Expansion in Lorentzian functions of spectra of quantum autocorrelations

Abstract

We show that in a quantum mechanical many-body system of Boltzmann particles having space inversion symmetry the spectrum of the autocorrelation function of a local observable can always be given, similarly to the classical case [Phys. Rev. E 85, 022102 (2012)], in terms of a series of Lorentzian functions multiplied by the proper quantum detailed balance factor. This is done by transforming the continued fraction representation, which is derived via recurrent relations and without the use of the generalized Langevin equation hierarchy, into a series expansion. In this way characteristic frequencies can be defined, also in quantum mechanics, which refer to the particular autocorrelation. These are the frequencies of the eigenmodes of the relaxation function connected to the observable. We also show that in practical cases of interest in experimental spectroscopy, and particularly in inelastic neutron and x-ray scattering, the use of a finite number of Lorentzian shapes for an approximate description of the data is related to a reduction of the number of the relevant dynamical variables taken into account, equivalent to the lowering of the dimensionality of the orthogonalized space onto which the dynamic of the system is projected. Examples of application are given for the spectrum of the velocity autocorrelation function of liquid parahydrogen, calculated with a quantum simulation algorithm (path-integral centroid molecular dynamics), and for the molecular center-of mass dynamic structure factor in liquid carbon dioxide as computed by means of classical molecular dynamics simulation.

Time dependence of the velocity autocorrelation function of a fluid: An eigenmode analysis of dynamical processes

“…We follow the theoretical approach recently presented [21][22][23], which states that the generalized Langevin equation for a normalized autocorrelation function C(t) of a classical many-body system has an exact solution written as an infinite sum of exponential functions (we need to consider positive t only, as C(t) is an even function), i.e.,…”

“…In this paper, we pursue this goal by applying a substantially different method, not attempted so far. This is based on a recently presented general theory [21][22][23] that describes the time dependence of any correlation function of dynamical variables in many-particle Hamiltonian systems as an infinite series of (complex and/or real) exponentials. This theoretical approach, which does not resort to any a priori hypotheses on what dynamical regimes occur in the fluid, also applies naturally to the VAF and evidently opens an interesting issue with respect to the current schemes of interpretation.…”

Neutron Brillouin scattering and ab initio simulation study of the collective dynamics of liquid silver

“…As mentioned, among our aims is trying to understand what information S(Q, ω) actually brings when some rigor is applied in the spectral analysis of both experimental and simulation data, and which models are appropriate for silver with varying Q, looking for analogies with or differences from the case of gold and other metals. We will show, as in the case of other important functions for the dynamics of the liquid state, that a multiexponential analysis [41][42][43] of the intermediate scattering function F (Q, t ) [or, equivalently, a multi-Lorentzian analysis of the dynamic structure factor S(Q, ω)] is once again extremely accurate, ensuring the fulfillment of the first few sum rules along with excellent descriptions of the addressed function, and leading to a clear characterization of the main dynamical properties of the system. Thus the multimode representation is demonstrated here to account very well not only for single-particle (self-) quantities like the VAF, Z (t ), [23,[44][45][46], or the self-intermediate scattering function, F self (Q, t ) [24], but also for a collective function as F (Q, t ).…”

“…In recent years we have shown the effectiveness of the exponential series representation [41][42][43] in accounting for the behavior of time correlation functions of classical [23,24,44] and quantum [45,46] fluids. The power of the method lies not only in its providing excellent descriptions of the most relevant functions in liquid state physics, but also in its merit to facilitate the imposition of, at least a few, physical constraints, thus largely increasing the reliability of the results.…”

“…Of course, the same holds for their respective spectra, Z (ω), S self (Q, ω), and S(Q, ω). For the sake of clarity and understanding of what follows, it is useful to recall that the mentioned GH and VE models, both corresponding to a (physically constrained) finite sum of Lorentzians, comply with realistic applications of the general multimode representation [41][42][43]. Indeed, most used models of the second-order memory function [3,25] can be shown to lead to a spectrum given by a sum of an appropriate number of Lorentzian lines [25], although this universal property of the line shape has long remained unnoticed.…”