Classical linear response theory

Classical linear response theory now yields ct(Z) = (for t > 0) with... 

The spectra s (v) will be described here in terms of a linear-response theory. We shall employ the specific form [GT, VIG] of this theory, called the ACF method, which previously was termed the dynamic method. The latter is based on the Maxwell equations and classical dynamics. A more detailed description of this method is given in Section II. Taking into attention the central role of the model suggested here, we, for the sake of completeness, give below a brief list of the main assumptions employed in our variant of the ACF method. 

The quantum mechanical forms of the correlation function expressions for transport coefficients are well known and may be derived by invoking linear response theory [64] or the Mori-Zwanzig projection operator formalism [66,67], However, we would like to evaluate transport properties for quantum-classical systems. We thus take the quantum mechanical expression for a transport coefficient as a starting point and then consider a limit where the dynamics is approximated by quantum-classical dynamics [68-70], The advantage of this approach is that the full quantum equilibrium structure can be retained. 

We begin our discussion with a survey of the quantum dynamics and linear response theory expressions for quantum transport coefficients. Since we wish to make a link to a partial classical description, the use of Wigner transforms... 

We may easily carry out a linear response theory derivation of transport properties based on the quantum-classical Liouville equation that parallels the... 

Supplementing this equation with an additional set of solvent oscillators one can incorporate a solvent environment. Notice that this does not necessarily imply harmonic solvent motions. In fact the full anharmonicity of the solvent can be accounted for in the context of linear response theory [39] where the interaction is described in terms of an effective harmonic oscillator bath. This allows calculation of relaxation rates from classical molecular dynamics simulations of the force fn(x) exerted by the solvent on the relevant system. This approach has found appli-... 

The notation A = iLA, A = (iL)M,... recalls that classical and quantum Liouvillians are time derivatives. The dissipative fluxes refer to the dynamically accessible microstates. They also provide the basis of the so-called extended thermodynamics [66]. For example, in the linear response theory [67], the dynamical susceptibilities are proportional to the time derivatives of the correlation functions [63]... 

Fig. 7. Schematic representation of the collective dipole spectra of sodium clusters obtained in linear response theory [57], The quantity plotted is a m) as the percentage of the total dipole strength, mi, normalized to 100% (see Eq. (42)). The lowest spectrum (Na ) represents the classical limit, where 100% of the strength lies in the surface plasmon (frequency (Ouu) and the volume plasmon (frequency co,) has zero strength. For finite clusters the surface plasmon is red-shifted and its missing strength is distributed over the remainder of the strongly fragmented volume plasmon.

Abstract. Kubo s paper on linear-response theory provided a unified language to describe a wide variety of transport phenomena, both quantum and classical, in a suitable microscopic language. The paper has been crucial for subsequent developments in numerical simulation. 

The linear-response approach has played an important role in the construction of most modern theories of transport processes (see, e.g. Ref. [3]). Moreover, it has had a profound impact on the development of classical molecular dynamics simulations - a field that was emerging at the same time. Linear-response theory showed how linear transport coefficients can be computed in a simulation, by studying the decay of fluctuations in equilibrium. More specifically, most transport coefficients can be expressed as time integrals of (au-to)correlation functions of microscopic fluxes (e.g. the... 

Let us first consider spectroscopy. Linear-response theory, in particular the fluctuation dissipation theorem - which relates the absorption of an incident monochromatic field to the correlation function of (e.g. dipole) fluctuations in equilibrium - has changed our perspective on spectroscopy of dense media. It has moved away from a static Schrodinger picture -phrased in terms of transitions between immutable (but usually incomputable) quantum levels - to a dynamic Heisenberg picture, in which the spectral line shape is related by Fourier transform to a correlation function that describes the decay of fluctuations. Of course, any property that cannot be computed in the Schrodinger picture, cannot be computed in the Heisenberg picture either however, correlation functions, unlike wave-functions, have a clear meaning in the classical limit. This makes it much easier to come up with simple (semi) classical interpretations and approximations. 

At T 0 the sharp lines corresponding to the harmonic modes are broadened by anharmonic effects until, at high temperature, the simple relationship between vibrational density of states and dynamical matrix is lost. In this regime, and especially for large aggregates, MD is the most suitable tool to compute the vibrational spectrum. Standard linear response theory within classical statistical mechanics shows that the spectrum f(co) is given by the Fourier transform of the velocity-velocity autocorrelation function... 

Microscopic theory of lattice dynamics studies the response of electron-ion systems to displacements of nuclei the "direct" method generally means an approach in which the undistorted crystal and the crystal with displacements are treated from the very beginning as two distinct and unrelated systems. The "direct" treatment of phonons is an alternative to the classical perturbation approach, in which phonons are viewed as a small perturbation of the ground state to be treated by linear response theory. This method, based on the inverse dielectric matrix, is explained in this Volume in the lecture-notes by J. T. Devreese, R. Resta and A. Baldereschl. 

Linear response theory [152] is perfectly suited to the study of fluid structures when weak fields are involved, which turns out to be the case of the elastic scattering experiments alluded to earlier. A mechanism for the relaxation of the field effect on the fluid is just the spontaneous fluctuations in the fluid, which are characterized by the equilibrium (zero field) correlation functions. Apart from the standard technique used to derive the instantaneous response, based on Fermi s golden rule (or on the first Bom approximation) [148], the functional differentiation of the partition function [153, 154] with respect to a continuous (or thermalized) external field is also utilized within this quantum context. In this regard, note that a proper ensemble to carry out functional derivatives is the grand ensemble. All of this allows one to gain deep insight into the equilibrium structures of quantum fluids, as shown in the works by Chandler and Wolynes [25], by Ceperley [28], and by the present author [35, 36]. In doing so, one can bypass the dynamics of the quantum fluid to obtain the static responses in k-space and also make unexpected and powerful connections with classical statistical mechanics [36]. 

Interaction between quantum systems and classical flelds is not problematic. It is the basis of almost all forms of optical spectroscopy where the transition dipole operator of the system interacts with the electric and magnetic flelds of light. It is a necessary ingredient of linear response theory, and also of the Redfleld relaxation mechanism. The starting point for all these examples is the quantum Liouville equation... 

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