Probabilistic equations

In addition, there is a relation between entropy and disorder disordered states have higher probabilities than ordered states. In general, the changes that are accompanied by an increase in entropy result in increased molecular disorder. Thus, entropy is also a measure of the molecular disorder of the state. Although disorder may be related to entropy qualitatively, the amount of disorder is a subjective concept and it is much better to relate entropy to probability rather than to disorder. Such concepts can be described in terms of thermodynamic probabilities (Q) in statistical mechanics. The entropy of a system is a function of the probability of the thermodynamic state of this system, S = /( 2). We know from statistical mathematics that only logarithmic functions satisfy probabilistic equations, so that we may use... 

This form may be easily assimilated to both a conservation principle (of chemical bond from adducts) and a probabilistic equation. In fact, one can interpret the chemical bond formation by the competition between the hard-soft and maximum hardness (hard-hard) bonding probabilities. The probability character is crucial to certify the quantum character of the chemical hardness involved in bonding, not only phenomenological but also analytically. Moreover, since the hard-soft term S rf) basically express the emerging HSAB principle and Y values associates with MH quantum effects, the unified chemical hardness of bonding equation may be formulated as (Putz, 2008c) ... 

The wave function T is a function of the electron and nuclear positions. As the name implies, this is the description of an electron as a wave. This is a probabilistic description of electron behavior. As such, it can describe the probability of electrons being in certain locations, but it cannot predict exactly where electrons are located. The wave function is also called a probability amplitude because it is the square of the wave function that yields probabilities. This is the only rigorously correct meaning of a wave function. In order to obtain a physically relevant solution of the Schrodinger equation, the wave function must be continuous, single-valued, normalizable, and antisymmetric with respect to the interchange of electrons. 

The price of flexibility comes in the difficulty of mathematical manipulation of such distributions. For example, the 3-parameter Weibull distribution is intractable mathematically except by numerical estimation when used in probabilistic calculations. However, it is still regarded as a most valuable distribution (Bompas-Smith, 1973). If an improved estimate for the mean and standard deviation of a set of data is the goal, it has been cited that determining the Weibull parameters and then converting to Normal parameters using suitable transformation equations is recommended (Mischke, 1989). Similar estimates for the mean and standard deviation can be found from any initial distribution type by using the equations given in Appendix IX. 

It can be seen from Table 4.3 that there is no positive or foolproof way of determining the distributional parameters useful in probabilistic design, although the linear rectification method is an efficient approach (Siddal, 1983). The choice of ranking equation can also affect the accuracy of the calculated distribution parameters using the methods described. Reference should be made to the guidance notes given in this respect. 

A popular way of determining the standard deviation for use in the probabilistic calculations is to estimate it by equation 4.21 which is based on the bilateral tolerance, t, and various empirical factors as shown in Table 4.7 (Dieter, 1986 Haugen, 1980 Smith, 1995). The factors relate to the fact that the more parts produced, the more confidence there will be in producing capable tolerances ... 

In the probabilistic design calculations, the value of Kt would be determined from the empirical models related to the nominal part dimensions, including the dimensional variation estimates from equations 4.19 or 4.20. Norton (1996) models Kt using power laws for many standard cases. Young (1989) uses fourth order polynomials. In either case, it is a relatively straightforward task to include Kt in the probabilistic model by determining the standard deviation through the variance equation. 

The formulations for the failure governing stress for most stress systems can be found in Young (1989). Using the variance equation and the parameters for the dimensional variation estimates and applied load, a statistical failure theory can be formulated for a probabilistic analysis of stress rupture. 

Using von Mises Theory from equation 4.58, the probabilistic requirement, P, to avoid yield in a ductile material, but under a biaxial stress system, is used to determine the reliability, R, as ... 

The shear stress, t, due to the assembly torque diminishes to zero with time, the preload, F, remaining constant, and so the stress on the solenoid section is only the direct stress,. v, as given in equation 4.75 (see Figure 4.41(b)) (Edwards and McKee, 1991). A second reliability can then be determined by considering the requirement that the pre-load stress remains above a minimum level to avoid loosening in service (0.5 S/)min from experiment) (Marbacher, 1999). The reliability, R, can then be determined from the probabilistic requirement, P, to avoid loosening ... 

Table 7.3 The four rules in the family of two-state/ one-absorbing-state probabilistic PCA defined in the text, along with their corresponding iterative maps (see equation 7.98).

The predicted product shown in Fig. 1.14 is different from that shown in Fig. 1.13, even though the reactants are the same. Of course it is nonsense to think that the product of a reaction between two species could depend upon the relative positions of their formulas in a representation of a reaction equation This is clear evidence that some students consider the juxtaposition of the written representations of the reactants on the page, rather than a visualisation of the reaction mixture at the sub-microscopic level. The quality of the predictions would have been greatly enhanced if the students had visualised a many-particle, probabilistic picture of the reaction mixture, as discussed earlier... 

In the first chapter several traditional types of physical models were discussed. These models rely on the physical concepts of energies and forces to guide the actions of molecules or other species, and are customarily expressed mathematically in terms of coupled sets of ordinary or partial differential equations. Most traditional models are deterministic in nature— that is, the results of simulations based on these models are completely determined by the force fields employed and the initial conditions of the simulations. In this chapter a very different approach is introduced, one in which the behaviors of the species under investigation are governed not by forces and energies, but by rules. The rules, as we shall see, can be either deterministic or probabilistic, the latter leading to important new insights and possibilities. This new approach relies on the use of cellular automata. 

K. Lindenberg, K. E. Shuler, V. Seshadri, and B. J. West, Langevin equations with multiplicative noise theory and applications to physical processes, in Probabilistic Analysis and Related Topics, Vol. 3, A. T. Bharucha-Reid (ed.), Academic Press, San Diego, 1983, pp. 81-125. 

Models of the above have been presented by various researchers of the U.S. Geological Survey (USGS) and the academia. The above equation has been solved principally (a) numerically over a temporal and spatial discretized domain, via finite difference or finite element mathematical techniques (e.g., 11) (b) analytically, by seeking exact solutions for simplified environmental conditions (e.g., 12) or (c) probabilistically (e.g., 13). 

Stochastic or probabilistic techniques can be applied to either the moisture module, or the solution of equation (3) — or for example the models of Schwartz Crowe (13) and Tang et al. (16), or can lead to new conceptual model developments as for example the work of Jury (17). Stochastic or probabilistic modeling is mainly aimed at describing breakthrough times of overall concentration threshold levels, rather than individual processes or concentrations in individual soil compartments. Coefficients or response functions and these models have to be calibrated to field data since major processes are studied via a black-box or response function approach and not individually. Other modeling concepts may be related to soil models for solid waste sites and specialized pollutant leachate issues (18). 

Thus, when the attention of the mathematicians of the time turned to the description of overdetermined systems, such as we are dealing with here, it was natural for them to seek the desired solution in terms of probabilistic descriptions. They then defined the best fitting equation for an overdetermined set of data as being the most probable equation, or, in more formal terminology, the maximum likelihood equation. 

To put equation 44-6 into a usable form under the conditions we wish to consider, we could start from any of several points of view the statistical approach of Hald (see [10], pp. 115-118), for example, which starts from fundamental probabilistic considerations and also derives confidence intervals (albeit for various special cases only) the mathematical approach (e.g., [11], pp. 550-554) or the Propagation of Uncertainties approach of Ingle and Crouch ([12], p. 548). In as much as any of these starting points will arrive at the same result when done properly, the choice of how to attack an equation such as equation 44-6 is a matter of familiarity, simplicity and to some extent, taste. 

The kernels of these integral equations, which are derived from simple probabilistic considerations, represent up to the factor 1 the product of two factors. The first of them, wa(r]), is equal to the fraction of a-th type blocks, whose lengths exceed rj. The second one, Vap(rj), is the rate with which an active center located on the end of a growing block of monomeric units M with length r) switches from a-th type to /i-lh type under the transition of this center from phase a into phase /3. The right-hand side of Eq. 74 comprises items equal to the product of the rate of initiation Ia of a-th type polymer chains and the Dirac delta function <5( ). 

Quantum mechanics is intrinsically probabilistic, but classical theory - as shown above by the existence of the delta-function limit for the classical distribution function - is not. Since Newton s equations provide an excellent description of observed classical systems, including chaotic systems, it is crucial to establish how such a localized description can arise quantum mechanically. We will call this the strong form of... 

Monte Carlo—A statistical technique commonly used to quantitatively characterize the uncertainty and variability in estimates of exposure or risk. The analysis uses statistical sampling techniques to obtain a probabilistic approximation to the solution of a mathematical equation or model. 

In Equation 3a, we take ej to be random, with zero mean and known distribution, in order to apply the probabilistic theory of hypothesis testing. Selection of the operator and the nature of ej are governed by (our perceptions of) the structure of Equation 2. Assumptions concerning and e are crucial. In the best of circumstances is linear (in the xj and B) and e is normal, independent and unbiased. Then,... 

Case II - Analyte Detection (A - assumed). Here, the analyte- rather than signal-detection limit is calculated, but the systematic error in A, applied in the estimation of x from Equation 2c imposes systematic error bounds which must be applied to the analyte detection limit. The limit is no longer purely probabilistic in nature ( ). 

The Sayre equation [12] is algebraic rather than probabilistic in origin and is derived from the expression relating the electron density and its square ... 

Equation 5.26 is the matrix form of all probabilistic linear models. For the model of Equation 5.21 this can be written... 

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