Einstein, definition

Although the assignment of a value to DM is arbitrary, it is by no means unrestricted. From its definition (Equation 20.5) it is clear that DM must be positive. To find the upper limit for DM, one need only substitute the Einstein definition [6] of the diffusion coefficient,... 

In the polymer literature each of the five quantities listed above is encountered frequently. Complicating things still further is the fact that a variety of concentration units are used in actual practice. In addition, lUPAC terminology is different from the common names listed above. By way of summary, Table 9.1 lists the common and lUPAC names for these quantities and their definitions. Note that when

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Substituting for the mobility using the Nemst-Einstein equation and the definition of the transport number... 

Sometimes an alternative definition of the Einstein eoefficients is employed. With this definition, equation (8) is replaced by... 

The operational approach to the definition of fundamental concepts in science has been emphasized by Mach, Poincare, and Einstein and has been expressed in a very clear form by Bridgman [2]. (Operational definitions had been used implicitly much earlier than the twentieth century. Boyle, for example, defined a chemical element in terms of the experiments by which it might be recognized, in order to avoid the futile discussions of his predecessors, who identified elements with qualities or properties.) In this approach, a concept is defined in terms of a set of experimental or mental operations used to measure or to recognize the quantity The concept is synonymous with the corresponding set of operations (Bridgman). An operational definition frequently may fail to satisfy us that we know what the concept really is. The question of scientific reality has been explored by many scientists and philosophers and is one that every student should examine. However, in the operational approach, we are not concerned with whether our definition has told us what the concept really is what we need to know is how to measure it. The operational approach has been stated succinctly by Poincare in the course of a discussion of the concept of force ... 

Whereas absorption spectra can be obtained at a given temperature via Monte-Carlo t q)e simulations, the reach of equilibrium in an excited state of an isolated cluster is less obvious, and even less is the definition of a relevant temperature. In any case, the final state may be strongly dependent on the excitation process. Here we will ignore the vibrations of the Na(3p)Arn cluster. We assume a Franck-Condon type approximation and that emission takes place from relaxed equilibrium geometry structures on the Na(3p)Arn excited PES. The Einstein coefficients of the lines of emission towards the ground state at energy AE are given by... 

We have in this way obtained a generalization of Einstein s theory of the interaction between matter and radiation including multiple photon processes and involving transition probabilities. But there is a basic difference. The operator definite positive. We no longer have a simple addition of transition probabilities. This corresponds exactly to the interference of probabilities discussed in Section IV. The process is not of the simple Chapman-Smoluchowski-Kolmogoroff type (Eq. (11)) the operator transition probability. As the result, the second of the two sequences discussed above may decrease the effect of the first one. It is very interesting that even in the limit of classical mechanics (which may be performed easily in the case of anharmonic oscillators) this interference of probabilities persists. This is in agreement with our conclusion in Section IV. 

In Chap. 1, we introduced the book with a quote from Albert Einstein (Schilpp 1949), which read in part that classical thermodynamics... is the only physical theory of universal content concerning which I am convinced that, within the framework of the applicability of its basic concepts, it will never be overthrown. An important qualification to this statement is the phrase within the framework of the applicability of its basic concepts. The laws of thermodynamics are based on laboratory-scale experiments. To assume that such laws are applicable to the Universe is a big assumption. However, we have no evidence yet that contradicts this assumption on the scales of problems relevant to life. Moreover, there remain vast cosmological questions with no answers and definitely no understanding of implications even if we knew the answers. For instance, does the proton have a very long but finite radioactive half-life Does the neutrino have a very small but finite mass Is the Universe opened or closed with respect to expansion and gravitational contraction Also, the Universe may not be isolated with respect to matter/energy or it could be isolated and cyclical. 

The major notations of scalars, vectors, and tensors and their operations presented in the text are summarized in Tables A1 through A5. Table A1 gives the basic definitions of vector and second-order tensor. Table A2 describes the basic algebraic operations with vector and second-order tensor. Tables A3 through A5 present the differential operations with scalar, vector, and tensor in Cartesian, cylindrical, and spherical coordinates, respectively. It is noted that in these tables, the product of quantities with the same subscripts, e.g., a b, represents the Einstein summation and < jj refers to the Kronecker delta. The boldface symbols represent vectors and tensors. 

Note that there are some variations in the literature about the definition of these quantities sometimes the definitions of 4> and 4 are swapped, and their sign is also sometimes different. Here we choose the convention that and 4 are equal in the absence of anisotropic stress (see below), and that is the quantity that appears in the Laplacian term of the 00 part of the Einstein equations (the general relativistic analog of the Poisson equation), thus following the Newtonian convention to note the gravitational potential by. ) It is of course possible to define other scalar gauge invariant quantities. For example one can define... 

With these definitions and the Einstein Equations of General Relativity, we can show that... 

SA = 0 subject to the energy constraint restates the principle of least action. When the external potential function is constant, the definition of ds as a path element implies that the system trajectory is a geodesic in the Riemann space defined by the mass tensor m . This anticipates the profound geometrization of dynamics introduced by Einstein in the general theory of relativity. 

The symmetry between curvature and matter is the most important result of Einstein s gravitational field equations. Both of these tensors vanish in empty euclidean space and the symmetry implies that whereas the presence of matter causes space to curve, curvature of space generates matter. This reciprocity has the important consequence that, because the stress tensor never vanishes in the real world, a non-vanishing curvature tensor must exist everywhere. The simplifying assumption of effective euclidean space-time therefore is a delusion and the simplification it effects is outweighed by the contradiction with reality. Flat space, by definition, is void. 

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