Markovian processes

Fig. 1.1. Time-dependence of the components of angular momentum J, (Markovian process) and the torque M, (white noise) in the impact approximation.

Maier-Saupe potential 275 Markovian process adiabatic interference 145 asymptotics 38 impact theory 38 noise 227... 

Knowing the functions (26) and(27) it is possible by means of the formalism of the theory of Markovian processes [53] to find any statistical characteristic in an ensemble of macromolecules with labeled units. A subsequent label erasing procedure is carried out by integration of the obtained expressions over time of the formation of monomeric units. Examples of the application of this algorithm are reported elsewhere [25]. 

V. I. Tikhonov and M. A. Mironov, Markovian Processes, Sovetskoe Radio, Moscow, 1978, in Russian. 

The Fourier transforms were performed in the standard way. No smoothing nor filtering was employed. Subtraction of the data from the least squares fit removes the constant or linear term characterizing a Markovian process. Fourier transform of the differences from the linear fit suppresses the enhancement of both the power and amplitude spectra at low frequencies. 

As an introduction to the peculiar properties of the spin Hamiltonians, we first give a short summary of the theory of spin relaxation in liquids where the problem is in fact a Brownian motion one. Then we consider the many-spin problem in solids and apply the general formalism of the theory of irreversible processes developed by Prigogine and his co-workers. We also analyse some aspects of the recent work of Caspers and Tjon on this subject. Finally, we indicate the special interest of spin-spin relaxation phenomena in connection with non-Markovian processes. 

Consider, on the other hand, a purely stochastic, Markovian process. The evolution of a dynamical function /(oa) is now determined by a transition probability / (co, r co, 0) from the state oa at time zero, to the state ca at time t > 0 This gives rise to a transformation Wt off ... 

The non-equivalence of the statistical and kinetic methods Is given by the fact that the statistical generation Is always a Markovian process yielding a Markovian distribution, e.g. In case of a blfunc-tlonal monomer the most probable or pseudo-most probable distributions. The kinetic generation Is described by deterministic differential equations. Although the Individual addition steps can be Markovian, the resulting distribution can be non-Markovian. An Initiated step polyaddltlon can be taken as an example the distribution Is determined by the memory characterized by the relative rate of the Initiation step ( ). ... 

The constant term T is, in part, by way of a normalising constant. The rapid and abrupt loss of memory about the magnitude and direction of this force is characteristic of a Markovian process. At a time f, the state of a system only depends on that at an infinitesimally short time earlier. 

Since the velocity relaxation time, m/J, is typically 0.1 ps, t is rather shorter than that estimated from the decay of the velocity autocorrelation function. As an operational convenience, rrel — mjl can be deduced from the decay time re of the velocity autocorrelation functions. However, this procedure still does not entirely adequately describe the details of Brownian motion of particles over short times. The velocity relaxes in a purely exponential manner characteristic of a Markovian process. Further comments on the reduction of the Fokker—Planck equation to the diffusion equation have been made by Harris [526] and Tituiaer [527]. 

In the previous section, the phenomenological description of Brownian motion was presented. The Langevin analysis leads to a velocity autocorrelation function which decays exponentially with time. This is characteristic of a Markovian process, as Doobs has shown (see ref. 490). Since it is known heyond question that the velocity autocorrelation function is far from such an exponential function, the effect that the solvent structure has on the progress of a chemical reaction cannot be assessed very reliably by means of phenomenological Langevin description. Since the velocity of a solute is correlated with its velocity a while before, a description which fails to consider solute and solvent velocities can hardly be satisfactory. Necessarily, the analysis requires a modification of the Langevin or Fokker—Plank description. In this section, some comments are made on this new and exciting area of research. 

Northrup and Hynes [103] solved these equations for the case of a diffusion model and found the same results as Collins and Kimball [4] of eqn. (25). This case is reasonably easy to solve because the diffusion and reaction of the pair can be separated. When the motion of the pair involves a non-Markovian process, that is the reactants recall which direction they were moving a moment before (i.e. have a memory ) and the process is not diffusional, this elegant separation becomes very difficult or impossible to effect. Under these circumstances, eqn. (368) can only be solved approximately for the pair probability. The initial condition term, l(t), is non-zero if the initial distribution p(0) is other than peq. 

Clearly, Eqs. (5) and (6) yield different results for p,(x, t) and the form for vv2 in Eq. (4) is not suitable. This is a reflection of a general property of the conditional probabilities for non-Markovian processes that we shall prove below. 

Here we will summarize some known results for the simplest example of random frequency modulation as defined by Eq. (2). Let us assume that the process 2( ) in Eq. (2) is a projection of a Markovian process characterized by the evolution operator T. This is possible in principle, because the dynamical motion of the environment can be described in terms of a Liouville operator. The set of variables defining the Markovian process is designated by X. If the variable fi itself is Markovian, X consists only of 2, but in general it has to be supplemented by additional variables to complete the set. Let the function W(x, X, t) be the probability or the probability density for finding the random variables x and X at the respective values at the time t. Then a systematic method of treating the problem, Eq. (2), is to rewrite it in the form... 

If we assume a Gaussian-Markovian process for the random field, the evolution operator in Eq. (42) becomes... 

The stochastic equation of motion of v(t), Eq. (77), can be transformed into a stochastic Liouville equation of the type Eq. (7) if a Markovian process can be properly defined to generate the process of H(t). Then we again obtain Eq. (63) for the conditional expectation V(t) defined by Eq. (60). The line shape function is then given by... 

H. Kramers Approach to Steady-State Rates of Reaction and Its Extension to Non-Markovian Processes... 

For a stationary Markovian process that is homogeneous in space, it yields... 

The transfer probability Pij (t0,t) gives the conditional probability that a given particle resident in compartment i at time ta will be in compartment j at time f. Because the particles move independently, the transfer probabilities do not depend on the number of other particles in the compartments. In this way, the (t0,t) serve to express the Markovian process. Indeed, the Markov process can be expressed in terms of the to X to transfer-intensity matrix H (/,) with (i,j)th element //,v (/,) given... 

Noteworthy is that only for the exponential distribution is the hazard rate h a) = f (a) /S (a) = k not a function of the age a, i.e., the molecule has no memory and this is the main characteristic of Markovian processes. In other words, the assumption of an exponential retention time is equivalent to the assumption of an age-independent hazard rate. One practical restriction of this model is that the transfer mechanism must not discriminate on the basis of the accrued age of a molecule in the compartment. In summary, it is clear that the formulations in the probabilistic transfer model and in the retention-time distribution model are equivalent. In the probabilistic transfer model we assume an age-independent hazard rate and derive the exponential distribution, whereas in the retention-time distribution model we assume an exponential distribution and derive an age-independent hazard rate. 

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