Memory kernel dynamic

The above expression has been used by Leutheusser [34] and Kirkpatrick [30] in the study of liquid-glass transition. Leutheusser [34] has derived the expression of the dynamic structure factor from the nonlinear equation of motion for a damped oscillator. In their expression they refer to the memory kernel as the dynamic longitudinal viscosity. 

Equation (210) when compared with the viscoelastic model of the dynamic structure factor we can identify the memory kernel in the viscoelastic model, which is written as... 

Thus we note that the memory kernel has a short-time and a long-time part. It is the long-time part which is not present in the viscoelastic model, becomes important in the supercooled-liquid-near-glass transition, and gives rise to the long-time tail of the dynamic structure factor. 

The barrier crossing dynamics is determined by the memory kernel K(c), and Russell and Sceats suggested that it be modeled such that it gave the same velocity autocorrelation function in the vicinity of the outer minimum Rq of K(R) as exjjected from a superposition of the damp>ed conformational normal modes of the polymer, j,J = 1, N — 1. The potential P(R) is expanded about Rq with a force constant ficoi, where coq is defined by an integral such as Eq. (7.20), but with the range of integration extended to infinity. The result is that... 

The dynamical contents ofEq. (14.89) is muchmore involved than its Markovian counterpart. Indeed, non-Markovian evolution is a manifestation of multidimensional dynamics, since the appearance of a memory kernel in an equation of motion signifies the existence of variables, not considered explicitly, that change on the same timescale. Still, the physical characteristics of the barrier crossing process remain the same, leading to similar modes of behavior ... 

Equations (5.1-5.3) are equivalent to the original set of molecular dynamics equations in Eq. (4.15c) in their ability to mimic an infinite solid. Indeed, the GLE equations are no easier to solve in their present form. The advantage of the GLE approach occurs because it is possible to approximate the memory kernel and random force (Tully 1980a DePristo 1984, 1989 Diestler and Riley 1985, 1987) to provide a reasonably accurate description of both the short-time and long-time (actually long-wavelength) response of the primary zone atoms to an external perturbation (e.g., a collision). 

The static and dynamic memory kernel matrix elements and have... 

Although it turned out that a number of essential features concerning dynamics of molecular liquids can be well captured by the theory of the previous subsection (see Secs. 3 and 5), an intense investigation through experimental, theoretical, and molecular-dynamics simulation studies for simple liquids has revealed that the microscopic processes underlying various time-dependent phenomena cannot be fully accounted for by a simplified memory-function approach [18, 19, 20]. In particular, the assumption that the decay of memory kernels is ruled by a simple exponential-type relaxation must be significantly revised in view of the results of the kinetic framework developed for dense liquids (see Sec. 5.1.4). This motivated us to further improve the theory for dynamics of polyatomic fluids. 

So far we have employed the exponential model for the memory kernel appearing in the GLE for the density correlation functions. In [60] the model based on the mode-coupling theory described in Sec. 5.2.4is applied to the calculation of the longitudinal current spectra of the same diatomic liquid as discussed here. It is found that the essential features of the results remained the same as far as the collective dynamics is concerned. It is also demonstrated that the results are in fair agreement with those determined from the molecular dynamics simulation. 

Longitudinal current spectra. Here we also employ a simple exponential model for the memory kernel described in Sec. 5.2.3 to account for the damping effect on the dynamics. Figures 5.16 and 5.17 exhibit the results based on this theory along with the MD simulation data performed on the same system. The results are reported in the... 

This decomposition is essentially the same as the one given in Eq. (5.184) which is discussed in Sec. 5.4.4 based on the solvent viewpointthe only difference comes from the fact that here the short-time behavior of the memory kernel also affects the friction. Thus and zz denote the terms which are essentially due to the couplings of the ion dynamics to the solvent acoustic and optical (OM-I) modes, respectively, and Cnz represents their cross term. (Explicit expressions for these terms can be found in [91].) Therefore we shall also regard here and Czz as the Stokes and dielectric parts of the friction, and Cnz as their cross term as in Sec. 5.4.4. 

We have already encountered Generalized Langevin dynamics in Chap. 6, where it appeared as an intermediate stage in the derivation of Langevin dynamics. In some cases, the properties of the heat bath and the resulting memory kernel are important... 

In principle the memory kernel in the mode-coupling equation contains contributions from three particle correlations, however, for all systems studied so far in computer simulations, these only slightly modified the predictions of the theory and helped improve agreement between simulation and theory [31]. Also, there has been an extension of the theory taking chain connectivity into account [32] which improved agreement with the simulations of the bead-spring model, but it remains to be seen whether an application of this theory to the two models presented here can account for their strongly different dynamic behavior. 

Abstract. This article reviews from both theoretical and numerical aspects three non-equivalent complete second-order formulations of quantum dissipation theory, in which both the reduced dynamics and the initial canonical thermal equilibrium are properly treated in the weak system-bath coupling limit. Two of these formulations are rather familiar as the time-local and the memory-kernel prescriptions, while another which can be termed as correlated driving-dissipation equations of motion will be shown to have the combined merits of the two conventional formulations. By exploiting the exact solutions to the driven Brownian oscillator system, we demonstrate that the time-local and correlated driving-dissipation equations of motion formulations are usually better than their memory-kernel counterparts, in terms of their applicability to a broad range of system-bath coupling, non-Markovian, and temperature parameters. Numerical algorithms are detailed for an efficient evaluation of both the reduced canonical thermal equilibrium state and the non-Markovian evolution at any temperature, in the presence of arbitrary time-dependent external fields. 

Here, can be easily evaluated via Eq. (3.7), but with TZs there being replaced by TZu [cf. Eq. (B.12b)] that may be considered as the Markovian dissipation superoperator. This statement may be supported by the arguments that the Markovian approximation amounts to the following two conditions (i) The bath correlation time is short compared with the reduced system dynamics (ii) The correlated effects of driving and dissipation can be neglected so that the Green s function G(t r) in the memory kernel can be replaced by its field-free counterpart Gs t t) = In this case, Eq. (B.9) reduces to... 

We suppose that the effective Hamiltonian is known. Let us first recall how it is directly related to the spectroscopical and dynamical observables [24,25]. Since the role of effective Hamiltonians in both line profiles and dynamics is already well documented the reader is referred to some review on this wide subject (see, e.g.. Refs. [16-18,26]). The reports [18] and [26] contain numerous references inside and outside chemical physics. Reference [17] is a review of time-dependent effective Hamiltonians. Earlier application can be found in references [27] and [28]. Memory kernels are discussed in references [14,15,18] with references to irreversible statistical mechanics. Here we briefly review the subject for introducing the basic concepts and the notations. In the second part of this section we will present corrections to the dynamics for taking into account the dependence on energy of the effective Hamiltonian. 

The mode coupling approximation for m (0 yields a set of equations that needs to be solved self-consistently. Hereby the only input to the theory is the static equilibrium structure factor 5, that enters the memory kernel directly and via the direct correlation function that is given by the Ornstein-Zernicke expression = (1 - l/5,)/p, with p being the average density. In MCT, the dynamics of a fluid close to the glass transition is therefore completely determined by equilibrium quantities plus one time scale, here given by the short-time diffusion coefficient. The theory can thus make rather strong predictions as the only input, namely, the equilibrium structure factor, can often be calculated from the particle interactions, or even more directly can be taken from the simulations of the system whose dynamics is studied. 

The key quantity in barrier crossing processes in tiiis respect is the barrier curvature Mg which sets the time window for possible influences of the dynamic solvent response. A sharp barrier entails short barrier passage times during which the memory of the solvent environment may be partially maintained. This non-Markov situation may be expressed by a generalized Langevin equation including a time-dependent friction kernel y(t) [ ]... 

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