Flux probability density

The selective flux maximization from the FOIST scheme shown in Fig. 2 is achieved by altering the spatial profile of the initial state to be subjected to the photolysis pulse and since changes in flux are due to the flow of probability density, it is useful to examine the attributes of the probability density profiles from the field optimized initial states. 

The probability density for the ground vibrational level peaks at 4.66 a.u., that for at 4.8 a.u. and for at 4.58 a.u. The optimal wave-function seems to maximize flux out of channel 2 (I -I- Br ) by localizing the probability density to the left of that given by the (po wavefiinction and... 

The dependence of T on energy for charged-particle decay can be found as follows. The decay rate is the probability density flux integrated over surface area at infinity, i.e. 

The conceptual difference between the flux and drift velocity may thus be illustrated further by considering a dilute colloidal suspension in thermal equilibrium in the presence of a gravitational field. In equilibrium, this system adopts a single-particle probability density /(R) oc with... 

One indication is given in the case of a macroscopically isolated system, i.e., one in which all the basic macroscopic fluxes of Section II are zero across the boundary of V. Then the total expected mass, momentum, and energy are constant. On the other hand, any initial probability density in lN consistent with these assumptions will evolve, during any time interval, in a manner which is deterministic for each given w(X, t)—and equally deterministic but different, for another vv (x, t), etc. In each case we are led to the functions etc., each of which corresponds to the... 

The rate constant is the flux normalized by the probability density in the initial well, that is,... 

For multichannel scattering where there are two or more open channels, the S matrix is a true matrix with elements Sy and the cross section for the transition from channel i to channel j is proportional to 5y - Sy 2. The symmetry of collision processes with respect to the time reversal leads to the symmetric property of the S matrix, ST = S, which, in turn, leads to the principle of detailed balance between mutually reverse processes. The conservation of the flux of probability density for a real potential and a real energy requires that SSf = SfS = I, i.e., S is unitary. For a complex energy or for a complex potential, in general, the flux is not conserved and S is non-unitary. 

Conversely, a coherent superposition of continuum states with a population closely reproducing an isolated peak in the density of states, which corresponds to a resonance, can be built in such a way to give rise to a localized state. From this localized state, there will be an outward probability density flux, i.e., it will have a finite lifetime. In the limit of a resonance position far from any ionization threshold and a narrow energy width, the decay rate will be exponential with the rate constant T/ft. The decay is to all the available open channels, in proportion to their partial widths. 

Fig. 7 also shows a temporal decrease in mass flux in graduated steps. The bound from one step to another is due to the passage of one layer to another in the numerical model and consequently to the decrement of the probability density function of the lift force. This is not directly representative of the reality. However, a general trend of the temporal decrease... 

The pavement modelling allows to introduce into the model the temporal evolution of the size distribution of materials at the bed surface. By a progressive decrease of the probability density function of the lift force, this model successfully predicts the temporal decrease in mass flux that occurs with the presence of coarse particles at the surface. The rate of this decrease depends on the flow velocity and the characteristics of the particles. In order to improve the accuracy of the estimation of fugitive particle emissions with a wide size distribution, it is necessary to take into account this temporal decrease. 

Although the shortest way to the tunneling gap 8 is the solution of Landau and Lifshits [27], here we consider the problem from a different perspective. Like in the theory of electric circuits, instead of a detailed consideration of each particle, one can apply some simple rules that provide enough equations to solve the problem. One is the junction rule. It is based upon the probability conservation law for a stationary state, PiQ, t). At any point Q in the domain of 77(2, t), the probability density, I PiQ, t) 2 remains constant, dl P(Q. f)P/df = 0. Consider the part of a vibronic state that is located in a potential well. In this region, the probability density, P(Q, t) 2, looks like an octopus with its tentacles extended into the restricted areas under the barriers.2 If we construct a closed surface S around the body of the octopus , then, due to conservation of probability density, the total flux of probability through the surface S must be equal to zero,... 

It depends on the position at the center x = xt, it is simply the speed times the probability density, jx = [p t / m ] (2 7r (A x)2) 1 /2. Note that the dimension of the flux density for one-dimensional motion is speed times an inverse distance, i.e., inverse time (for three-dimensional motion, the dimension of j is speed/volume). 

C Speed of light F Probability density function F Probability distribution function I Intensity ijk Cell indices q" Heat flux vector M Cell index... 

Equations (8.10)—(8.13) are merely Stokes equations rewritten in a suggestive form chosen to emphasize transport of the momentum tracer density pv, as well as to exploit the analogy between Eq. (8.10) for momentum transport and the comparable equation (Brenner, 1980b) for transport of the scalar probability density P, which is equivalent to the material tracer density. The absence of a convective term vp from the flux expression in Eq. (8.12)... 

By far, the most widely employed models for reactive flow processes are based on Reynolds-averaged Navier Stokes (RANS) equations. As discussed earlier in Chapter 3, Reynolds averaging decomposes the instantaneous value of any variable into a mean and fluctuating component. In addition to the closure equations described in Chapter 3, for reactive processes, closure of the time-averaged scalar field equations requires models for (1) scalar flux, (2) scalar variance, (3) dissipation of scalar variance, and (4) reaction rate. Details of these equations are described in the following section. Broadly, any closure approach can be classified either as a phenomenological, non-PDF (probability density function) or as a PDF-based approach. These are also discussed in detail in the following section. 

In this chapter, we have shown why the recent transition path theory (TPT) offers the correct probabilistic framework to understand the mechanism by which rare events occur by analyzing the statistical properties of the reactive trajectories involved in these events. The main results of TPT are the probability density of reactive trajectories and the probability current (and associated streamlines) of reactive trajectories, which also allows one to compute the probability flux of these trajectories and the rate of the reaction. It was also shown that TPT is a constructive theory under the assumption that the reaction channels are local, TPT naturally leads to algorithms that allow to identify these channels in practice and compute the various quantities that TPT offers. 

The flux of a beam of particles is the product of their velocity and density, v p, and the probability density is ... 

Time-evolution could conveniently be described in terms of the probability density p r, t) = F(r,0l2 and flux... 

Nusselt number pressure (Pa) capillary pressure (Pa) probability density function Legendre polynomial Peclet number Prandtl number heat flux (W/m2)... 

In forced convective systems, the bubble departure diameter can be critically affected by the presence of a velocity field. Studies of bubble departure diameters in forced convection include those of Al-Hayes and Winterton [66], Winterton [67], Kandlikar et al. [68], and Klausner et al. [69]. The results obtained by Klausner et al. for the probability density function of departure diameter as a function of mass flux and heat flux, respectively, are shown in Figs. 15.28 and 15.29. The most probable departure diameter decreases strongly with increasing flow rate and decreases (less strongly) with decreasing heat flux at a fixed flow rate. It is clear that these velocity effects have to be taken into account in predicting forced convective boiling systems. 

FIGURE 15.28 Probability density function of bubble departure diameter in flow boiling as a function of mass flux for a constant heat flux (from Klausner et al. [69], with permission of Elsevier Science). 

Since the survival probability may be ditficult to measure, some decay analyses discuss other quantities, such as the nonescape probability from a region of space [57, 58], the probability density at chosen points of space [25, 59, 60], the flux [61-63], and the arrival time [64]. For initially localized wave packets, there is no major discrepancy between survival probability and the nonescape probability [3, 57, 59, 65-67]. Examination of densities, fluxes, or arrival time distributions may be interesting since a new variable is introduced (we shall see later some applications), but at the price of losing the simplicity and directness of the survival probability. 

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