Time scale dynamics associated with

As demonstrated by the power spectra in Figs. 12.2a and 12.3b, regulation of the blood flow to the individual nephron involves several oscillatory modes. The two dominating time scales are associated with the period Tsiow 30—40 s of the slow TGF-mediated oscillations and the somewhat shorter time scale Tjast 5—10 s defined by the myogenic oscillations of the afferent arteriolar diameter. The two modes interact because they both involve activation of smooth muscle cells in the arteriolar wall. Our model describes these mechanisms and the coupling between the two modes, and it also reproduces the observed multi-mode dynamics. We can, therefore, use the model to examine some of the phenomena that can be expected to arise from the interaction between the two modes. 

Large-scale numerical simulation for samples that are many times os large as the critical wavelength is perhaps the only way to develop a quantitative understanding of the dynamics of solidification systems. Even for shallow cells, such calculations will be costly, because of the fine discretizations needed to be sure the dynamics associated with the small capillary length scales are adequately approximated. Such calculations may be feasible with the next generation of supercomputers. 

Even if we consider a single solvent, e g., water, at a single temperature, say 298K, depends on the solute and in fact on the coordinate of the solute which is under consideration, and we cannot take xF as a constant. Nevertheless, in the absence of a molecular dynamics simulation for the solute motion of interest, XF for polar solvents like water is often approximated by the Debye model. In this model, the dielectric polarization of the solvent relaxes as a single exponential with a relaxation time equal to the rotational (i.e., reorientational) relaxation time of a single molecule, which is called Tp) or the Debye time [32, 347], The Debye time may be associated with the relaxation of the transverse component of the polarization field. However the solvent fluctuations and frictional relaxation occur on a faster scale given by [348,349]... 

The dynamics associated with the Hamiltonian Eq. (8) or its variants Eq. (11) and Eq. (14) can be treated at different levels, ranging from the explicit quantum dynamics to non-Markovian master equations and kinetic equations. In the present context, we will focus on the first aspect - an explicit quantum dynamical treatment - which is especially suited for the earliest, ultrafast events at the polymer heterojunction. Here, the coherent vibronic coupling dynamics dominates over thermally activated events. On longer time scales, the latter aspect becomes important, and kinetic approaches could be more appropriate. 

The electronic absorption spectra of complex molecules at elevated temperatures in condensed matter are generally very broad and virtually featureless. In contrast, vibrational spectra of complex molecules, even in room-temperature liquids, can display sharp, well-defined peaks, many of which can be assigned to specific vibrational modes. The inverse of the line width sets a time scale for the dynamics associated with a transition. The relatively narrow line widths associated with many vibrational transitions make it possible to use pulse durations with correspondingly narrow bandwidths to extract information. For a vibration with sufficiently large anharmonicity or a sufficiently narrow absorption line, the system behaves as a two-level transition coupled to its environment. In this respect, time domain vibrational spectroscopy of internal molecular modes is more akin to NMR than to electronic spectroscopy. The potential has already been demonstrated, as described in some of the chapters in this book, to perform pulse sequences that are, in many respects, analogous to those used in NMR. Commercial equipment is available that can produce the necessary infrared (IR) pulses for such experiments, and the equipment is rapidly becoming less expensive, more compact, and more reliable. It is possible, even likely, that coherent IR pulse-sequence vibrational spectrometers will... 

We first consider the intermolecular modes of liquid CS2. One of the details that two-dimensional Raman spectroscopy has the potential to reveal is the coupling between intermolecular motions on different time scales. We start with the one-dimensional Raman spectrum. The best linear spectra are based on time domain third-order Raman data, and these spectra demonstrate the existence of three dynamic time scales in the intermolecular response. In Fig. 3 we have modeled the one-dimensional time domain spectrum of CS2 for 3 cases (A) a single mode represented by the sum of three Brownian oscillators, (B) three Brownian oscillators, and (C) a distribution of 20 arbitrary Brownian oscillators. Case (A) represents the fully coupled, or isotropic case where the liquid is completely homogeneous on the time scales of the simulation. Case (B) deconvolutes the linear response into the three time scales that are directly evident in the measured response and is in the limit that the motions associated with each of the three timescales are uncoupled. Case (C) is an example where the liquid is represented by a large distribution of uncoupled motions. 

Cell dynamics simulations are based on the time dependence of an order parameter, (i) (Eq. 1.23), which varies continuously with coordinate r. For example, this can be the concentration of one species in a binary blend. An equation is written for the time evolution of the order parameter, dir/dt, in terms of the gradient of a free energy that controls, for example, the tendency for local diffusional motions. The corresponding differential equation is solved on a lattice, i.e. the order parameter V (r) is discretized on a lattice, taking a value at lattice point i. This method is useful for modelling long time-scale dynamics such as those associated with phase separation processes. 

A set of molecular dynamics simulations was carried out on samples prepared using the direct method described in Section 5.4. In all, four different sample sets of polymer glass were produced by using different preparation procedures. All of the samples were relaxed at 200 K with P = 1 bar for 1 ns. The final densities were all the same to within 1% and we expect all of the samples to have glass transformations in the range 300-400 K on the simulation time scale. The associated correlation lengths and fractions of trans conformers obtained from the final 200 ps of these runs are shown in Table 5.1. 

Before embarking on a discussion of the results of these studies let us add one historical note. The difficulty with swinging the polymer tails in a conformational transition has been recognized for many years. A means of circumventing was proposed by Schatzki. Verdier and Stockmayer had earlier invoked a similar principle but used it only to produce Rouse modes. We know now that slow Rouse modes are insensitive to the details of the faster time-scale dynamics. The proposed motions are completely local, and involve going from one equilibrium rotational isomeric state to another by moving only a finite, small number of atoms. Mechanisms of this class have come to be known as crankshaft motions (a term applicable in the strictest sense only to the Schatzki proposal). Because of the limited amount of motion and the simplicity of the dynamics these models are easy to understand, analyze, and simulate. This probably contributes to the continued attention devoted to them. The crankshaft idea has helped to focus attention on the necessity to localize the motion associated with conformational transitions, but complete localization is too restrictive. There are theoretical objections that can be raised to the crankshaft mechanism, but the bottom line is that no signs of it are found in our simulations. 

The molecular dynamics associated with the glass transition of polymers are cooperative segmental dynamics. The relaxation process of the cooperative segmental motions is known as the a-relaxation process. At the glass transition, the length scale of a cooperative segmental motion is believed to be 1-4 nm, and the average a-relaxation time is 100 s [56]. The a-relaxation process is represented by a distribution of relaxation times. In time-domain measurements, the a-relaxation is non-exponential and can be described by a stretched-exponential function. The most common function used to describe the a-process is that of the Kohlrausch-Williams-Watts (KWW) [57, 58] equation ... 

---