Reduced probability density

Following [33] closely, we now take the time evolution equation for the reduced probability density in just as the average over q obtained by computing the expectation value with respect to 0) that is... 

Let, for example, Ri denote the reaction coordinate of interest in a multi-dimensional system. The corresponding reduced probability density is defined as... 

In the case of a reduced density-matrix description of the system dynamics, the reduced probability density of coordinate Ri is given by... 

Since the coiTelation between opposite spins has both intra- and inter-orbital contributions, it will be larger than the correlation between electrons having the same spin. The Pauli principle (or equivalently the antisymmetry of the wave function) has the consequence that there is no intraorbital conelation from electron pairs with the same spin. The opposite spin correlation is sometimes called the Coulomb correlation, while the same spin correlation is called the Fermi correlation, i.e. the Coulomb correlation is the largest contribution. Another way of looking at electron correlation is in terms of the electron density. In the immediate vicinity of an electron, here is a reduced probability of finding another electron. For electrons of opposite spin, this is often referred to as the Coulomb hole, the corresponding phenomenon for electrons of the same spin is the Fermi hole. 

It is often important to be able to extend our present notion of conditional probability to the case where the conditioning event has probability zero. An example of such a situation arises when we observe a time function X and ask the question, given that the value of X at some instant is x, what is the probability that the value of X r seconds in the future will be in the interval [a,6] As long as the first order probability density of X does not have a Dirac delta function at point x, P X(t) = x = 0 and our present definition of conditional probability is inapplicable. (The reader should verify that the definition, Eq. (3-159), reduces to the indeterminate form in this case.)... 

Two electrons occupy the in-phase combined orbital. The probability density inaeases in the overlap region. Two more electrons occupy the out-of-phase combined orbital and reduce the density there. The decrease is greater than the increase. The electrons are expelled from the overlap region. 

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, ... 

Figure 4.7 Assumed probability density function for the degree of melting F (top). Resulting probability density functions for the reduced solid concentration of element i upon fractional melting (middle) and batch melting (bottom) for different solid-liquid partition coefficients D,.

In SIMCA the distribution of the object in the inner model space is not considered, so the probability density in the inner space is constant and the overall PD appears as shown in Figs. 29, 30 for the enlarged and reduced SIMCA models. In CLASSY, Kernel estimation is used to compute the PD in the inner model space, whereas the errors in the outer space are considered, as in SIMCA, uncorrelated and with normal multivariate distribution, so that the overall distribution, in the inner and outer space of a one-dimensional model, looks like that reported in Fig. 31. Figures 32, 33 show the PD of the bivariate normal distribution and Kernel distribution (ALLOC) for the same data matrix as used for Fig. 31. Although in the data set of French wines no really important differences have been detected between SIMCA (enlarged model), ALLOC and CLASSY, it seems that CLASSY should be chosen when the number of objects is large and the distribution on the components of the inner model space is very different from a rectangular distribution. 

Fig. 30. Probability density function for SIMCA (reduced model)...

So, MCBA builds a covariance matrix of the residuals around the inner model and from this matrix it obtains a probability density function as bayesian analysis does, taking into account that the dimensionality of the inner space correspondingly reduces the rank of the covariance matrix from which a minor must be extracted. 

In the special case that only one X corresponds to each Y (and hence necessarily r = s one may invert (5.1) to give X = g(Y). In that case the transformation of the probability density reduces to... 

Example. As a model for two-stage diffusion take i = 1,2 and F as in (7.1). Then 7i,2 = 2 and 72,1 =7i- F°r computing the cross-section for neutron scattering one needs to know the probability density Gs(r, t) that a molecule that, at t = 0, was at r = 0 will, at time t, be at r. The differential cross-section is its double Fourier transform GS(A , co). It is convenient to apply the Fourier transformation in space right away to (7.4) so that both operators Ff reduce to factors,... 

On the other hand, random errors do not show any regular dependence on experimental conditions, since they are generated by many small and uncontrolled causes acting at the same time, and can be reduced but not completely eliminated. Thus, random errors are observed when the same measurement is repeatedly performed. In the simplest case, the universe of random errors is described by a continuous random variable e following a normal distribution with zero mean, i.e., for a univariate variable, the probability density function is given by... 

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