Ensemble average mean square displacement

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 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. 

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]... 

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. 

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)) ... 

Fig. 2 Mean square displacement of individual microspheres dispersed in solution of glycerol 85% (A), Sterocoll FD 1% (B) and Sterocoll D 1% (C). The white curve is the ensemble-average MSD...

Equilibrium is a state of matter that results from spatial uniformity. In contrast, when there are concentration differences or gradients, particles will flow. In these cases, the rate of flow is proportional to the gradient. The proportionality constant between the flow rate and the gradient is a transport property for particle flow, this property is the diffusion constant. Diffusion can be modelled at the microscopic level as a random flight of the particle. The diffusion constant describes the mean square displacement of a particle per unit time. The fluctuation-dissipation theorem describes how transport properties are related to the ensemble-averaged fluctuations of the system in equilibrium. 

We note that parameters of this phenomenological description can be inferred from experimental observations. The drift, k, is directly given by the ensemble-averaged growth curves. From time series growth data, the mean squared displacement of x with time can be computed to confirm that the behavior is diffusive, and to read off the diffusion strength, B. The scale invariance of the FPT, when external parameters are tuned, provides an additional check on whether these two parameters are then found to be related to each other as predicted. 

Dynamic propensity has been introduced to study structural influences on the heterogeneous dynamics of supercooled liquids (Rodriguez Fris et al, 2009 Widmer-Cooper Harrowell, 2007 Widmer-Cooper et al, 2004). Dynamic propensity of the ion i, denoted by p, is defined as the mean squared displacement of the ion i for t, which is averaged over the trajectories starting from a given initial configuration with different initial momenta, i.e., the isoconfigurational (IC) ensemble. 

---