Completely random process

Because coalescence is a completely random process, the total number of coalescence events per unit time is assumed to be proportional to the total surface area A of the droplets ... 

Since the random-walk approach is successful in molecular diffusion (K5) and Brownian motion studies (C14), it would seem that it might also be useful for the dispersion process. This has been considered by Baron (B2), Ranz (Rl), Reran (B5), Scheidegger (S6), Latinen (L4) and more recently by de Josselin de Jong (D14) and Salfman (SI, S2, S3). The latter two did not strictly use random-walk since a completely random process was not assumed. Methods based on statistical mechanics have been proposed by Evans et al. (E7), Prager (P8), and Scheidegger (S7). 

The energy of an electronically excited state may be lost in a variety of ways. A radiative decay is a process in which a molecule discards its excitation energy as a photon. A more common fate is non-radiative decay, in which the excess energy is transferred into the vibration, rotation, and translation of the surrounding molecules. This thermal degradation converts the excitation energy into thermal motion of the environment (i.e., to heat). Two radiative processes are possible spontaneous emission, just like radioactivity, which is a completely random process where the excited state decays ... 

Exercise. The trivial exception to Doob s theorem mentioned above is the completely random process defined by... 

In an acid process the slurry, containing about 35-45% starch solids, is pumped into a pressure vessel called a converter and acidified to a pH of about 2.0 with dilute hydrochloric acid at 140-160°C and a pressure of 80 psi (5.4 atm). Although acid-catalyzed hydrolysis is a rather (but not completely) random process,19,20 carefully controlled hydrolysis produces syrups in the 25 to 45 DE range with very predictable carbohydrate profiles as shown in Table 21.2.21... 

The important point is that the determination of the crystallization conditions for various polymorphic forms need not be a completely random process. The combination of keen thoughtful observation with consideration of all the available crystal structures and thermodynamic information can provide extremely useful guidelines, if not for success, then at the very least for further experiments. Crystallization is almost never a sure-fire procedure, especially when one is trying to selectively produce a particular polymorph, and one that has proven consistently or suddenly elusive. 

Radioactive decay is a completely random process. Thus, although no prediction can be made concerning the lifetime of an individual nucleus, the behavior of a large ensemble of identical nuclei can be described by the first-order rate expression... 

Because this type of polymerization is a completely random process, with all molecules having equal probability of reacting, the distribution of molecular weights corresponds to the most probable, or binomial, distribution, which is related to the extent of polymerization as follows (Flory, 1953d) ... 

The white noise idealisation is motivated by the assumption that the environmental state varies on a much faster time than the macroscopic state of the system. In consequence of this assumption the notion of white noise, i.e. a completely random process characterised by zero correlation time has been adopted. 

The simplest random process is completely stochastic so that one may write, for example, Pjivih yih) = d [(y h)P y2h)- However, here we are concerned with a slightly more complex process known as the Markov process, characterized by... 

The important point we wish to re-emphasize here is that a random process is specified or defined by giving the values of certain averages such as a distribution function. This is completely different from the way in which a time function is specified i.e., by giving the value the time function assumes at various instants or by giving a differential equation and boundary conditions the time function must satisfy, etc. The theory of random processes enables us to calculate certain averages in terms of other averages (known from measurements or by some indirect means), just as, for example, network theory enables us to calculate the output of a network as a function of time from a knowledge of its input as a function of time. In either case some information external to the theory must be known or at least assumed to exist before the theory can be put to use. 

There is one further point that is worth mentioning in connection with the random variable concept. We have repeatedly stressed the fact that the theory of random processes is primarily concerned with averages of time functions and not with their detailed structure. The same comment applies to random variables. The distribution function of a random variable (or perhaps some other less complete information about averages) is the quantity of interest not its functional form. The functional form of the random variable is only of interest insofar as it enables us to derive its distribution function from the known distribution function of the underlying time function X(t). It is the relationship between averages of various time functions that is of interest and not the detailed relationship between the time functions themselves. 

A random process can be (and often is) defined in terms of the random variable terminology introduced in Section 3.8. We include this alternate definition for completeness. Limiting ourselves to a single time function X( ), it is seen that X(t) is completely specified as a random process by the specification all possible finite-order joint distribution functions of the infinite set of random variables T, — oo < t < oo, defined by the equations... 

The complete specification of a random process requires us to have some way of writing down an infinite number of distribution functions. For practical reasons, this is an impossible task unless all the distribution functions can be specified by means of a rule that enables one to calculate any distribution function of interest in terms of a finite amount of prespecified information. The following examples will illustrate these ideas by showing howr some particular stochastic processes of interest can be defined. 

Equation (3-325), along with the fact that Y(t) has zero mean and is gaussian, completely specifies Y(t) as a random process. Detailed expressions for the characteristic function of the finite order distributions of Y(t) can be calculated by means of Eq. (3-271). A straightforward, although somewhat tedious, calculation of the characteristic function of the finite-order distributions of the gaussian Markov process defined by Eq. (3-218) now shows that these two processes are in fact identical, thus proving our assertion. 

Coordinating centers that prefer this approach use it because first, they feel they can ensure the validity of the subject-consenting process before randomizing, and second, they feel they can exert full control over the randomization processes to ensure their validity. This is achieved by requiring participating sites to fax signed informed consent forms to the coordinating center to enable review for completeness and validity before randomization. 

Obviously, the nucleation is a randon process which is amplified by subsequent deposition of many thousands of silver atoms before the surface is completely covered (if integrated over the time interval of monolayer formation the current in each pulse corresponds to an identical charge). Such an amplification of random processes is the only way they can be observed. This situation is quite analogous, for example, to radioactive decay where a single disintegration is followed, in a Geiger tube, by the flow of millions of electrons. ... 

Most kinetic treatments of the photo-oxidation of solid polymers and their stabilization are based on the tacit assumption that the system behaves in the same way as a fluid liquid. Inherent in this approach is the assumption of a completely random distribution of all species such as free radicals, additives and oxidation products. In all cases this assumption may be erroneous and has important consequences which can explain inhibition by the relatively slow radical scavenging processes (reactions 7 and 9) discussed in the previous section. 

If we will consider arbitrary random process, then for this process the conditional probability density W xn,tn x, t, ... x i,f i) depends on x1 X2,..., x . This leads to definite temporal connexity of the process, to existence of strong aftereffect, and, finally, to more precise reflection of peculiarities of real smooth processes. However, mathematical analysis of such processes becomes significantly sophisticated, up to complete impossibility of their deep and detailed analysis. Because of this reason, some tradeoff models of random processes are of interest, which are simple in analysis and at the same time correctly and satisfactory describe real processes. Such processes, having wide dissemination and recognition, are Markov processes. Markov process is a mathematical idealization. It utilizes the assumption that noise affecting the system is white (i.e., has constant spectrum for all frequencies). Real processes may be substituted by a Markov process when the spectrum of real noise is much wider than all characteristic frequencies of the system. 

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. 

Consider two liquid substances that are rather similar, such as benzene and toluene or water and ethylene glycol. When moles of the one are mixed with B moles of the other, the composition of the liquid mixture is given by specification of the mole fraction of one of them [e.g., Xa, according to Eq. (2.2)]. The energy or heat of the mutual interactions between the molecules of the components is similar to that of their self interactions, because of the similarity of the two liquids, and the molecules of A and B are distributed completely randomly in the mixture. In such mixtures, the entropy of mixing, which is a measure of the change in the molecular disorder of the system caused by the process of mixing the specified quantities of A and B, attains its maximal value ... 

Essentially, any optimisation process requires good judgement. This is basically because the errors which are incurred during measurement are not completely random. There are systematic errors which can arise from, for example, faulty calibration or unsuitability of a particular measurement technique. It is worth, therefore, considering a question and answer session which succeeded a paper presented by one of die present authors at the Royal Society (Saunders 1995) ... 

In Sections 2.2 and 2.3 we considered the application of response surface methodology to the investigation of the robustness of a product or process to environmental variation. The response surface designs discussed in those sections are appropriate if all of the experimental runs can be conducted independently so that the experiment is completely randomized. This section will consider the application of an alternative class of experimental designs, called split-plot designs, to the study of robustness to environmental variation. A characteristic of these designs is that, unlike the response surface designs, there is restricted randomization of the experiment. 

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