Fluctuating moment method

Finally, it is worth remarking that retarded van der Waals dispersion potentials between molecules in ground and excited electronic states may also be calculated [12,51] using the fluctuating moment method. Because dispersion forces arise from the perturbation induced by the zero-point energy associated with the vacuum electromagnetic field, the expectation value of... 

Thus, the turbulent-reacting-flow problem can be completely closed by assuming independence between Y and 2, and assuming simple forms for their marginal PDFs. In contrast to the conditional-moment closures discussed in Section 5.8, the presumed PDF method does account for the effect of fluctuations in the reaction-progress variable. However, the independence assumption results in conditional fluctuations that depend on f only through Tmax(f ) The conditional fluctuations thus contain no information about local events in mixture-fraction space (such as ignition or extinction) that are caused by the mixture-fraction dependence of the chemical source term. 

Surface diffusion can be studied with a wide variety of methods using both macroscopic and microscopic techniques of great diversity.98 Basically three methods can be used. One measures the time dependence of the concentration profile of diffusing atoms, one the time correlation of the concentration fluctuations, or the fluctuations of the number of diffusion atoms within a specified area, and one the mean square displacement, or the second moment, of a diffusing atom. When macroscopic techniques are used to study surface diffusion, diffusion parameters are usually derived from the rate of change of the shape of a sharply structured microscopic object, or from the rate of advancement of a sharply defined boundary of an adsorption layer, produced either by using a shadowed deposition method or by fast pulsed-laser thermal desorption of an area covered with an adsorbed species. The derived diffusion parameters really describe the overall effect of many different atomic steps, such as the formation of adatoms from kink sites, ledge sites... 

These methods have been applied to calculate the polarizabilities of atoms,31 and the long-ranged forces between atoms,33 with a typical calculated accuracy of 10 % or less. Thus, we have been able to estimate successfully the significant features of zero-point fluctuations of atomic dipole moments, without actually solving the quantum equations of motion to obtain all the excited state energies and wave functions. 

Only in rare cases is it possible to solve the master equation explicitly. For instance, we have seen that one-step master equations can be solved when the step probabilities rn and gn are constant or linear in n, but not otherwise. It is therefore important to find approximation methods, of which the Fokker-Planck approximation is the best known. Many other methods have been suggested in the literature, usually consisting of ad hoc prescriptions for cutting off higher moments of the fluctuations, and often determined by the needs or taste of the author rather than by logic. As a consequence they have led to the unreliable and contradictory results mentioned in VIII. 1. One thing they have in common, however, namely the idea that somehow the fluctuations are small. 

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