Macroscopic system

Oyy/Ais of the order of hT, as is Since a macroscopic system described by themiodynamics probably has at least about 10 molecules, the uncertainty, i.e. the typical fluctuation, of a measured thennodynamic quantity must be of the order of 10 times that quantity, orders of magnitude below the precision of any current experimental measurement. Consequently we may describe thennodynamic laws and equations as exact . 

Macroscopic systems contain a large number, N, of microscopic constituents. Typically A is of the order of... 

Wlien = N/2, the value of g is decreased by a factor of e from its maximum atm = 0. Thus the fractional widtii of the distribution is AOr/A i M/jV)7 For A 10 the fractional width is of the order of 10 It is the sharply peaked behaviour of the degeneracy fiinctions that leads to the prediction that the thennodynamic properties of macroscopic systems are well defined. 

In equilibrium statistical mechanics, one is concerned with the thennodynamic and other macroscopic properties of matter. The aim is to derive these properties from the laws of molecular dynamics and thus create a link between microscopic molecular motion and thennodynamic behaviour. A typical macroscopic system is composed of a large number A of molecules occupying a volume V which is large compared to that occupied by a molecule ... 

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. 

A macroscopic system maintains constant pressure by changing its volume. A simulation i the isothermal-isobaric ensemble also maintains constant pressure by changing the volurr... 

Now, in principle, the angle of contact between a liquid and a solid surface can have a value anywhere between 0° and 180°, the actual value depending on the particular system. In practice 6 is very difficult to determine with accuracy even for a macroscopic system such as a liquid droplet resting on a plate, and for a liquid present in a pore having dimensions in the mesopore range is virtually impossible of direct measurement. In applications of the Kelvin equation, therefore, it is almost invariably assumed, mainly on grounds of simplicity, that 0 = 0 (cos 6 = 1). In view of the arbitrary nature of this assumption it is not surprising that the subject has attracted attention from theoreticians. 

The energy E is said to be quantized in discrete packets, or quanta, each of energy hv. It is because of the extremely small value of h that quantization of energy in macroscopic systems had escaped notice, but, of course, it applies to all systems. 

Thermodynamics is a deductive science built on the foundation of two fundamental laws that circumscribe the behavior of macroscopic systems the first law of thermodynamics affirms the principle of energy conservation the second law states the principle of entropy increase. In-depth treatments of thermodynamics may be found in References 1—7. 

Thermodynamics is the branch of science that embodies the principles of energy transformation in macroscopic systems. The general restrictions which experience has shown to apply to all such transformations are known as the laws of thermodynamics. These laws are primitive they cannot be derived from anything more basic. 

To close this chapter we emphasize that Hie statistical mechanical definition of macroscopic parameters such as temperature and entropy are well designed to describe isentropic equilibrium systems, but are not immediately applicable to the discussion of transport processes where irreversible entropy increase is an essential feature. A macroscopic system through which heat is flowing does not possess a single tempera-... 

Nakhleh Krajcik, 1994) macroscopic system microscopic system symbolic system algebraic system... 

Another example of slight conceptual inaccuracy is given by the Wigner function(12) and Feynman path integral(13). Both are useful ways to look at the wave function. However, because of the prominence of classical particles in these concepts, they suggest the view that QM is a variant of statistical mechanics and that it is a theory built on top of NM. This is unfortunate, since one wants to convey the notion that NM can be recovered as an integral part of QM pertaining to for macroscopic systems. 

The incomplete and active subject of describing macroscopic systems. 

The method described above for the atomic system can be extended to a macroscopic system shown in Fig. 7 where a spherical body is connected via the spring ktoa supporter in relative motion with respect to a stationary plane. 

The mechanical instability, jump-in and pull-off phenomenon, can also be observed in a macroscopic system, and both the trajectory and force curves exhibit similar patterns to those in Fig. 6. As a comparison, Fig. 9 shows a force curve obtained from SFA experiments of mica surface separation in diy air [8]. The pattern of the force variation, the... 

The building blocks of all materials in any phase are atoms and molecules. Their arrangements and how they interact with one another define many properties of the material. The nanotechnology MBBs, because of their sizes of a few nanometers, impart to the nanostructures created from them new and possibly preferred properties and characteristics heretofore unavailable in conventional materials and devices. These nanosize building blocks are intermediate in size, lying between atoms and microscopic and macroscopic systems. These building blocks contain a hmited and countable number of atoms. They constitute the basis of our entry into new realms of bottom-up nanotechnology [97, 98]. 

While the condition of stoichiometric neutrality invariably must hold for a macroscopic system such as a space-network polyelectrolyte gel, its application to the poly electrolyte molecule in an infinitely dilute solution may justifiably be questioned. In a polyelectrolyte gel of macroscopic size the minute excess charge is considered to occur in the surface layer (the gel being conductive), which is consistent with the assumption that the potential changes abruptly at the surface. This change is never truly abrupt, for it must take place throughout a layer extending to a depth which is of the order of magnitude of the... 

Micro reactors permit high-throughput screening of process chemistries imder controlled conditions, unlike most conventional macroscopic systems [2], In addition, extraction of kinetic parameters from sensor data is possible, as heat and mass transfer can be fully characterized due to the laminar-flow condihons applied. More uniform thermal condihons can also be utilized. Further, reactor designs can be developed in this way that have specific research and development funchons. 

The term numerical diffusion describes the effect of artificial diffusive fluxes which are induced by discretization errors. This effect becomes visible when the transport of quantities with small diffusivities [with the exact meaning of small yet to be specified in Eq. (42)] is considered. In macroscopic systems such small diffusivities are rarely found, at least when being looked at from a phenomenological point of view. The reason for the reduced importance of numerical diffusion in many macroscopic systems lies in the turbulent nature of most macro flows. The turbulent velocity fluctuations induce an effective diffusivity of comparatively large magnitude which includes transport effects due to turbulent eddies [1]. The effective diffusivity often dominates the numerical diffusivity. In contrast, micro flows are often laminar, and especially for liquid flows numerical diffusion can become the major effect limiting the accuracy of the model predictions. 

In many macroscopic systems, the massive behavior is a convoluted answer to many microscopic features of the system. For example, the catalysis of the electrooxidation of an organic molecule may be generated by some local arrangement of atoms on a catalyst, defined at the atomic level. If some hypotheses are available to explain the enhancement of the reaction, this can be checked by inserting these hypotheses in the model. In a first approximation, a qualitative explanation is often sought. If this is... 

One of the drawbacks with such a macroscopic system is the increased time for the diffusion of molecules relative to that in nanoscale systems. Molecules will clearly take longer to pass through thick barriers and to diffuse great distances than in the nanoscale regime. Therefore, the nanoencapsulation of such systems is desirable as it potentially reduces these limitations very significantly. Our attention now proceeds to various potential methods for nanoencapsulation. 

The Gibbs Ensemble MC simulation methodology [17-19] enables direct simulations of phase equilibria in fluids. A schematic diagram of the technique is shown in Fig. 10.1. Let us consider a macroscopic system with two phases coexisting at equilibrium. Gibbs ensemble simulations are performed in two separate microscopic regions, each within periodic boundary conditions (denoted by the dashed lines in Fig. 10.1). The thermodynamic requirements for phase coexistence are that each... 

Molecular calculations provide approaches to supramolecular structure and to the dynamics of self-assembly by extending atomic-molecular physics. Alternatively, the tools of finite element analysis can be used to approach the simulation of self-assembled film properties. The voxel4 size in finite element analysis needs be small compared to significant variation in structure-property relationships for self-assembled structures, this implies use of voxels of nanometer dimensions. However, the continuum constitutive relationships utilized for macroscopic-system calculations will be difficult to extend at this scale because nanostructure properties are expected to differ from microstructural properties. In addition, in structures with a high density of boundaries (such as thin multilayer films), poorly understood boundary conditions may contribute to inaccuracies. 

Now — L is the Landau-Ginzburg free energy, where m2 = a(T — Tc) near the critical temperature, is a macroscopic many-particle wave function, introduced by Bardeen-Cooper-Schrieffer, according to which an attractive force between electrons is mediated by bosonic electron pairs. At low temperature these fall into the same quantum state (Bose-Einstein condensation), and because of this, a many-particle wave function (f> may be used to describe the macroscopic system. At T > Tc, m2 > 0 and the minimum free energy is at = 0. However, when T [Pg.173]

For systems close to equilibrium the non-equilibrium behaviour of macroscopic systems is described by linear response theory, which is based on the fluctuation-dissipation theorem. This theorem defines a relationship between rates of relaxation and absorption and the correlation of fluctuations that occur spontaneously at different times in equilibrium systems. 

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