Description macroscopic

The term macroscopic description refers to the long-time and large-scale limit, t - oo and x - oo, of mesoscopic equations where the details of the microscopic movement are irrelevant. In particular, it refers to the diffusive limit where balance equations such as (3.13), (3.41), and (3.74) are approximated by the diffusion equation (2.1). The standard derivation of the diffusion equation involves the assumption that the typical microscopic jumps and times are small compared to the characteristic macroscopic space and time scales. Let us illustrate this using the mesoscopic transport equation (3.74). If the jump density w(z) is a rapidly decaying function for large z, one can expand p(x - z, t) in z and truncate the Taylor series at the second moment  

This truncation is a well-defined procedure, if the higher moments become progressively smaller. If the jump density w z) is even, then we obtain the standard diffusion equation. However, this naive Taylor series expansion is not valid for heavy-tailed probability density functions, such as a Cauchy PDF, 

3 Random Walks and Mesoscopic Reaction-Transport Equations 

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We have considered briefly the important macroscopic description of a solid adsorbent, namely, its speciflc surface area, its possible fractal nature, and if porous, its pore size distribution. In addition, it is important to know as much as possible about the microscopic structure of the surface, and contemporary surface spectroscopic and diffraction techniques, discussed in Chapter VIII, provide a good deal of such information (see also Refs. 55 and 56 for short general reviews, and the monograph by Somoijai [57]). Scanning tunneling microscopy (STM) and atomic force microscopy (AFT) are now widely used to obtain the structure of surfaces and of adsorbed layers on a molecular scale (see Chapter VIII, Section XVIII-2B, and Ref. 58). On a less informative and more statistical basis are site energy distributions (Section XVII-14) there is also the somewhat laige-scale type of structure due to surface imperfections and dislocations (Section VII-4D and Fig. XVIII-14). 

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. 

The following qualitative picture emerges from these considerations in weak flow where the molecular coils are essentially undeformed, the polymer solution should behave approximately as a Newtonian fluid. In strong flow of a highly dilute polymer solution where the macroscopic velocity field can still be approximated by the Navier-Stokes equation, it should be expected, nevertheless, that in the immediate proximity of a chain, the fluid will be slowed down because of the energy intake to stretch the molecular coil thus, the local velocity field may deviate from the macroscopic description. In the general case of polymer flow,... 

Many other, less obvious physical consequences of miniaturization are a result of the scaling behavior of the governing physical laws, which are usually assumed to be the common macroscopic descriptions of flow, heat and mass transfer [3,107]. There are, however, a few cases where the usual continuum descriptions cease to be valid, which are discussed in Chapter 2. When the size of reaction channels or other generic micro-reactor components decreases, the surface-to-volume ratio increases and the mean distance of the specific fluid volume to the reactor walls or to the domain of a second fluid is reduced. As a consequence, the exchange of heat and matter either with the channel walls or with a second fluid is enhanced. 

There are situations where such a macroscopic description of reaction dynamics will break down. For instance, biochemical reactions in the cell may involve only small numbers of molecules of certain species that participate in the mechanism. An example is gene transcription where only tens of free... 

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. 

G Rossi, KA Mazich. Macroscopic description of the kinetics of swelling for a cross-linked elastomer or a gel. Phys Rev E 48 1182-1191, 1993. 

Macroscopic descriptions of matter and radiation are adequate without taking the discontinuous nature of matter and/or radiation into account. However, when dealing with particles approaching the size of elementary quanta, the quantum effects become increasingly important and must be taken into account explicitly in the mechanical description of these particles. Unlike relativistic mechanics, quantum mechanics cannot be used to describe macroscopic events. There is a fundamental difference between classical and non-classical, or quantum, phenomena and the two systems are complimentary rather than alternatives. 

In the preceding chapters, we are primarily concerned with an empirical macroscopic description of reaction rates, as summarized by rate laws. This is without regard for any description of reactions at the molecular or microscopic level. In this chapter and the next, we focus on the fundamental basis of rate laws in terms of theories of reaction rates and reaction mechanisms. ... 

Equations (257) through (266) provide us with a closed set of equations which allow us, in principle, to calculate yaB and w . However, an exact solution of these equations is very difficult to obtain and is moreover not very useful. Indeed, we expect our macroscopic description to be valid only at very small ionic concentrations and it is thus not necessary to derive an exact result only the leading term in an asymptotic expansion at small C will be relevant. The following approximations will thus be used ... 

This completes our macroscopic description of the limiting conductance in dilute electrolytes. 

Postmortem Procedures. Rabbits are euthanized by lethal dose of a barbiturate soon after the last vaginal irritation scores are collected. The vagina is opened by longitudinal section and examined for evidence of mucosal damage such as erosion, localized hemorrhage, and so on. No other tissues are examined. No tissues are collected. After the macroscopic description of the vagina is recorded, the animal is discarded. 

Surface-complexation models require a high degree of detail about the heterogeneous systems. Unfortunately, the chemical detail required to use surface-complexation models will often exceed our knowledge of interactions taking place in natural systems. Consequently, geochemists have often resorted to semi-empirical, macroscopic descriptions, which are more easily utilized. 

Macroscopic Descriptions of Solute Adsorption and the Net Proton Coefficient. The macroscopic proton coefficient plays two important roles in our macroscopic descriptions of surface processes. First,... 

Another reason for describing surface reaction kinetics in more detail is that we need to examine the processes on a microscopic scale. While we are interested primarily in the macroscopic description of catalytic reactor behavior, we cannot do this intelligently until we understand these processes at a molecular level. 

The limit cycle is an attractor. A slightly different kind occurs in the theory of the laser Consider the electric field in the laser cavity interacting with the atoms, and select a single mode near resonance, having a complex amplitude E. One then derives from a macroscopic description laced with approximations the evolution equation... 

In the present section we are concerned with genuine internal noise. We consider a closed, isolated many-body system, whose evolution is given by a Schrodinger equation. Remember that in the classical case in III.2 we gave a macroscopic description in terms of a reduced set of macroscopic variables, which obey an autonomous set of differential equations. These equations are approximate and deviations appear in the form of fluctuations, which are a vestige of the large number of eliminated microscopic variables. Our task is to carry out this program in the framework of quantum mechanics. 

Understanding chemical reactions has been a major preoccupation since the historical origins of chemistry. A main difficulty is to reconcile the macroscopic description in which reactions are rate processes ruling the time evolution of populations of chemical species with the microscopic Hamiltonian dynamics governing the motion of the translational, vibrational, and rotational degrees of freedom of the reacting molecules. 

The transition from a macroscopic description to the microscopic level is always a complicated mathematical problem (the so-called many-particle problem) having no universal solution. To illustrate this point, we recommend to consider first the motion of a single particle and then the interaction of two particles, etc. The problem is well summarized in the following remark from a book by Mattuck [18] given here in a shortened form. For the Newtonian mechanics of the 18th century the three-body problem was unsolvable. The general theory of relativity and quantum electrodynamics created unsolvable two-body and single-body problems. Finally, for the modem quantum field... 

In terms of the master equation for the Markov process the formal kinetics is nothing but the mean-field theory where the fluctuation terms like that on the r.h.s. of equation (2.2.43) are neglected. Strictly speaking, the macroscopic description, equation (2.1.2), were correct if the fluctuation terms vanished as V —> oo. In a general case the function P(N, t) does not satisfy the Poisson distribution [16, 27] in particular, °N (N> ... 

Despite recent promising strategies, the principle of micro-process engineering is still not widely used in combinatorial catalysis. One drawback certainly is the increasing distance from industrial applications with decreasing dimensions. However, the small structures possess laminar flow conditions that are fully accessible by analytical as well as numerical macroscopic descriptions. This offers the chance to describe thoroughly the fluidic, diffusive and reactive phenomena in catalysis to find intrinsic kinetics on using, for example, non-porous sputtered catalysts. 

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