linear chain of coupled harmonic oscillators. Finally, the question of knowing whether the continued fraction expansion is convergent for any Harmitian dynamical variable and any system is discussed, with its implications for the general behaviour of time autocorrelation functions.
We consider properties of the ground state density for the [Formula: see text]-dimensional Fermi gas in an harmonic trap. Previous work has shown that the [Formula: see text]-dimensional Fourier transform has a very simple functional form. It is shown that this fact can be used to deduce that the density itself satisfies a third-order linear differential equation, previously known in the literature but from other considerations. It is shown too how this implies a closed form expression for the [Formula: see text]th non-negative integer moments of the density, and a second-order recurrence. Both can be extended to general Re[Formula: see text]. The moments, and the smoothed density, permit expansions in [Formula: see text], where [Formula: see text], with [Formula: see text] denoting the shell label. The moment expansion substituted in the second-order recurrence gives a generalization of the Harer–Zagier recurrence, satisfied by the coefficients of the [Formula: see text] expansion of the moments of the spectral density for the Gaussian unitary ensemble in random matrix theory.