Phenomenological microscopic approach

Models of a second type (Sec. IV) restrict themselves to a few very basic ingredients, e.g., the repulsion between oil and water and the orientation of the amphiphiles. They are less versatile than chain models and have to be specified in view of the particular problem one has in mind. On the other hand, they allow an efficient study of structures on intermediate length and time scales, while still establishing a connection with microscopic properties of the materials. Hence, they bridge between the microscopic approaches and the more phenomenological treatments which will be described below. Various microscopic models of this type have been constructed and used to study phase transitions in the bulk of amphiphihc systems, internal phase transitions in monolayers and bilayers, interfacial properties, and dynamical aspects such as the kinetics of phase separation between water and oil in the presence of amphiphiles. 

No attempt will be made here to extend our results beyond the simple lowest-order limiting laws the often ad hoc modifications of these laws to higher concentrations are discussed in many excellent books,8 11 14 but we shall not try to justify them here. As a matter of fact, for equilibrium as well as for nonequilibrium properties, the rigorous extension of the microscopic calculation beyond the first term seems outside the present power of statistical mechanics, because of the rather formidable mathematical difficulties which arise. The main interests of a microscopic theory lie both in the justification qf the assumptions which are involved in the phenomenological approach and in the possibility of extending the mathematical techniques to other problems where a microscopic approach seems necessary in the particular case of the limiting laws, obvious extensions are in the direction of other transport coefficients of electrolytes (viscosity, thermal conductivity, questions involving polyelectrolytes) and of plasma physics, as well as of quantum phenomena where similar effects may be expected (conductivity of metals and semi-... 

The two standard approaches in any treatment of kinetics [28] are to explain the system in terms of me thermodynamic driving forces (namely, VjJ.) or in terms of the fnndamental rate eqnations. The rate equations can be fnrther subdivided into an atomistic, or microscopic, approach that accounts for individual molecules as they go through the various processes (adsorption, desorption, diffusion, capture, and release) or a phenomenological, or macroscopic, explanation that looks for correlations and the so-called scaling laws over large distances (much larger than the lattice spacing). 

In deformed systems, it is preferable to work in the principal axes system of the deformation tensor and to consider the component of the mean square deviations in the directions of the principal axes. In phenomenological as well as in microscopic approaches almost exclusively the power law... 

The microscopic approach to nucleation problems has apparently not yet been carried out. There have been a number of mesoscopic developments for homogeneous nucleation [2.19,36-38]. The mesoscopic approach is successful in giving information on fluctuations, which are, of course, central to the process of nucleation. In this, the mesoscopic approach improves on the macroscopic approach. However, the transition probability is not known from "first principles" and, therefore, must retain some phenomenological elements. 

Unfortunately, the need to take higher excited electronic bands into account makes it unlikely that the mostly phenomenologically theoretical approaches used so far are applicable to the results reported here (Levenson and Bloembergen, 1974). It is particularly unlikely that these models would be able to account for the observed anisotropy in Xg On the other hand, it is possible that the more microscopic, ab initio, calculations of semiconductor systems currently being developed (Martin, 1981) will in time become theoretical bases for a comparison to the experimentally observed components of reported here. 

In summary, continuum models of membranes have been instrumental to understand the physical properties of lipid assemblies and they have dramatically helped the development of this research field. They nowadays represent a gold standard to interpret both in vitro and in vivo experiments and to test the accuracy of bottom-up microscopic approaches on large-scale membrane properties. Yet, despite massively elaborated mathematical attempts to incorporate microscopic details into the equations, the inevitable assumptions that are intrinsic of phenomenological approaches prevent a faithful and accurate description of the chemical properties of lipid assemblies. 

The crossover between these two extremes of behavior is found to be quite extensive. These results involve many approximations, which are not always well controlled, but they represent one of the first microscopic approaches to dynamics in semi-dilute solutions and they seem to confirm the more phenomenological tube model. It should also be noticed that the confinement of the chain in what would be the tube is not due to topological entanglements, as in Edwards approach, but rather to the excluded volume interactions with the other chains. The relative role of these two effects has not yet been studied. 

The significance of this coefficient consists of the fact that it connects the macroscopic (technical) value a with microscopic one (1 and t>). Notice that this result is obtained from the comparison of macroscopic (phenomenological) and microscopic approaches. 

The two sources of stochasticity are conceptually and computationally quite distinct. In (A) we do not know the exact equations of motion and we solve instead phenomenological equations. There is no systematic way in which we can approach the exact equations of motion. For example, rarely in the Langevin approach the friction and the random force are extracted from a microscopic model. This makes it necessary to use a rather arbitrary selection of parameters, such as the amplitude of the random force or the friction coefficient. On the other hand, the equations in (B) are based on atomic information and it is the solution that is approximate. For ejcample, to compute a trajectory we make the ad-hoc assumption of a Gaussian distribution of numerical errors. In the present article we also argue that because of practical reasons it is not possible to ignore the numerical errors, even in approach (A). 

To improve the accuracy of the solution, the size of the time step may be decreased. The smaller is the time step, the smaller are the assumed errors in the trajectory. Hence, in contrast (for example) to the Langevin equation that includes the friction as a phenomenological parameter, we have here a systematic way of approaching a microscopic solution. Nevertheless, some problems remain. For a very large time step, it is not clear how relevant is the optimal trajectory to the reality, since the path variance also becomes large. Further-... 

If it cannot be guaranteed that the adsorbate remains in local equilibrium during its time evolution, then a set of macroscopic variables is not sufficient and an approach based on nonequihbrium statistical mechanics involving time-dependent distribution functions must be invoked. The kinetic lattice gas model is an example of such a theory [56]. It is derived from a Markovian master equation, but is not totally microscopic in that it is based on a phenomenological Hamiltonian. We demonstrate this approach... 

Previous theoretical kinetic treatments of the formation of secondary, tertiary and higher order ions in the ionization chamber of a conventional mass spectrometer operating at high pressure, have used either a steady state treatment (2, 24) or an ion-beam approach (43). These theories are essentially phenomenological, and they make no clear assumptions about the nature of the reactive collision. The model outlined below is a microscopic one, making definite assumptions about the kinematics of the reactive collision. If the rate constants of the reactions are fixed, the nature of these assumptions definitely affects the amount of reaction occurring. 

In the following the present status of calculations of the SE is reviewed first the ones in which a microscopic nucleon-nucleon (NN) interaction is used, followed by the phenomenological mean field approaches. In section 4 we discuss possible constraints that can be obtained from empirical information. 

As noted above at higher densities the EoS is sensitive to 3NF contributions. Whereas the 3NF for low densities seems now well understood its contribution to nuclear matter densities remains unsettled. In practice in calculations of the symmetry energy in the BHF approach two types of 3NF have been used in calculations in ref.[4] the microscopic 3NF based upon meson exchange by Grange et al. was used, and in ref. [15] as well in most VCS calculations the Urbana interaction. The latter has in addition to an attractive microscopic two-pion exchange part a repulsive phenomenological part constructed in such a way that the empirical saturation point for SNM is reproduced. Also in practice in the BHF approach to simplify the computational efforts the 3NF is reduced to a density dependent two-body force by averaging over the position of the third particle. 

Nowadays attention is turned also to the supermolecular level, that is, to the morphologic aspects, to the nature of interfaces, to the formation of new phases, or of particular aggregates (liquid crystals, gels, etc.). Interest has also been directed to the study of chain mobility for its influence on frictional properties of polymers. In recent years there have been many successful approaches to a microscopic theory (in contrast to a phenomenological approach) of the physi-comechanical behavior of macromolecular materials. 

The mechanism for cross-linking of thermosetting resins is very complex because of the relative interaction between the chemical kinetics and the changing of the physical properties [49], and it is still not perfectly understood. The literature is ubiquitous with respect to studies of cure kinetic models for these resins. Two distinct approaches are used phenomenological (macroscopic level) [2,5,50-72] and mechanistic (microscopic level) [3,73-85]. The former is related to an overall reaction (only one reaction representing the whole process), the latter to a kinetic mechanism for each elementary reaction occurring during the process. 

All that remains to be done for determining the fluctuation spectrum is to compute the conditional average, Eq. (31). However, this involves the full equations of motion of the many-body system and one can at best hope for a suitable approximate method. There are two such methods available. The first method is the Master Equation approach described above. Relying on the fact that the operator Q represents a macroscopic observable quantity, one assumes that on a coarse-grained level it constitutes a Markov process. The microscopic equations are then only required for computing the transition probabilities per unit time, W(q q ), for example by means of Dirac s time-dependent perturbation theory. Subsequently, one has to solve the Master Equation, as described in Section TV, to find both the spectral density of equilibrium fluctuations and the macroscopic phenomenological equation. 

Of course, the macroscopic equations cannot actually be derived from the microscopic ones. In practice they are pieced together from general principles and experience. The stochastic mesoscopic description must be obtained in the same way. This semi-phenomenological approach is remarkably successful in the range where the macroscopic equations are linear, see chapter VIII. In the nonlinear case, however, difficulties appear, which can only be resolved by the improved, but still mesoscopic, method of chapter X. 

In this tribute and memorial to Per-Olov Lowdin we discuss and review the extension of Quantum Mechanics to so-called open dissipative systems via complex deformation techniques of both Hamiltonian and Liouvillian dynamics. The review also covers briefly the emergence of time scales, the definition of the quasibosonic pair entropy as well as the precise quantization relation between the temperature and the phenomenological relaxation time. The issue of microscopic selforganization is approached through the formation of certain units identified as classical Jordan blocks appearing naturally in the generalised dynamical picture. 

The main goal of simulation methods is to obtain information on the spatial and temporal behavior of a complex system (a material), that is, on its structure and evolution. Simulation methods are subdivided into atomistic and phenomenological methods. Atomistic methods directly consider the evolution of the system of interest at the atomic level with regard to the microscopic structure of the substance. These methods include classical and quantum MD and various modifications of the MC technique. Phenomenological methods are based on macroscopic equations in which the atomistic nature of the material is not directly taken into account. Within the multiscale approach, both groups of methods mutually complement each other, which permits the physicochemical system under study to be described most comprehensively. 

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