Pure death process

Just as in the unimolecular cases, the basis for the stochastic approach is to consider the reaction 2A-> B as being a pure death process with a continuous time parameter and transition probabilities for the elementary events that make up the reaction process. Letting the random variable X(t) be the number of A molecules in the system at time t, the stochastic model is then completely defined by the following assumptions ... 

Observe that the steps only go in one direction, as in radioactive decay ( pure death process ). As a consequence it is, in principle, possible to obtain... 

N(t) can be interpreted as the size of the population at time t. The most simple kind of birth-death process is that for which the rates of birth and death are time independent. A birth-death process N(t) is called a pure birth process (respectively pure death process) if Dm = 0 (Bm = 0) for any m. 

The pure death process is exactly analogous to the pure birth process, except that the pure death process X(t) is decreased rather than increased by the occurrence of an event. 

Clearly, in the thermal explosion case [ ] the underlying stochastic process is a pure death process, since the concentration of X can only decrease. We can therefore write the following Master equation for the probability P(X,t) ... 

For the pure linear death process, the forward Kolmogorov equation [see Eq. (4.57)] is... 

In the following, we derive the Kolmogorov differential equation on the basis of a simple model and report its various versions. In principle, this equation gives the rate at which a certain state is occupied by the system at a certain time. This equation is of a fundamental importance to obtain models discrete in space and continuous in time. The models, later discussed, are the Poisson Process, the Pure Birth Process, the Polya Process, the Simple Death Process and the Birth-and-Death Process. In section 2.1-3 this equation, i.e. Eq.2-30, has been derived for Markov chains discrete in space and time. 

To illustrate this problem a pure breakage process is considered, i.e., Dq = Be = 0, while Db 7 0 and Bb 7 0. In addition, as an example only, a generalized formulation of the breakage death term Db) is analyzed in the following. Assuming that the breakage frequency, 6b(d), can be represented... 

Deep-bed filtration involves the flow of particles through randomly distributed passages thus, it tends to be stochastic in nature. The filtration process has been modeled as a pure birth process [12,13], a birth-death process [14-17], a random-walk process [18], and a stochastic diffusion process [19]. 

In Litwiniszyn s pure birth model, the entire bed is considered as on state, and the number of blocked pores in a unit volume of the bed is considered as a random variable. In his birth-death model, the number of trapped particles over the entire bed is a random variable. Fan and his co-workers have extended the pure birth and birth-death process models by incorporating... 

In this section pure birth-death processes will be discussed. Although such models have been dealt with extensively in the literature (see, for instance [4.1-8]), there still exist some seldom considered problems referring to the relation between the exact stationary or quasi-stationary solution of the master equation and the deterministic approach. This relation will be treated in Sect. 4.3.1 including an appUcation of the results obtained to observations on animal populations [4.15,17]. Further, the generalization of the stochastic standard model by including multistep birth or death processes will be investigated in Sect. 4.3.2. 

DETAILS - A crude extract of hemlock was used by the ancient Greeks to execute enemies of the state. Plato s description of the death of Socrates is a very detailed account of the symptoms and their progression. Pure coniine has a burning taste similar to low-grade hot sauce and a characteristic odor which should be masked. Hard liquor or pungent spicy foods are best for this. Coniine poisoning is a relatively painless process, but takes several hours and the victim is conscious and aware to the very end. 

Arnaldian treatises, such as the On the Secrets of Nature, compare the great work of the alchemists to the life and death ofjesus. The author says that mercury, likejesus, is to be taken and beaten and scourged lest by reason of pride he perish. Elaborating on this theme, he continues Therefore take the son [i.e., mercury] after he has been beaten and put him to bed to enjoy himself for a while, and when you feel that he is enjoying himself, then take him pure and extinguish in cold water. And when you have repeated this process, hand him over to the Jews to be crucified. 121... 

In 1923, at age 48, he consolidated his knowledge of thermodynamics in the great classic Thermodynamics and the Free Energy of Chemical Substances with Merle Randal as coauthor. After that his interests changed to other subjects. He was the first to prepare pure deuterium and D2O (1933), he formulated his generalized definitions of acids and bases (Lewis acids and bases, 1938), and at the time of his death he was doing research on photochemical processes. 

In technical safety, risk is a function of accident frequency and accident consequences. Pure quantified risk assessment (QRA, also called probabilistic risk assessment or PRA) typically uses mortality (i.e., the number of assessed deaths in any given potential accident situation) as a measure of consequences. The overall risk for a variety of potential accidents at a particular process plant can be plotted against recognized risk criteria in a graph of accident frequency against mortality, to help make judgments about whether the plant is acceptably safe or not. 

---