Theoretic-numerical approach systems

In this chapter, we discuss three theoretical-numerical approaches for studying heat flow in molecular systems. The different methods were developed in order to focus on various aspects of heat flow at the nanoscale, and are applicable at different parameters regimes quantum, semiclassical and classical. Ultimately, developing a unified approach is of great interest [16,17]. 

The LICS, produced by an idealized Continuous-Wave (CW) laser (steady amplitude and single-frequency laser), can differ substantially from the structure produced by a pulsed laser, since the AC Stark shifts produce time-dependent detunings relative to one- and two-photon resonance. The time-dependent pulse and frequency effects in population trapping in LICS have received attention in theoretical works [93]. Using numerical approaches, as well as approximate analytical solution, it was shown that the trapped population in realistic atomic systems can be sufficiently decreased, to the point when no population remains in the system, by the increase in laser energies. Furthermore, the use of properly chirped laser pulses not only helps to increase the trapped population but also makes the system more stable against increases in the pulse energy. 

The hydrodynamics of the experimental system can be described theoretically. Such approach is very important for correct interpretation of the experimental results, and for their extrapolation for the conditions not attainable in the existing experimental system. With the mathematical model the parametric study of the system is also possible, what can reveal the most important factors responsible for the occurrence of the specific transport phenomena. The model was presented in details elsewhere [2]. It was based on the equations of the momentum and mass transfer in the simplified two-dimensional geometry of the air-water-surfactant system. Those basic equations were supplemented with the equation of state for the phopsholipid monolayer. The resultant set of equations with the appropriate initial and boundary conditions was solved numerically and led to temporal profiles of the surface density of the surfactant, T [mol m ], surface tension, a [N m ], and velocity of the interface. Vs [m s ]. The surface tension variation and velocity field obtained from the computations can be compared with the results of experiments conducted with the LFB. 

From theoretical and fundamental aspects to recent advances and novel developments in characterization and analysis of polymers, spectroscopy has, over the years, proved itself to be the most popular family of techniques in providing information at molecular levels. MS, ESR, and NMR applications highlighted in this chapter belong to the numerous approaches in the spectroscopic characterization and analysis of polymer systems. 

This equation has been derived as a model amplitude equation in several contexts, from the flow of thin fluid films down an inclined plane to the development of instabilities on flame fronts and pattern formation in reaction-diffusion systems we will not discuss here the validity of the K-S as a model of the above physicochemical processes (see (5) and references therein). Extensive theoretical and numerical work on several versions of the K-S has been performed by many researchers (2). One of the main reasons is the rich patterns of dynamic behavior and transitions that this model exhibits even in one spatial dimension. This makes it a testing ground for methods and algorithms for the study and analysis of complex dynamics. Another reason is the recent theory of Inertial Manifolds, through which it can be shown that the K-S is strictly equivalent to a low dimensional dynamical system (a set of Ordinary Differentia Equations) (6). The dimension of this set of course varies as the parameter a varies. This implies that the various bifurcations of the solutions of the K-S as well as the chaotic dynamics associated with them can be predicted by low-dimensional sets of ODEs. It is interesting that the Inertial Manifold Theory provides an algorithmic approach for the construction of this set of ODEs. 

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