Non-Markovian process

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. 

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. 

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

Dekker has studied multiplicative stochastic processes. In his work the stochastic Liouville equation was solved explicitly through first order in an expansion in terms of correlation times of the multiplicative Gaussian colored noise for a general multidimensional weakly non-Markovian process. He followed the suggestions of refs. 17 and 18 and applied, Novikov s theorem. In the general multidimensional case, however, he improved the earlier work by San Miguel and Sancho. ... 

Equations (5.7) were introduced so as to treat the non-Markovian process of Eq. (5.8) in the frame of the time-independent Fokker-Planck formalism. The equivalence has been shown to require that the fluctuation-dissipation relationship (5.10) holds the white noise limit can then be recovered by making t vanish for a fixed value of D. If we substitute Eq. (5.10) into Eqs. 

Since Eq. (49) takes into account only the term of order Dt, the term of order in Eq. (51) is meaningless and the term linear in t in vanishes exactly. For T = 0, our result equals the well-known Smoluchowski rate. The main conclusion we can draw is that the activation rates for non-Markovian processes like Eq. (44) decrease as t increases the exact result of ref. 44 can thus be extended to the case of Gaussian random forces of finite correlation time as well. However, if we take Eq. (50) seriously, we obtain an Arrhenius factor, exp(A /Z)), of T(x) which does not exhibit a dependence on T. This is in contrast to the result found for telegr hic noises, where the Arrhenius factor increases with increasing autocorrelation time r (see ref. 44). The result of a numerical simulation for J(x) based on the bi-... 

Equation (19) describes a non-Markovian process in the CV space. In fact, the forces acting on the CVs depend explicitly on their history. Due to this non-Markovian nature, it is not clear if, and in which sense, the system can reach a stationary state under the action of this dynamics. In [32] we introduced a formalism that allows to map this history-dependent evolution into a Markovian process in the original variable and in an auxiliary field that keeps track of the visited configurations. Defining... 

T (62, 63, 64), and internal viscosity is negligible. Thus, the absence of internal viscosity at solution pHs 6.0, 6.5, and 7.0 and its presence at 7.5, 8.0, and 8.5 is due to the adiabatic elimination of internal viscosity at the lower pHs, by which slowly varying degrees of freedom are exhausted (65). Whereas intrinsic viscosity is essentially low frequency viscosity based on Markovian processes, internal viscosity is essentially high frequency viscosity based on non-Markovian processes with memory (65, 66). Internal viscosity is correlated with the magnitude... 

Perhaps, it was Hynes who initiated two of the most popular so far semi classical non-Markovian approximations [84]. The first approximation was inspired by the success of the [1,0]-Pade approximant, which turns out to be exact in the Markovian limit. This approximation is sometimes referred to as the substitution approximation, because effectively one substitutes non-Markovian two-point distribution function (9.46)-(9.47) into the Markovian expressions (9.50)-(9.51) for the rate kernel. The substitution approximation was shown to work rather well for the case of biexponential relaxation with similar decay times [102]. However, as Bicout and Szabo [142] recently demonstrated, it considerably overestimates the reaction rate when the two relaxation timescales become largely different (see Fig. 9.14). They also showed that for a non-Markovian process with a multiexponential correlation function, which can be mapped onto a multidimensional Markovian process [301], the substitution approximation is equivalent to the well-known Wilemski-Fixman closure approximation [302-304]. A more serious problem arises when we try to deal with the... 

It turns out that the inclusion of the kinetic term F p) is not a trivial procedure for non-Markovian processes. In particular, the transport term might depend on the reaction kinetics. We discuss this problem later in detail, see Sect. 3.4. 

As p 0", we have v oo, and as p oo, we find that v grows with p. In consequence, a minimum velocity exists, but no maximum velocity. In the previous section we considered the cases when the microscopic transport processes are described by Markovian random walks. The great advantage of the Hamilton-Jacobi formulation of the front propagation problem in general, and the formulas (4.46) in particular, is that they allow us to study quite complicated transport operators for the evolution of the scalar field p and the underlying random walk model, including non-Markovian processes [118,121,125]. 

In literature, various approaches are used to deal with non-Markovian processes ... 

Since white noise is qualified by a single parameter, namely by its intensity, the characterisation of real noise requires at least another parameter, its correlation time. The theoretical treatment of nonlinear systems subject to real, i.e. coloured , external noise has two newer difficulties. First, only the white noise idealisation leads to a Markov process. Second, from the practical point of view the Gaussian and Poissonian distributions are relevant only to describe white noise, and real noise can have a richer description. The disadvantage due to the loss of Markov property is partially compensated by the fact that non-Markovian processes have smoother realisations than Markov processes. Therefore no particular (Ito or Strato-... 

In addition to destroying the isotropy of space, the introduction of the SCF field implies a Markovian (albeit self-consistent) approximation to the inherently non-Markovian process described by (6.16). Thus, if EjiCK... 

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