Macroscopic systems, descriptions

A statistical ensemble can be viewed as a description of how an experiment is repeated. In order to describe a macroscopic system in equilibrium, its thennodynamic state needs to be specified first. From this, one can infer the macroscopic constraints on the system, i.e. which macroscopic (thennodynamic) quantities are held fixed. One can also deduce, from this, what are the corresponding microscopic variables which will be constants of motion. A macroscopic system held in a specific thennodynamic equilibrium state is typically consistent with a very large number (classically infinite) of microstates. Each of the repeated experimental measurements on such a system, under ideal... 

How are fiindamental aspects of surface reactions studied The surface science approach uses a simplified system to model the more complicated real-world systems. At the heart of this simplified system is the use of well defined surfaces, typically in the fonn of oriented single crystals. A thorough description of these surfaces should include composition, electronic structure and geometric structure measurements, as well as an evaluation of reactivity towards different adsorbates. Furthemiore, the system should be constructed such that it can be made increasingly more complex to more closely mimic macroscopic systems. However, relating surface science results to the corresponding real-world problems often proves to be a stumbling block because of the sheer complexity of these real-world systems. 

Molecular mechanics calculations are a very useful tool for the spatial and energetic description of small molecules as well as macroscopic systems like proteins or DMA. 

If quantum mechanics is really the fundamental theory of our world, then an effectively classical description of macroscopic systems must emerge from it - the so-called quantum-classical transition (QCT). It turns out that this issue is inextricably connected with the question of the physical meaning of dynamical nonlinearity discussed in the Introduction. The central thesis is that real experimental systems are by definition not isolated, hence the QCT must be viewed in the relevant physical context. 

For a system composed of N particles, the complete velocity distribution function is denoted f(N> (r(N>, p(/V), t). It is a function of 6N variables, that is, the three vector coordinates for each of the N molecules rW) and the three components of the momentum of each molecule p(-V). Of course, for a macroscopic system, where IV is a very large number, on the order of Avogadro s number A, it is impossible to obtain f(N). One usually attempts to find a less complete description of the system by looking at f(h which depends on the positions and momentum of a smaller number of molecules h and integrates over the effects of the remaining N — h molecules. 

This Ansatz is the essential step. The -expansion is not just one out of a plethora of approximation schemes, to be judged by comparison with experimental or numerical results 0. It is a systematic expansion in and is the basis for the existence of a macroscopic deterministic description of systems that are intrinsically stochastic. It justifies as a first approximation the standard treatment in terms of a deterministic equation with noise added, as in the Langevin approach. It will appear that in the lowest approximation the noise is Gaussian, as is commonly postulated. In addition, however, it opens up the possibility of adding higher approximations. 

Until now we assumed that we have the maximum information on the many-particle system. Now we will consider a large many-body system in the so-called thermodynamic limit (N- °o, V—> >, n = NIV finite) that means a macroscopic system. Because of the (unavoidable) interaction of the macroscopic many-particle system with the environment, the information of the microstate is not available, and the quantum-mechanical description is to be replaced by the quantum-statistical description. Thus, the state is characterized by the density operator p with the normalization... 

When other kinds of work are involved, it is necessary to specify more variables, but the point is that when a small number of properties are specified, all the other properties of the system are fixed. This is in contrast with the very large number of properties that have to be specified to describe the microscopic state of a macroscopic system. In classical physics the complete description of a mole of an ideal gas would require the specification of 3NA components in the three directions of spatial coordinates and 3NA components of velocities of molecules, where NA is the Avogadro constant. 

Evaluating the performance of a gas-solid transport system usually requires a means of macroscopic field description of the distribution of basic flow properties such as pressure, mass fluxes, concentrations, velocities, and temperatures of phases in the system. To conduct such an evaluation, the Eulerian continuum or multifluid approach is usually the best choice among the available approaches. 

Besides, let us note the automatic observance (certainly with correctly set initial data) and, hence, needlessness of the formalized descriptions in equilibrium modeling of such important regularities of macroscopic system behavior as the Gibbs phase rule, the Le Chatelier-Brown principle, mass action laws, the Henry law, the Raoult law, etc. 

Choice of the mathematical apparatus of macroscopic equilibrium descriptions. Problems in modeling the nonholonomic, nonscleronomous, and nonconservative systems. Possibilities for using differential equations (autonomous and nonautonomous) and MP. 

Experimental observations of the time evolution of externally unforced macroscopic systems on the level meSo l show that the level eth of classical equilibrium thermodynamics is not the only level offering a simplified description of appropriately prepared macroscopic systems. For example, if Cmeso is the level of kinetic theory (Sections 2.2.1, starting point. In order to see the approach 2.2.2, and 3.1.3) then, besides the level, also the level of fluid mechanics (we shall denote it here Ath) emerges in experimental observations as a possible simplified description of the experimentally observed time evolution. The preparation process is the same as the preparation process for Ath (i.e., the system is left sufficiently long time isolated) except that we do not have to wait till the approach to equilibrium is completed. If the level of fluid mechanics indeed emerges as a possible reduced description, we have then the following four types of the time evolution leading from a mesoscopic to a more macroscopic level of description (i) Mslow/ (ii) Aneso 2 -> Ath, (ui) Aneso l -> Aneso 2, and (iv) Aneso i —> Aneso 2 —> Ath- The first two are the same as (111). We now turn our attention to the third one, that is,... 

Koelman and Hoogerbrugge (1993) have developed a particle-based method that combines features from molecular dynamics (MD) and lattice-gas automata (LGA) to simulate the dynamics of hard sphere suspensions. A similar approach has been followed by Ge and Li (1996) who used a pseudo-particle approach to study the hydrodynamics of gas-solid two-phase flow. In both studies, instead of the Navier-Stokes equations, fictitious gas particles were used to represent and model the flow behavior of the interstial fluid while collisional particle-particle interactions were also accounted for. The power of these approaches is given by the fact that both particle-particle interactions (i.e., collisions) and hydrodynamic interactions in the particle assembly are taken into account. Moreover, these modeling approaches do not require the specification of closure laws for the interphase momentum transfer between the particles and the interstitial fluid. Although these types of models cannot yet be applied to macroscopic systems of interest to the chemical engineer they can provide detailed information which can subsequently be used in (continuum) models which are suited for simulation of macroscopic systems. In this context improved rheological models and boundary condition descriptions can be mentioned as examples. 

Molecules are small and light typical linear dimensions are 10 to ICr m, and typical masses are 10 to kg. Hence the number of molecules in a macroscopic system is enonuous. For example, one mole of matter contains 6.022 x 10 molecules (Avogadro s number). Because of these features— smallness, lightness, and numerical abundance— the proper description of behavior at the molecular level and its extrapolation to a macroscopic scale require the special methods of quairtum mechanics aird statistical mechanics. We pursue neitherof these topics here. Instead, we present material nseful for relating molecular concepts to observed thenrrodynamicproperties. 

Statistical mechanics is the branch of physical science that studies properties of macroscopic systems from the microscopic starting point. For definiteness we focus on the dynamics ofan A-particle system as our underlying microscopic description. In classical mechanics the set of coordinates and momenta, (r, p ) represents a state of the system, and the microscopic representation of observables is provided by the dynamical variables, v4(r, p, Z). The equivalent quantum mechanical objects are the quantum state [/ ofthe system and the associated expectation value Aj = of the operator that corresponds to the classical variable A. The corresponding observables can be thought of as time averages... 

By introducing the notion of various statistical physical ensembles in Section 2.2.1, we saw that wc can make the quantum iiurhaitical treatment consistent with several constraints imposed at the macroscopic level of description. That way we obtain an understanding of a thermal system at the microscopic level that is, we can interpret thermodynamic properties in terms of the interaction between the micro.scopic constituents forming a macroscopic system. 

Descriptions of individual corrosion processes can be assembled and used to predict materials degradation in macroscopic systems. However, the computations required are usually so lengthy and complex as to require access to large scale computational facilities. Expansion of this approach to the analysis and prediction of corrosion behavior on a wider scale requires the development of more efficient mathematical techniques and algorithms and of methods for simplifying the calculations without loss of significant factors. 

The specific examples chosen in this section, to illustrate the dynamics in condensed phases for the variety of system-specific situations outlined above, correspond to long-wavelength and low-frequency phenomena. In such cases, conservation laws and broken symmetry play important roles in the dynamics, and a macroscopic hydrodynamic description is either adequate or is amenable to an appropriate generalization. There are other examples where short-wavelength and/or high-frequency behaviour is evident. If this is the case, one would require a more microscopic description. For fluid systems which are the focus of this section, such descriptions may involve a kinetic theory of dense fluids or generalized hydrodynamics which may be linear or may involve nonlinear mode coupling. Such microscopic descriptions are not considered in this section. 

Decay-scheme studies have provided theoretical physicists with a broad range of experimental data for the development of models to interpret nuclear properties. Interestingly, such models take up ideas established for the description of atomic, molecular, or macroscopic systems, in spite of the many orders of magnitude difference in dimensions. 

In Chapters 1 through 4, we focused on a desaiption of matter at the molecular and atomic levels. In such a description, the state of the system is described quantum mechanically in terms of the wave function, which is a function of the positions of all the particles. However, without highly specialized equipment, the observable world is far removed from the molecular realm both in terms of the number of atoms or molecules ( 10 instead of just a few) and length scale (centimeters and meters instead of Angstroms and nanometers). Objects that are very large compared to the molecular scale are referred to as macroscopic It is both inconvenient and impossible to describe a macroscopic system in terms of the detailed atomic-scale variables of the constituent molecules—there are simply too many. Instead we characterize the state of macroscopic systems using a relatively small set of quantities, called macroscopic properties (or thermodynamic properties). Two important examples of such properties are pressure and temperature. 

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