Probability density terms

When providing input for the STOMP calculation a range of values of porosity (and all of the other input parameters) should be provided, based on the measured data and estimates of how the parameters may vary away from the control points. The uncertainty associated with each parameter may be expressed in terms of a probability density function, and these may be combined to create a probability density function for STOMP. 

In addition to operators eorresponding to eaeh physieally measurable quantity, quantum meehanies deseribes the state of the system in terms of a wavefunetion F that is a funetion of the eoordinates qj and of time t. The funetion F(qj,t)p = gives the probability density for observing the eoordinates at the values qj at time t. For a many-partiele system sueh as the H2O moleeule, the wavefunetion depends on many eoordinates. For the H2O example, it depends on the x, y, and z (or r,0, and ([)) eoordinates of the ten... 

The shape of the Normal distribution is shown in Figure 3 for an arbitrary mean, /i= 150 and varying standard deviation, ct. Notice it is symmetrical about the mean and that the area under each curve is equal representing a probability of one. The equation which describes the shape of a Normal distribution is called the Probability Density Function (PDF) and is usually represented by the term f x), or the function of A , where A is the variable of interest or variate. 

The basic requirement is that the time variation of Eq. (10) should vanish in the thermodynamic equilibrium. A sufficient condition for this is the principle of the detailed balance, which means that each term in the sum on the right-hand side of Eq. (10) vanishes for an equilibrium probability density t)eq... 

The function px derived from Fx is called a first order probability density function. The reason for this terminology can be appreciated by noting the form that Eqs. (3-10), (3-12), and (3-13) assume when they are written in terms of px instead of Fx... 

The properties of joint distribution functions can be stated most easily in terms of their associated probability density functions. The n + mth order joint probability density function px. . , ( > ) is defined by the equation... 

This conditional probability can also be written in terms of the joint probability density function for lt , n+m as follows ... 

In terms of these functions, we define all possible joint probability density functions for a time function X(t) by writing... 

In other words, if we assume that the counting function N(t) has statistically independent increments (Eq. (3-237)), and has the property that the probability of a single jump occurring in a small interval of length h is approximately nh but the probability of more than one jump is zero to within terms of order h, (Eq. (3-238)), then it can be shown 51 that its probability density functions must be given by Eq. (3-231). It is the existence of theorems of this type that accounts for the great... 

Our first objective is going to be the determination of the finite order probability density function of Y(t) in terms of the known finite order probability densities for the increments of N(t). In preparation for this, we first note that, since N(s) — N(t) is Poisson distributed with parameter.n(s — t) for s > t, it follows that... 

The form of the scalar product in terms of Schrddinger amplitudes indicates that if we want to introduce into the theory a probability density it is x(x,f)la which must play this role. In fact, the reason for introducing the Schrddinger amplitude stems precisely from the circumstances that in terms of the latter the scalar product takes the ample form (9-95), and as a consequence of this jY (x.QPd3 may directly be interpreted as the probability of finding the particle within the volume V at (hue K... 

Joint distribution functions, in terms of associated probability density functions, 133 notation, 143... 

The distribution of stretches can be quantified in terms of the probability density function F (X)=dN(X)/dX, where dN(A) is the number of points that have values of stretching between A and (X+dX) at the end of period n. Another possibility is to focus on the distribution of log A. In this case we define the measure //n(logA)=dA(logA)/d(logA). 

But what does this probability density mean in physical terms The electron should not be thought of as intrinsically smeared out as originally conceived by Schrodinger. In the Born... 

The multimedia model present in the 2 FUN tool was developed based on an extensive comparison and evaluation of some of the previously discussed multimedia models, such as CalTOX, Simplebox, XtraFOOD, etc. The multimedia model comprises several environmental modules, i.e. air, fresh water, soil/ground water, several crops and animal (cow and milk). It is used to simulate chemical distribution in the environmental modules, taking into account the manifold links between them. The PBPK models were developed to simulate the body burden of toxic chemicals throughout the entire human lifespan, integrating the evolution of the physiology and anatomy from childhood to advanced age. That model is based on a detailed description of the body anatomy and includes a substantial number of tissue compartments to enable detailed analysis of toxicokinetics for diverse chemicals that induce multiple effects in different target tissues. The key input parameters used in both models were given in the form of probability density function (PDF) to allow for the exhaustive probabilistic analysis and sensitivity analysis in terms of simulation outcomes [71]. 

The key input parameters used in the 2 FUN model were given in the form of probability density function (PDF) to allow the exhaustive probabilistic analysis and sensitivity analysis in terms of simulation outcomes. 

The term 11 (0) 2 is the square of the absolute value of the wavefunction for the unpaired electron, evaluated at the nucleus (r = 0). Now it should be recalled that only s orbitals have a finite probability density at the nucleus whereas, p, d, or higher orbitals have nodes at the nucleus. This hyperfine term is isotropic because the s wavefunctions are spherically symmetric, and the interaction is evaluated at a point in space. 

The term scar was introduced by Heller in his seminal paper (Heller, 1984), to describe the localization of quantum probability density of certain individual eigenfunctions of classical chaotic systems along unstable periodic orbits (PO), and he constructed a theory of scars based on wave packet propagation (Heller, 1991). Another important contribution to this theory is due to Bogomolny (Bogomolny, 1988), who derived an explicit expression for the smoothed probability density over small ranges of space and energy... 

In Section 8.4.2, we considered the problem of the reduced dynamics from a standard DFT approach, i.e., in terms of single-particle wave functions from which the (single-particle) probability density is obtained. However, one could also use an alternative description which arises from the field of decoherence. Here, in order to extract useful information about the system of interest, one usually computes its associated reduced density matrix by tracing the total density matrix p, (the subscript t here indicates time-dependence), over the environment degrees of freedom. In the configuration representation and for an environment constituted by N particles, the system reduced density matrix is obtained after integrating pt = T) (( over the 3N environment degrees of freedom, rk Nk, ... 

A theoretical framework based on the one-point, one-time joint probability density function (PDF) is developed. It is shown that all commonly employed models for turbulent reacting flows can be formulated in terms of the joint PDF of the chemical species and enthalpy. Models based on direct closures for the chemical source term as well as transported PDF methods, are covered in detail. An introduction to the theory of turbulence and turbulent scalar transport is provided for completeness. 

In order to compare various reacting-flow models, it is necessary to present them all in the same conceptual framework. In this book, a statistical approach based on the one-point, one-time joint probability density function (PDF) has been chosen as the common theoretical framework. A similar approach can be taken to describe turbulent flows (Pope 2000). This choice was made due to the fact that nearly all CFD models currently in use for turbulent reacting flows can be expressed in terms of quantities derived from a joint PDF (e.g., low-order moments, conditional moments, conditional PDF, etc.). Ample introductory material on PDF methods is provided for readers unfamiliar with the subject area. Additional discussion on the application of PDF methods in turbulence can be found in Pope (2000). Some previous exposure to engineering statistics or elementary probability theory should suffice for understanding most of the material presented in this book. 

The temporal luminescence of a highly heterogeneous sensor-carrier mixtures cannot be uniquely represented by sums of exponentials (Eq. (9.23)) due to the lack of orthogonality of the exponential function. In this case it becomes appropriate to express equations (9.17) or (9.23) in terms of probability density functions or lifetime distribution functions 5t(8 14)... 

---