Ensemble averaged velocity

The ensemble averaged velocity for each voxel is then evaluated by... 

If all possible particle-particle approaches are considered, the ensemble average velocities after the collision will be equal to G = G = Constant, because all directions of reflection or rebound are assumed to occur with equal likelihood in kinetic theory of dilute gases [77]. By ensemble averaging, we may write ... 

By assuming that an ensemble average velocity equals a Maxwellian average velocity, we might introduce (2.374) into (2.368). In this way we get ... 

Noting that the spectral density of the ensemble-averaged velocity auto-correlation function is the diffusion tensor... 

F. 5.2 Ensemble average velocity field in grey scale within an aqueous plug at mixture velocity of 0.01 m s (flow rate ratio equal to 1) in a channel of a 0.5 nun ID, and b 1 mm ID. Thirty instantaneous fields were averaged. TBP/[C4mim] [NTf2] (30 %, v/v) as carrier fluid... 

Once position versus time data are obtained and stored on a PC hard disk, time differentiation of displacements yields local instantaneous velocities. Ensemble averaged velocity, rms velocity and other turbulence quantities are computed after acquiring the data for a sufficient length of time (e.g. several hours) at fixed operating conditions. 

The initial energy - E XoA t), VoA(t)) - is a function of the coordinates and the velocities. In principle, the use of momenta (instead of velocities) is more precise, however, we are using only Cartesian coordinates, making the two interchangeable. We need to sample many paths to compute ensemble averages. Perhaps the most direct observable that can be computed (and measured experimentally) is the state conditional probability - P A B,t) defined below ... 

The most useful of such properties would appear to be an ensemble-averaged momentum and density (their ratio being velocity) or elastic modulus, weighting the likelihood... 

TFL is essentially a transition lubrication regime between EHL and boundary lubrication. A new postulation based on the ordered model and ensemble average (rather than bulk average) was put forward to describe viscosity in the nanoscale gap. In TFL, EHL theories cannot be applied because of the large discrepancies between theoretical outcomes and experimental data. The effective viscosity model can be applied efficiently to such a condition. In thin him lubrication, the relation between Him thickness and velocity or viscosity accords no longer with an exponential one. The studies presented in this chapter show that it is feasible to use a modi-Hed continuous scheme for describing lubrication characteristics in TFL. 

In the complete Eulerian description of multiphase flows, the dispersed phase may well be conceived as a second continuous phase that interpenetrates the real continuous phase, the carrier phase this approach is often referred to as two-fluid formulation. The resulting simultaneous presence of two continua is taken into account by their respective volume fractions. All other variables such as velocities need to be averaged, in some way, in proportion to their presence various techniques have been proposed to that purpose leading, however, to different formulations of the continuum equations. The method of ensemble averaging (based on a statistical average of individual realizations) is now generally accepted as most appropriate. 

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. 

The Cartesian drift velocity V ( ) may also be defined as the ensemble average... 

On multiplying eqn. (284) by u0, integrating over time, and taking the ensemble average, the velocity autocorrelation function is obtained [271]. 

By taking the Laplace transform of eqn. (290) and then multiplying throughout by the initial velocity u(0) and finally taking an ensemble average, the Laplace transformed velocity autocorrelation function is [490]... 

In a classical Bohr orbit, the electron makes a complete journey in 0.15 fs. In reactions, the chemical transformation involves the separation of nuclei at velocities much slower than that of the electron. For a velocity 105 cm/s and a distance change of 10 8 cm (1 A), the time scale is 100 fs. This is a key concept in the ability of femtochemistry to expose the elementary motions as they actually occur. The classical picture has been verified by quantum calculations. Furthermore, as the deBroglie wavelength is on the atomic scale, we can speak of the coherent motion of a single-molecule trajectory and not of an ensemble-averaged phenomenon. Unlike kinetics, studies of dynamics require such coherence, a concept we have been involved with for some time. 

The important point to note here is that the 2nd moment of Ky(t) depends on the 2nd and 4th moments of y(t). The 2nd moments of each of the three previously mentioned autocorrelation functions may be calculated from ensemble averages of appropriate functions of the positions, velocities, and accelerations created in the dynamics calculations. Likewise, the 4th moment of the dipolar autocorrelation function may also be calculated in this manner. However the 4th moments of the velocity and angular momentum correlation functions depend on the derivative with respect to time of the force and torque acting on a molecule and, hence, cannot be evaluated directly from the primary dynamics information. Therefore, these moments must be calculated in another manner before Eq. (B.3) may be used. 

The quadruplet contributions to q (0) vanishes because rjW(0) contains the ensemble averages of the scalar products of velocities of four different particles. 

Let us see how such an equation is solved. First we must define the random function F(t) quantitatively. The average of F(t) over an ensemble of Brownian particles vanishes. This condition ensures that the average velocity of the Brownian particle obeys the macroscopic law (Eq. (11.4)), that is, that the fluctuations cancel each other on average. This is written as follows ... 

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