Discrete probability distributions model systems

A.4.11 Discrete probability distributions model systems with finite, or countably infinite, values, while a continuous probability distribution model systems with infinite possible values within a range. 

The simplest of these models which permits a detailed discussion of the decay of correlations is a random walk model in which a set of random walkers whose positions are initially correlated is allowed to diffuse the motion of any single random walker being independent of any other member of the set. Let us assume that there are r particles in the set and motion occurs on a discrete lattice. The state of the system is, therefore, completely specified by the probabilities Pr(nlf n2,..., nr /), (tij = — 1, 0, 1, 2,. ..) in which Pr(n t) is the joint probability that particle 1 is at n1( particle 2 is at n2, etc., at time l. We will also use the notation Nj(t) for the random variable that is the position of random walker j at time t. Reduced probability distributions can be defined in terms of the Pr(n t) by summation. We will use the notation P nh, rth,..., ntj I) to denote the distribution of random walkers iu i2,..., i at time t. We define... 

Maximum Entropy Method (MEM). MEM is based on information theory that the information content of a physical system is related to its entropy in a logarithmic formalism. The induced probability distribution of the system is the only one having the maximum entropy and corresponds to that with the minimum assumption about the system. In a PCS application, reconstruction of q(F) is based on the Shannon-Jaynes entropy model. The most probable solution ofq(F) or q(Fi) will be the one that will maximize the entropy function. In its discrete form, the function is... 

Except the kinetic equations, now various numerical techniques are used to study the dynamics of surfaces and gas-solid interface processes. The cellular automata and MC techniques are briefly discussed. Both techniques can be directly connected with the lattice-gas model, as they operate with discrete distribution of the molecules. Using the distribution functions in a kinetic theory a priori assumes the existence of the total distribution function for molecules of the whole system, while all numerical methods have to generate this function during computations. A success of such generation defines an accuracy of simulations. Also, the well-known molecular dynamics technique is used for interface study. Nevertheless this topic is omitted from our consideration as it requires an analysis of a physical background for construction of the transition probabilities. This analysis is connected with an oscillation dynamics of all species in the system that is absent in the discussed kinetic equations (Section 3). 

It should be pointed out, however, that there are at least two shortcomings to our data evaluation procedure i) The two types of H sites involved in the diffusion process must unfortunately remain unspecified until we have the necessary structural information. A corresponding neutron spectroscopic study of the local environment of interstitial hydrogen in these materials is presently in progress, ii) Due to the disorder in the host system we cannot, of course, expect two discrete jump rates in amorphous PdgQSi2QH but rather two well separated distributions of jump rates or linewidths. That our two-state-model takes into account only two distinct jump rates, is certainly a crucial simplification and most probably obscures the T dependence of the jump rates. 

The use of the embedded Discrete Time Markov Chain in a continuous stochastic process for deter-mining the events probability makes assumption that the system is in a stationary state characterizing by stationary distribution probabihties over its states. But the embedded DTMC is not limited to Continuous Time Markov Chain a DTMC can also be defined from semi-Markov or under some hypothesis from more generally stochastic processes. Another advantage to use the DTMC to obtain the events probability is that the probability of an event is not the same during the system evolution, but can depends on the state where it occurs (in other words the same event can be characterized by different occurrence probabilities). The use of the Arden lemma permits to formally determine the whole set of events sequences, without model exploring. Finally, the probability occurrence for relevant or critical events sequences and for a sublanguage is determined. 

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