Initial probability density

If the initial probability density W(xq. to) is known and the transition probability density W(x, t xo, to) has been obtained, then one can easily get the one-dimensional probability density at arbitrary instant of time ... 

Let continuous one-dimensional Markov process x(f) at initial instant of time t = 0 have a fixed value x(0) = xo within the interval (c, d) that is, the initial probability density is the delta function ... 

Relaxation of initial probability density functions in the turbulent convection of scalar fields. The Physics of Fluids 22, 20-30. 

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... 

These results clearly show that the time development of the system is entirely determined by the transition probability densities f(xk,tkl xk x,tk x) and the initial probability density of states /(x, tx). 

Considering ifo R) as the initial probability density, using the reflection method and setting yo = yg - ZPE, the Absorption Cross Section (XS) can be written ... 

As shown in Figure 3, the excitation energy specific, "delta function or "monoenergetic, unimolecular rate constants (ks) are strongly energy dependent. Hie magnitude of ks diminishes rapidly as the excess internal energy of the approaches Eo. The initial probability density... 

Another important observation results from the fact that, for the initial probability density of the Dirac distribution form... 

These updated models are based on measurements obtained during a period in operation of 15 years. Designating the initial probability density of the floater offset by f ( o) and the initial probability density for the internal pressure by, p ixip), the corresponding updated probability densities can be obtained by... 

We also notice that although the scattering amplitudes depend on 0a, the differential cross sections do not. The reason is that the initial probability density is cylindrically symmetrical around the quantization axis, and therefore so must the final one be, in the absence of external fields. In addition, since m m(O) = m m and d, tt) = we conclude from (5.69)... 

Averaging the formal solution for p(t) given by Eq. (270) over the initial probability density for the particle plus bath, assuming the bath is at thermal equilibrium, we find that the average values of the Cartesian components of the initial average momentum p 0)) decay exponentially to zero with the same relaxation time t =. Thus, we think of the inverse of the friction coefficient f as providing a measure of the time scale for momentum relaxation. 

Figure 12.45 shows some of the most interesting and important results of die transient calculation. The initial probability density as seen for the t = 0.0 curve is a Gaussian type distribution centered at x = -.5. The wavepacket moves with time... 

Figure A3.13.10. Time-dependent probability density of the isolated CH clnomophore in CHF. Initially, tlie system is in a Fenni mode with six quanta of stretching and zero of bending motion. The evolution occurs within the multiplet with chromophore quantum number A = 6 = A + 1 = 7). Representations are given...

In addition to initial conditions, solutions to the Schrodinger equation must obey eertain other eonstraints in form. They must be eontinuous funetions of all of their spatial eoordinates and must be single valued these properties allow T T to be interpreted as a probability density (i.e., the probability of finding a partiele at some position ean not be multivalued nor ean it be jerky or diseontinuous). The derivative of the wavefunetion must also be eontinuous exeept at points where the potential funetion undergoes an infinite jump (e.g., at the wall of an infinitely high and steep potential barrier). This eondition relates to the faet that the momentum must be eontinuous exeept at infinitely steep potential barriers where the momentum undergoes a sudden reversal. 

Human actions can initiate accident sequences or cause failures, or conversely rectify or mitigate an accident sequence once initiated. The current methodology lacks nuclear-plant-based data, an experience base for human factors probability density functions, and a knowledge of how this distribution changes under stress. 

Several colloidal systems, that are of practical importance, contain spherically symmetric particles the size of which changes continuously. Polydisperse fluid mixtures can be described by a continuous probability density of one or more particle attributes, such as particle size. Thus, they may be viewed as containing an infinite number of components. It has been several decades since the introduction of polydispersity as a model for molecular mixtures [73], but only recently has it received widespread attention [74-82]. Initially, work was concentrated on nearly monodisperse mixtures and the polydispersity was accounted for by the construction of perturbation expansions with a pure, monodispersive, component as the reference fluid [77,80]. Subsequently, Kofke and Glandt [79] have obtained the equation of state using a theory based on the distinction of particular species in a polydispersive mixture, not by their intermolecular potentials but by a specific form of the distribution of their chemical potentials. Quite recently, Lado [81,82] has generalized the usual OZ equation to the case of a polydispersive mixture. Recently, the latter theory has been also extended to the case of polydisperse quenched-annealed mixtures [83,84]. As this approach has not been reviewed previously, we shall consider it in some detail. 

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. 

Formula (2.2) contains only one-dimensional probability density W(xi, t ) and the conditional probability density. The conditional probability density of Markov process is also called the transition probability density because the present state comprehensively determines the probabilities of next transitions. Characteristic property of Markov process is that the initial one-dimensional probability density and the transition probability density completely determine Markov random process. Therefore, in the following we will often call different temporal characteristics of Markov processes the transition times, implying that these characteristics primarily describe change of the evolution of the Markov process from one state to another one. 

For the solution of real tasks, depending on the concrete setup of the problem, either the forward or the backward Kolmogorov equation may be used. If the one-dimensional probability density with known initial distribution deserves needs to be determined, then it is natural to use the forward Kolmogorov equation. Contrariwise, if it is necessary to calculate the distribution of the mean first passage time as a function of initial state xo, then one should use the backward Kolmogorov equation. Let us now focus at the time on Eq. (2.6) as much widely used than (2.7) and discuss boundary conditions and methods of solution of this equation. 

The solution of Eq. (2.6) for infinite interval and delta-shaped initial distribution (2.8) is called the fundamental solution of Cauchy problem. If the initial value of the Markov process is not fixed, but distributed with the probability density Wo(x), then this probability density should be taken as the initial condition ... 

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