Markov process first order

If, instead, the dyad probability depends on the nature (m or r) of the preceding dyad, the distribution follows a first-order Markov process, with two independent statistical parameters and Pr , (the probability that after a m dyad a r dyad follows and vice versa, respectively). The corresponding equations are listed in Table 4, column 3. They correspond to those of a nonideal copolymerization and are reduced to the previous case when p + p = 1 ... 

The exact computation of P W) in this simple one-dipole model is already a very arduous task that, to my knowledge, has not yet been exactly solved. We can, however, consider a limiting case and try to elucidate the properties of the work (heat) distribution. Here we consider the limit of large ramping speed r, where the dipole executes just one transition from the down to the up orientation. A few of these paths are depicted in Fig. 13b. This is also called a first-order Markov process because it only includes transitions that occur in just one direction (from down to up). In this reduced and oversimplified description, a path is fully specified by the value of the field H at which the dipole reverses orientation. The work along one of these paths is given by... 

In Section V.B.l we have evaluated the path entropy s q) (Eq. (163)) for an individual dipole N = 1) in the approximation of a first-order Markov process. The following result has been obtained (Eq. (142)) ... 

They correspond to a first-order Markov process for the stereocontrol— i.e., a penultimate effect of the last diad on stereocontrol. 

More convenient and entirely sufficient for the present purpose is the calculation of ratio of rate constants. The calculation will be reviewed for a one-way first-order Markov process. A one-way mechanism is chosen because it is intuitively the most appropriate model for a free radical mechanism. Furthermore it has some experimental support. The assumption of a first-order Markov process does not rule out higher Markov processes. The differentiation between a first-order Markov process and higher order Markov processes is however possible experimentally in very rare cases because it involves the determination of tetrad, pentad, etc. fractions (11, 12, 13, 14). A Bemoullian process is ruled out by the analysis of the data of Table I. 

To analyse bond breakage under steady loading, we take advantage of the enormous gap in time scale between the ultrafast Brownian diffusion (r 10 — 10 s) and the time frame of laboratory experiments ( 10 s to min). This means that the slowly increasing force in laboratory experiments is essentially stationary on the scale of the ultrafast kinetics. Thus, dissociation rate merely becomes a function of the instantaneous force and the distribution of rupture times can be described in the limit of large statistics by a first-order (Markov) process with time-dependent rate constants. As force rises above the thermal force scale, i.e. rj-t> k T/x, the forward transition... 

A (first-order) Markov process is defined as a finite-state probability model in which only the current state and the probability of each possible state change is known. Thus, the probability of making a transition to each state of the process, and thus the trajectory of states in the future, depends only upon the current state. A Markov process can be used to model random but dependent events. Given the observations from a sequence of events (Markov chain), one can determine the probability of one element of the chain (state) being followed by another, thus constructing a stochastic model of the system being observed. For instance, a first-order Markov chain can be defined as... 

First-order Markov processes are therefore defined by two independent addition probabilities. Although the propagation steps shown above depict free radical polymerisation, the statistical models are equally applicable to other types of chain growth as found, for example, in ionic and Ziegler-Natta polymers (see section 2.3.4). 

The stochastic version of a memory-free deterministic process is a Markov process — more precisely, a first-order Markov process. It is interesting to remark that in the theory of stochastic processes the concept of history-dependent processes, had been adopted by the time the theory was established (i.e. in the mid-thirties). 

An (A, (p) dynamic system is deterministic if knowing the state of the system at one time means that the system is uniquely specified for all r 6 T. In many cases, the state of a system can be assigned to a set of values with a certain probability distribution, therefore the future behaviour of the system can be determined stochastically. Discrete time, discrete state-space (first order) Markov processes (i.e. Markov chains) are defined by the formula... 

Of particular importance are first-order Markov processes,... 

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