Macroscopic average value

Molecules are in continuous random motion, and as a result of this, small volume elements within the liquid continuously experience compression or rarefaction such that the local density deviates from the macroscopic average value. If we represent by 6p the difference in density between one such domain and the average, then it is apparent that, averaged over all such fluctuations, 6p = 0 Equal contributions of positive and negative 6 s occur. However, if we consider the average value of 6p, this quantity has a nonzero value. Of these domains of density fluctuation, the following statements can be made ... 

We define the concentration of fluctuation domains at any instant by the symbol N. In addition, we assume that the polarizability associated with one of these domains differs from the macroscopic average value for the substance... 

The extraction of a homogeneous process from a stationary Markov process is a familiar procedure in the theory of linear response. As an example take a sample of a paramagnetic material placed in a constant external magnetic field B. The magnetization Y in the direction of the field is a stationary stochastic process with a macroscopic average value and small fluctuations around it. For the moment we assume that it is a Markov process. The function Px (y) is given by the canonical distribution... 

According to the assumptions on which classical kinetics is based, deterministic models are adequate as long as deviations from the macroscopic average values remain negligible. A number of situations can be listed to argue for relevance of fluctuations in chemical sytems ... 

As already addressed in section 2.3.3, an inherent problem that is faced when studying bimolecular termination kinetics is the troublesome relationship between the microscopic and macroscopic kinetics. Microscopic (chain-length dependent) termination rate coefficients cannot readily be transformed into macroscopic (average) values and vice versa. Many... 

In the self-consistent model the matrix material outside the inclusion is assumed as possessing the effective macroscopic properties of the composite. Moreover, two consecutive problems were solved by assuming either phase of the composite as occupying its position and surrounded by this average material. In both cases the average values of the composite are determined from the values of the characteristic... 

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. 

It should be possible to distinguish between eigenvalues of different energy shells, but not between those of the same shell, by a macroscopic measurement that corresponds to a diagonal matrix whose elements for all ij)n of a shell [AE]K are equal to an average value E. The associated operator H is defined with the same accuracy with which the measurement can be carried out. 

In most problems involving boundary conditions, the boundary is assigned a specific empirical or deterministic behavior, such as the no-slip case or an empirically determined slip value. The condition is defined based on an averaged value that assumes a mean flow profile. This is convenient and simple for a macroscopic system, where random fluctuations in the interfacial properties are small enough so as to produce little noise in the system. However, random fluctuations in the interfacial conditions of microscopic systems may not be so simple to average out, due to the size of the fluctuations with respect to the size of the signal itself. To address this problem, we consider the use of stochastic boundary conditions that account for random fluctuations and focus on the statistical variability of the system. Also, this may allow for better predictions of interfacial properties and boundary conditions. 

The best fit is obtained by a search, which iterates through a sequence of trials to minimize the error between the calculated overall deuterium shift, AD, and the experimentally measured shift by varying the unknown parameters (/ , Do, AD ). The average data (at least two runs) are used for the curve fitting to give mean values for the unknown parameters (yS , AD , Do). The macroscopic JC values are calculated from yS values. Finally, a resampling statistical analysis is used to evaluate the precision for each parameter in the search. 

The key point of these methods is the concept of self-averaging in accordance with this concept, the observable value of an additive function, for example - of the free energy, F, of macroscopic sample of spin glass, practically coincides with the averaged value over the ensemble of disorder realizations ... 

Figure 2 The dependence of O2 binding on Hb concentration. Binding curves are shown (solid black lines) for Hb concentrations of 0.005, 0.04, 0.10, 0.27,1.0, 5.4, and 38 pM (from left to right). Theoretical binding curves (broken red lines) are shown for a pure tetramer solution and a pure dimer solution. The macroscopic, thermodynamic linkage scheme relates the dimer tetramer assembly constants to the O2 binding constants for free dimer and assembled tetramer. The brackets around figurines indicate that the O2 ligand may be bound at any one of the available deoxy hemesites. Thus, the macroscopic constants are average values for multiple microscopic processes.

The microcanonical analog of the macroscopic phases is quite often defined by means of the energy dependence of Lindemann index. This quantity is designed to detect the stiffness of a molecule by measuring the deviation of the bond lengths from their averaged values as... 

The measurements presented here illustrate impressively that it is essential to consider the effect of substrate heterogeneities on passive film growth. Measurements of passive films on macroscopic surfaces can, in principle, not be quantitatively understood in terms of classical models without taking texture effects into account, as macroscopic measurements can yield only averaged values. 

The macroscopic behavior of physical systems is determined by the microscopic behavior of these systems. Usually the microscopic fluctuations are averaged, and on larger scales the averaged values satisfy the classical equations. 

Considering a dilute gas containing n molecules per unit volume inducing a macroscopic transfer flux of property tp. The symbol tp denotes any property of a single molecule that can be changed by collisions, and ( 0)m represents the average value of tp for the gas. The molecules are assumed to move in a... 

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