Particle fluctuations

Physically, as we go to larger masses during the A integration, the widths of the Gaussians in the kinetic-energy piece of the sampling function become very narrow. This means that the distributions of the a/., are essentially Gaussian due to no influence from the potential on the scale of the very small particle fluctuations, the... 

According to standard NMR theory, the spin-lattice relaxation is proportional to the spectral density of the relevant spin Hamiltonian fluctuations at the transition frequencies coi. The spectral density is given by the Fourier transform of the auto-correlation fimction of the single particle fluctuations. For an exponentially decaying auto-correlation function with auto-correlation time Tc, the well-known formula for the spectral density reads as ... 

The correlation function of similar particles, Xv, could be easily related to the simplest characteristics of spatial particle fluctuations, namely, dispersion of a number of particles within arbitrary volume V. The mean number of u-kind particles inside V is just... 

Furthermore, it is assumed that there is no particle fluctuation on the wall so that the boundary condition of particle turbulent kinetic energy at the wall is given by... 

The first term on the LHS of Eq. (31) represents the Reynolds stress associated with particle fluctuations, and j o is the mobility. Thus, the contribution to Z may be considered to be made up of two parts one arising from the mean square velocity fluctuations, and the other due to dispersion. 

B. Study of Particle Fluctuations by Proteolysis and Mass Spectrometry. 200... 

In this paper we will concentrate on the diffraction techniques (SANS and reflectometry), and hence static measurements. However, it should be pointed out that through inelastic scattering, aspects of polymer dynamics are accessible. In particular, it has been possible to access single chain dynamics in bulk systems, deformation and relaxation of polymer melts under shear, shed new light on viscoelasticity in polymer melts, and obtain direct information on polymer reputation and particle fluctuations. 

Following ref. 8, let us expand the total energy loss scattering cross section in a correlational series. The terms of this series represent contributions to the cross section by isolated many-particle fluctuational complexes of different ranges. Using the identical representation for /g,iv(e),... 

Now we determine 2,=<(R)> which is equal to E% R) — Eqq R), where E%(R) and oo (R) are the ground- and excited-state interatomic interaction energies of the two-particle fluctuational complexes in the adiabatic approximation. 

The transport equation for the gas-particle fluctuation covariance is given by [126] ... 

The dissipation rate of the gas-particle fluctuation covariance Cgp and the gas-particle turbulent viscosity i/ are defined by ... 

Proper boundary conditions are generally required for the primary variables like the gas and particle velocities, pressures and volume fractions at all the vessel boundaries as these model equations are elliptic. Moreover, boundary conditions for the granular temperature of the particulate phase is required for the PT, PGT and PGTDV models. For the models including gas phase turbulence, i.e., PGT and PGTDV, additional boundary conditions for the turbulent kinetic energy of the gas phase, as well as the dissipation rate of the gas phase and the gas-particle fluctuation covariance are required. The... 

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