Macroscopic dynamics

The ability to create and observe coherent dynamics in heterostructures offers the intriguing possibility to control the dynamics of the charge carriers. Recent experiments have shown that control in such systems is indeed possible. For example, phase-locked laser pulses can be used to coherently amplify or suppress THz radiation in a coupled quantum well [5]. The direction of a photocurrent can be controlled by exciting a structure with a laser field and its second harmonic, and then varying the phase difference between the two fields [8,9]. Phase-locked pulses tuned to excitonic resonances allow population control and coherent destruction of heavy hole wave packets [10]. Complex filters can be designed to enhance specific characteristics of the THz emission [11,12]. These experiments are impressive demonstrations of the ability to control the microscopic and macroscopic dynamics of solid-state systems. 

Throughout this chapter we focus on the extended hydrodynamic description for smectic A-type systems presented in [42,43], We discuss the possibility of an undulation instability of the layers under shear flow keeping the layer thickness and the total number of layers constant. In contrast to previous approaches, Auernhammer et al. derived the set of macroscopic dynamic equations within the framework of irreversible thermodynamics (which allows the inclusion of dissipative as well as reversible effects) and performed a linear stability analysis of these equations. The key point in this model is to take into account both the layer displacement u and the director field ft. The director ft is coupled elastically to the layer normal p = in such a way that ft and p are parallel in equilibrium z is the coordinate perpendicular to the plates. 

Following the lines proposed above will give a prediction of the pattern formed above onset. For a transition from undulating lamellae to reorientated lamellae or to multilamellar vesicles, defects have to be created for topological reasons. Since the order parameter varies spatially in the vicinity of the defect core, a description of such a process must include the full (tensorial) nematic order parameter as macroscopic dynamic variables. 

The dielectric behavior of PMCHI was studied by Diaz Calleja et al. [210] at variable frequency in the audio zone and second, by thermal stimulated depolarization. Because of the high conductivity of the samples, there is a hidden dielectric relaxation that can be detected by using the macroscopic dynamic polarizability a defined in terms of the dielectric complex permittivity e by means of the equation ... 

On the other hand, the properties of the system as a whole can be calculated and the macroscopic dynamic modulus can be determined. Here the question of the relation between the postulated micro-viscoelasticity and the resulting macro-viscoelasticity appears. The answer requires a properly formulated self-consistency condition. Simple speculations show that equality of the micro- and macro-viscoelasticity cannot be obtained. Nevertheless, it is natural to require the equality of relaxation times of micro- and macroviscoelasticities. It will be shown in this section that this condition can be satisfied. 

The "laminar" macroscopic flow equations contain phenomenological terms which represent averages over the macroscopic dynamics to include the effects of turbulence. Examples of these terms are eddy viscosity and diffusivity coefficients and average chemical heat release terms which appear as sources in the macroscopic flow equations. Besides providing these phenomenological terms, the turbulence model must use the information provided by the large scale flow dynamics self-consistently to determine the energy which drives the turbulence. The model must be able to follow reactive interfaces on the macroscopic scale. 

Reduction of kinetic theory to fluid mechanics is historically the first example of a successful reduction of a mesoscopic dynamical theory to a more macroscopic dynamical theory. The method (called the Chapman-Enskog method) that was invented by Chapman and Enskog for this particular reduction remains still a principal inspiration for all other types of reduction (see, e.g.,. Gorban and Karlin, 2003, 2005), Yablonskii et al., 1991). In this example we briefly recall the geometrical viewpoint of the Chapman-Enskog method. We shall also illustrate the point (IV)... 

For most macroscopic dynamic systems, the neglect of correlations and fluctuations is a legitimate approximation [383]. For these cases the deterministic and stochastic approaches are essentially equivalent, and one is free to use whichever approach turns out to be more convenient or efficient. If an analytical solution is required, then the deterministic approach will always be much easier than the stochastic approach. For systems that are driven to conditions of instability, correlations and fluctuations will give rise to transitions between nonequihbrium steady states and the usual deterministic approach is incapable of accurately describing the time behavior. On the other hand, the stochastic simulation algorithm is directly applicable to these studies. 

Among possible macroscopic dynamical quantities of interest, the most useful is the macroscopic stress,... 

In addition to the microstructural geometrical features described above, macroscopic, dynamical, rheological properties of the suspensions are derived by Brady and Bossis (1985). Dual calculations are again performed, respectively with and without DLVO-type forces. When such forces are present, an additional contribution (the so-called elastic stress) to the bulk stress tensor exists. In such circumstances, the term (Batchelor, 1977 Brady and Bossis, 1985)... 

I. Horenko, E. Dittmer, F. Lankas, J. Maddocks, P. Metzner, and C. Schiitte (2005) Macroscopic dynamics of complex metastable systems Theory, algorithms, and application to b-dna. J. Appl. Dyn. Syst, submitted... 

In the previous chapter we have seen how spatial correlation functions express useful structural infonnation about our system. This chapter focuses on time correlation functions (see also Section 1.5.4) tlrat, as will be seen, convey important dynamical information. Time correlation functions will repeatedly appear in our future discussions of reduced descriptions of physical systems. A typical task is to derive dynamical equations for the time evolution of an interesting subsystem, in which only relevant information about the surrounding thermal environment (bath) is included. We will see that dynamic aspects of this relevant information usually enter via time correlation functions involving bath variables. Another type of reduction aims to derive equations for the evolution of macroscopic variables by averaging out microscopic information. This leads to kinetic equations that involve rates and transport coefficients, which are also expressed as time correlation functions of microscopic variables. Such functions are therefore instrumental in all discussions that relate macroscopic dynamics to microscopic equations of motion. 

The contaminant is passive. This means the contaminant will not change the macroscopic dynamic parameters of the carrier gas (usually air). The limited amount of the toxic compound will not change gas parameters such as viscosity and density, hence one can use the known parameters of dry or humid air to calculate the dynamics of the gas flow. 

For one-dimensional flow in the z direction, where v (r) is a function of radial position only, the final expression for the macroscopic dynamic force... 

Using X-ray investigations and optical measurements the reorientation behavior of nematic monodomain samples has been studied in detail by Kundler and Finkelmann [12]. This reorientation behavior has been modeled using a bifurcation analysis of the macroscopic dynamic description by Wei-... 

Since there are very few dynamic experimental investigations of pretransitional effects [8], not much modeling has been reported to date either. Based on work for the macroscopic dynamics of the nematic-isotropic transition in sidechain polymers [27 -29], it has been suggested [28] that the non-meanfield exponent observed in dynamic stress-optical experiments [8] can be accounted for at least qualitatively by the mode-coupling model [28, 29]. Intuitively this qualitatively new dynamic behavior can be traced back to static nonlinear coupling terms between the nematic order parameter and the strain tensor. 

Ultrasonic experiments using laser induced phonon spectroscopy have been performed in a nematic liquid single crystal elastomer [48]. The experiments reveal a dispersion step for the speed of sound and a strong anisotropy for the acoustic attenuation constant in the investigated frequency range (100 MHz -1 GHz). These results are consistent with a description of LCEs using macroscopic dynamics [54-56] and reflect a coupling between elastic effects and the nematic order parameter as analyzed in detail previously [48]. 

In previous sections we showed that the macroscopic dynamics of propagating fronts depend on the statistical characteristics of the underlying random walk model for the mesoscopic transport process. Since the dynamics of fronts are nonuniversal, it is an important problem to find universal rules relating both levels of description. The goal of this section is to address this problem. We are interested in exploring the physical properties of systems of particles that disperse according to a general CTRW. 

Givon, D., Kupferman, R., Stuart, A. Extracting macroscopic dynamics model problems and algorithms. Nonlinearity 17(6), R55 (2004)... 

Experiments on supramolecular networks formed with multiple types of crosslinkers show that the response to an applied stress occurs through discrete contributions of each type of cross-linker rather than being an average of the contributing species [149-151], Frequency-dependent measurements of such networks exhibit multiple plateau values in G (inverse frequencies of which can be associated with the individual time constants of dissociation of the different cross-links, as demmistrated in Fig. 8d. Because the macroscopic dynamic response is controllable at the level of molecular associations, this effect has been called the macromolecular analogue of the kinetic isotope effect [147]. 

There is also a very specific scaling for the macroscopic dynamic contact angle 6. This was first observed by Hoffman [5], who investigated the variation in the contact angle of an advancing meniscus front in a capillary tube subject to different fluid driving velocities, as well as Tanner [6], who observed similar behavior for drops spreading on a horizontal substrate. In both cases, it was observed that for small Ca,... 

We shall discuss here the macroscopic dynamics of liquid crystals that is an area of hydrodynamics or macroscopic properties related to elasticity and viscosity. With respect to the molecular dynamics, which deals, for example, with NMR, molecular diffusion or dipolar relaxation of molecules, the area of hydrodynamics is a long scale, both in space and time. The molecular dynamics deals with distances of about molecular size, a 10 A, i.e., with wavevectors about 10 cm , however, in the vicinity of phase transitions, due to critical behaviour, characteristic lengths of short-range correlations can be one or two orders of magnitude larger. Therefore, as a limit of the hydrodynamic approach we may safely take the range of wavevectors q 10 cm and corresponding frequencies (O c q 10 - 10 = 10"s (c is sound velocity). 

The macroscopic structure of matter can be assessed, for example, by optical microscopy and can then be linked to its microscopic origin through X-ray, neutron, or electron diffraction experiments and the various forms of electron and atomic-force microscopy. A factor of 10 -10 separates the atomic, nanometer scale from the macroscopic, micrometer scale. Macroscopic dynamic techniques ultimately linked to molecular motion are, for example, dynamic mechanical and dielectric analyses and calorimetry. In order to have direct access to the details of the underlying microscopic motion, one must, however, use computational methods. A realistic microscopic description of motion has recently become possible through accurate molecular dynamics simulations and will be described in this review. It will be shown that the basic large-ampHtude molecular motion exists on a picosecond time scale (1 ps = 10 s), a ffictor at... 

Wolframl discovered analogous behavior from simulation studies with cellular automata. His work shows that, notwithstanding well-defined short-range interaction rules between components on a microscopic level, macroscopic dynamic behavior can become unpredictable. This implies that external disturbances can have an important outcome on both temporal and structural events. This is consistent with a condition of life, where there is change due to evolutionary response. Examples of the four basic classes of behavior Wolfram discovered are shown in Fig. 9.11. 

Knowledge of the eigenfunctions, Eq. 102, allows us to get analytical expressions not only for the macroscopic dynamic characteristics (averaged over all monomers), but also for local quantities of interest, related to individual cross-links or individual chain monomers [25,66]. hi view of space restrictions we will not provide these details here, instead we turn to the discussion of the macroscopic viscoelastic properties of the system. The reader interested in the details is referred to the original papers [25,66]. 

The air-water interface has interesting features as a medium for molecular recognition. For example, (1) a molecularly flat environment is formed at the interface, (2) a boundary region is facing the two phases with different dielectric constants, (3) macroscopically dynamic changes can be taken place... 

What are the implications of these studies on the calculation of macroscopic dynamical properties of ionic liquids At the very least, they suggest that one should be careful when applying standard computational techniques used for simple liquids to ionic liquids. Most of these techniques assume ergodic behavior, but the work described above shows this may not always be the case. Due to the sluggish dynamics of ionic liquid systems, one should carry out very long simulations to ensure adequate sampling. 

Based on these experimental findings, a refined theory of spreading has been developed (10). The fundamental result of this theory is that the energy associated with the spreading parameter S is presumably nearly entirely spent through viscous dissipation in this film. Therefore, the macroscopic dynamics of the drop is independent of the microscopic behaviour of the precursor film. This explains why the macroscopic dynamics of these systems do not contain the parameter S. 

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