Ensemble average mean square

F. 6 Time- and ensemble-averaged mean square displacements of hydrogen molecules in the Co stmcture at different temperatures and 100 % cage occupancy... 

One useful approach to the description of internal motion is to follow time dependence of the ensemble-average mean-squared displacement, Z, along the laboratory frame gradient axis as tl PGSE pulse separation time, A, is varied. The characteristic behaviour of as respectively dependent on 1/4 1/2 various time regimes represents a signature for... 

Figure 9.9 Ensemble-averaged mean squared displacement versus time for the statistical sub-units of polystyrene in 9% volume fraction semi-dilute solution with CCI4, as obtained using PGSE NMR. (o) 1.8 x 10 Da (a) 3.0 x 10 Da ( ) 15 x 10 Da. The data are compared with asymptotic lines for and scaling where t corresponds to the PGSE...

The depolarized scattering in the RGD regime from a solute with anisotropic scattering elements may be expressed as a function of the ensemble-averaged mean-square optical anisotropy per molecule (j ) [5,10, 24, 39, 84] ... 

The microcanonical ensemble is a certain model for the repetition of experiments in every repetition, the system has exactly the same energy, Wand F but otherwise there is no experimental control over its microstate. Because the microcanonical ensemble distribution depends only on the total energy, which is a constant of motion, it is time independent and mean values calculated with it are also time independent. This is as it should be for an equilibrium system. Besides the ensemble average value (il), another coimnonly used average is the most probable value, which is the value of tS(p, q) that is possessed by the largest number of systems in the ensemble. The ensemble average and the most probable value are nearly equal if the mean square fluctuation is small, i.e. if... 

The average value and root mean square fluctuations in volume Vof the T-P ensemble system can be computed from the partition fiinction Y(T, P, N) ... 

With the total number of monomers and the volume of the system fixed, a number of statistical averages can be sampled in the course of canonical ensemble averaging, like the mean squared end-to-end distance Re), gyration radius R g), bond length (/ ), and mean chain length (L). 

The Bragg scattering of X-rays by a periodic lattice in contrast to a Mossbauer transition is a collective event which is short in time as compared to the typical lattice vibration frequencies. Therefore, the mean-square displacement (x ) in the Debye-Waller factor is obtained from the average over the ensemble, whereas (r4) in the Lamb-Mossbauer factor describes a time average. The results are equivalent. 

The diffusion coefficient D is one-third of the time integral over the velocity autocorrelation function CvJJ). The second identity is the so-called Einstein relation, which relates the self-diffusion coefficient to the particle mean square displacement (i.e., the ensemble-averaged square of the distance between the particle position at time r and at time r + f). Similar relationships exist between conductivity and the current autocorrelation function, and between viscosity and the autocorrelation function of elements of the pressure tensor. 

Each average value of velocity can be used to best describe some particular property of the ensemble of molecular velocities. For example, in a gas all molecules have the same average kinetic energy. Hence, the root-mean-square velocity is the best estimate of velocity to use for computing parameters that are a function of kinetic energy... 

The mean-squared, end-to-end distance in Equation [9] is the simplest average property of interest for a polymer chain. Among other physical properties, this quantity appears in the equations of statistical mechanical theories of rubber-like elasticity. In Equation [9], the angle brackets denote the ensemble (or time) average over all possible conformations. The subscript 0 indicates that the average pertains to an unperturbed chain (theta conditions no excluded volume effects are present). (See Figure... 

Apart from trapping, there also exist situations where, as far as ensemble average ( ) is concerned, the mean square displacement does not exist. This corresponds to a jump length distribution X(x) emerging from an Levy stable density for independent identically distributed random variables of the symmetric jump length x, whose second moment diverges. The characteristic function of this Levy stable density is [14,33,34]... 

For polymers and other fluctuating objects, the square radius of gyration is usually averaged over the ensemble of allowed conformations giving the mean-square radius of gyration ... 

It is evident from the above argument that the temporal variation of the mean-squared displacement of the reactants determines the asymptotic decay rates. For instance, if P (where the notation )) is used to denote ensemble averages over the different particles of a reactant species), then the concentrations decay as no t . Therefore it behooves us to determine the exponent 5 for diffusion in the fluctuating potential field. 

A method for calculating observables resulting from incoherent excitation transport among chromophores randomly tagged in low concentration on isolated, flexible polymer chains is described. The theory relates the ensemble average root-mean-square radius of gyration ) of a polymer coil to the rate... 

Due to the sensitivity of electronic excitation transport to the separation and orientation of chromophores, techniques which monitor the rate of excitation transport among chromophores on polymer chains are direct probes of the ensemble average conformation (S). It is straightforward to understand qualitatively the relationship between excitation transport dynamics and the size of an isolated polymer coil which is randomly tagged in low concentration with chromophores. An ensemble of tagged coils in a polymer blend will have some ensemble averaged root-mean-squared radius of gyration,... 

However, the fully extended conformation is only one of a great many it would be more meaningful to consider an average size of the macromolecule such as the mean square end-to-end distance, r2. As the name implies, the end-to-end distance is just the length of the vector connecting the two ends of the ideal chain. This average can be that for a given molecule at a number of times or that of an ensemble of identical molecules at the same time.5 Thus, for p chains that do not interact with one another,... 

Although this random migration does not change the average position of the particle ensemble, it does tend to spread the particles over the axis. The extent of spread can be determined by examining the mean square displacement of the particle ensemble, (x (n)) ... 

Figure 20. Center-of-mass fluctuations of N2 on graphite with respect to the nearest hexagon center (registry point) near the melting transition obtained from Monte Carlo simulations. Left-hand scales root-mean-square fluctuations perpendicular (circles) and parallel (triangles) to the surface plane. Right-hand scales average center-of-mass position above the surface plane (squares). Fluctuations are obtained (a) in the NPT ensemble and (ft) NVT ensemble at coverage unity, both with deformable periodic boundary conditions. (Adapted from Figs. 8 and 14 of Ref. 301.)...

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