Macroscopic microscopic model

More realistic percolation models of catalyst deactivation in which diffusional limitations are present are also now available (1.3,34,49,50). Most of these models have utilized the integrated macroscopic-microscopic modelling approach discussed above (33,49, 50). Completely microscopic models are also available (1,3), in which diffusion is modeled by a random walk process. 

Both deterministic and stochastic models can be defined to describe the kinetics of chemical reactions macroscopically. (Microscopic models are out of the scope of this book.) The usual deterministic model is a subclass of systems of polynomial differential equations. Qualitative dynamic behaviour of the model can be analysed knowing the structure of the reaction network. Exotic phenomena such as oscillatory, multistationary and chaotic behaviour in chemical systems have been studied very extensively in the last fifteen years. These studies certainly have modified the attitude of chemists, and exotic begins to become common . Stochastic models describe both internal and external fluctuations. In general, they are a subclass of Markovian jump processes. Two main areas are particularly emphasised, which prove the importance of stochastic aspects. First, kinetic information may be extracted from noise measurements based upon the fluctuation-dissipation theorem of chemical kinetics second, noise may change the qualitative behaviour of systems, particularly in the vicinity of instability points. 

By using the macroscopic-microscopic model, shell correction energies can be extracted from the experimental mass M and a macroscopic (spherical) mass ATld taken from a theoretical model. The shell correction is negative and it stabilizes the nucleus ... 

Probing superheavy element space by " Ca-induced hot-fusion reactions is characterized by advancing beyond the = 162 deformed subshell closure toward nuclei that are spherical and tightly bound. The macroscopic-microscopic model characterizes the ground-states of nuclei with > 175 as having a prolate deformation parameter 62 < 0.1, making them nearly spherical [8, 58, 60]. At neutron numbers below N — 175, any cross section benefit of the " Ca-induced hot-fusion approach to the Island of Stability is expected to decrease, as the shell stabilization of the ground state of the compound nucleus decreases. If a coldfusion path to a particular superheavy nuclide is available, it is expected to be the better one however, very little experimental evidence of this is available. As an example, the attempt to produce Cn (Z — 112) in the UC Ca,4n) reaction was unsuccessful (cross section limit <0.6 pb) [8, 316], in contrast to its production in the ° Pb( °Zn,n) reaction (cross section 0.5 pb) [263, 271]. 

Attempts have been made to use the a-decay properties of the " Ca-induced reaction products to benchmark the nuclear mass evaluations arising from the various model calculations and to determine the location of the closed proton shell in transhassium space. Globally, the decay properties most closely match the predictions of the macroscopic-microscopic model, which predicts a spherical shell closure at Z = 114 [52, 58, 371-373]. However, the resiliency of theory is such that the a-decay Q values are also adequately reproduced by other models that predict a higher Z closed shell. Discrimination among the model calculations will only come about through the measurements of the decay properties of nuclides with higher Z and/or N than are currently known [8, 66]. 

Baran, A., Lojewski, Z., Sieja, K., Kowal, M. Global properties of even-even superheavy nuclei in macroscopic-microscopic models. Phys. Rev. C72, 044310(13) (2005)... 

Fig. 19 Q-values calculated with the macroscopic-microscopic model versus neutron number. Data taken from [36, 37]...

In this section we consider electromagnetic dispersion forces between macroscopic objects. There are two approaches to this problem in the first, microscopic model, one assumes pairwise additivity of the dispersion attraction between molecules from Eq. VI-15. This is best for surfaces that are near one another. The macroscopic approach considers the objects as continuous media having a dielectric response to electromagnetic radiation that can be measured through spectroscopic evaluation of the material. In this analysis, the retardation of the electromagnetic response from surfaces that are not in close proximity can be addressed. A more detailed derivation of these expressions is given in references such as the treatise by Russel et al. [3] here we limit ourselves to a brief physical description of the phenomenon. 

The relation between the microscopic friction acting on a molecule during its motion in a solvent enviromnent and macroscopic bulk solvent viscosity is a key problem affecting the rates of many reactions in condensed phase. The sequence of steps leading from friction to diflfiision coefficient to viscosity is based on the general validity of the Stokes-Einstein relation and the concept of describing friction by hydrodynamic as opposed to microscopic models involving local solvent structure. In the hydrodynamic limit the effect of solvent friction on, for example, rotational relaxation times of a solute molecule is [ ]... 

Monte Carlo simulations generate a large number of confonnations of tire microscopic model under study that confonn to tire probability distribution dictated by macroscopic constrains imposed on tire systems. For example, a Monte Carlo simulation of a melt at a given temperature T produces an ensemble of confonnations in which confonnation with energy E. occurs witli a probability proportional to exp (- Ej / kT). An advantage of tire Monte Carlo metliod is tliat, by judicious choice of tire elementary moves, one can circumvent tire limitations of molecular dynamics techniques and effect rapid equilibration of multiple chain systems [65]. Flowever, Monte Carlo... 

The low-temperature chemistry evolved from the macroscopic description of a variety of chemical conversions in the condensed phase to microscopic models, merging with the general trend of present-day rate theory to include quantum effects and to work out a consistent quantal description of chemical reactions. Even though for unbound reactant and product states, i.e., for a gas-phase situation, the use of scattering theory allows one to introduce a formally exact concept of the rate constant as expressed via the flux-flux or related correlation functions, the applicability of this formulation to bound potential energy surfaces still remains an open question. 

M. Seesselberg, G. J. Schmitz, B. Nestler, I. Steinbach. Macroscopic and microscopic modeling of the growth of YBaCuO bulk material. IEEE Trans Appl Supercond 7 11>9, 1997. 

Their theory can also be regarded as the begiiming of micro-macro thinking in the written history of science in a philosophical manner, macroscopic properties are projected, but not transferred on a pttre hypothetical microscopic model (Weillbach, 1971). 

On the continuum level of gas flow, the Navier-Stokes equation forms the basic mathematical model, in which dependent variables are macroscopic properties such as the velocity, density, pressure, and temperature in spatial and time spaces instead of nf in the multi-dimensional phase space formed by the combination of physical space and velocity space in the microscopic model. As long as there are a sufficient number of gas molecules within the smallest significant volume of a flow, the macroscopic properties are equivalent to the average values of the appropriate molecular quantities at any location in a flow, and the Navier-Stokes equation is valid. However, when gradients of the macroscopic properties become so steep that their scale length is of the same order as the mean free path of gas molecules,, the Navier-Stokes model fails because conservation equations do not form a closed set in such situations. 

In most of the gas lubrication problems in nano-gaps, gas flow usually locates in the slip flow or the transient flow regime, depending on working conditions and local geometry. Therefore, both of the macroscopic and microscopic models are introduced to analyze the gas lubrication problems. 

The structure of the interface between two immiscible electrolyte solutions (ITIES) has been the matter of considerable interest since the beginning of the last century [1], Typically, such a system consists of water (w) and an organic solvent (o) immiscible with it, each containing an electrolyte. Much information about the ITIES has been gained by application of techniques that involve measurements of the macroscopic properties, such as surface tension or differential capacity. The analysis of these properties in terms of various microscopic models has allowed us to draw some conclusions about the distribution and orientation of ions and neutral molecules at the ITIES. The purpose of the present chapter is to summarize the key results in this field. 

In physical chemistry the most important application of the probability arguments developed above is in the area of statistical mechanics, and in particular, in statistical thermodynamics. This subject supplies the basic connection between a microscopic model of a system and its macroscopic description. The latter point of view is of course based on the results of experimental measurements (necessarily carried out in each experiment on a very large number of particle ) which provide the basis of classical thermodynamics. With the aid of a simple example, an effort now be made to establish a connection between the microscopic and macroscopic points of view. 

The first difficulty derives from the fact that given any values of the macroscopic expected values (restricted only by broad moment inequality conditions), a probability density always exists (mathematically) giving rise to these expected values. This means that as far as the mathematical framework of dynamics and probability goes, the macroscopic variables could have values violating the laws of phenomenological physics (e.g., the equation of state, Newton s law of heat conduction, Stokes law of viscosity, etc.). In other words, there is a macroscopic dependence of macroscopic variables which reflects nothing in the microscopic model. Clearly, there must exist a principle whereby nature restricts the class of probability density functions, SF, so as to ensure the observed phenomenological dependences. 

Macroscopic stick-slip motion described above applies to the center of mass movement of the bodies. However, even in situations where the movement of the overall mass is smooth and steady, there may occur local, microscopic stick-slip. This involves the movement of single atoms, molecular groups, or asperities. In fact, such stick-slip events form the basis of microscopic models of friction and are the explanation why the friction force is largely independent of speed (see Section 11.1.9). 

In this connection kinetic models can also be separated into microscopic and macroscopic models. The relations between these models are established through statistical physics equations. Microscopic models utilize the concepts of reaction cross-sections (differential and complete) and microscopic rate constants. An accurate calculation of reaction cross-sections is a problem of statistical mechanics. Macroscopic models utilize macroscopic rates. 

One can think that this situation, described by equations (3.1), can be visualised as a picture of interacting (and connected in chains) Brownian particles suspended in anisotropic viscoelastic segment liquid . Introduction of macroscopic concepts is unavoidable consequence of transition from microscopic to mesoscopic approach, or better to say, from the microscopic model of interacting Kuhn-Kramers chains to mesoscopic model of interacting chains of Brownian particles. 

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