Markovian processes limits

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. 

The above relation is valid also for (3=1. Indeed, in that limit, we have i(u) = e /T, whence /(f) = 8(f — x). This sharp distribution of the waiting time is but one possible definition of a Markovian process. In the remainder of this review, we solely focus on processes with (3=1. 

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... 

We can also relate these two approximations through the stochastic theory of the line shape developed by Kubo [11] and applied to molecular line shapes by Saven and Skinner [10]. As shown by Kubo, the overlap function given by Eq. (13) is a general result for a Gaussian-dis-tributed random variable in a Markovian process [11]. In the limit of a very slow decay of the time-correlation function of this random variable, the overlap function reduces to Eq. (9) and the line has a Lorentzian shape. In the limit of a very fast decay of the time-correlation function, the overlap function reduces to Eq. (15) and the line has a Gaussian shape. Employing molecidar dynamics simulations of chromophores within non-polar fluids. 

Section III is devoted to Prigogine s theory.14 We write down the general non-Markovian master equation. This expression is non-instantaneous because it takes account of the variation of the velocity distribution function during one collision process. Such a description does not exist in the theories of Bogolubov,8 Choh and Uhlenbeck,6 and Cohen.8 We then present two special forms of this general master equation. On the one hand, when one is far from the initial instant the Variation of the distribution functions becomes slower and slower and, in the long-time limit, the non-Markovian master equation reduces to the Markovian generalized Boltzmann equation. On the other hand, the transport coefficients are always calculated in situations which are... 

To calculate the drift velocity and diffusivity for such a process, we imagine a discrete sequence of coordinates values X = X (t ) sampled at times ti, t2, ri, separated by At, and then take the limit At 0. Such a sequence could be generated by integrating Eq. (2.244) over each timestep to obtain the change AX for that step. Such a sequence will be approximately Markovian... 

T = J0°° w(t)tdt —> oo, manifesting the self-similar nature of this waiting process that has also prompted the coinage of fractal time processes [48]. Note that in the limit a —> 1, this waiting time pdf reduces to the singular form Lf (t/x) — 5(t - x) with finite T = % that leads back to the temporally local Markovian formulation of classical Brownian transport. In fact, for any waiting time pdf with a finite characteristic time T, one recovers the Brownian picture, such as for the Poissonian form w(t) = x xe t x. 

They may be obtained by means of the matrix IET but only together with the kernel E(f) = F(t) specified by its Laplace transformation (3.244), which is concentration-independent. However, from the more general point of view, Eqs. (3.707) are an implementation of the memory function formalism in chemical kinetics. The form of these equations shows the essentially non-Markovian character of the reversible reactions in solution the kernel holds the memory effect, and the convolution integrals entail the prehistoric evolution of the process. In the framework of ordinary chemical kinetics S(/j = d(t), so that the system (3.707) acquires the purely differential form. In fact, this is possible only in the limit when the reaction is entirely under kinetic control. 

Recent simulations by Straub et al. show that the non-Markovian theory of Section V may break down for very large friction and very large correlation times of the thermal bath. This failure is due to the fact that in such extreme (and unphysical) limits of the parameters, the well motion may become again the rate-limiting step in the process, in contrast to the theoretical assumption. - ... 

The Kramers theory and its extensions have found many applications since the original work by Kramers. Recent application of the non-Markovian theory in the low-friction limit to thermal desorption was described by Nitzan and Carmeli. Another novel application of the Markovian theory is to transition from a nonequilibrium state of a Josephson junction. In what follows we shall briefly review the recent application of the generalized Kramers theory to chemical rate processes. More detailed reviews of the exjjerimental and theoretical status of this field may be found in Hynes. ... 

While Markovian stochastic processes play important role in modeling molecular dynamics in condensed phases, their applicability is limited to processes that involve relatively slow degrees of freedom. Most intramolecular degrees of freedom are characterized by timescales that are comparable or faster than characteristic environmental times, so that the inequality (7.53) often does not hold. Another class of stochastic processes that are amenable to analytic descriptions also in non-Markovian situations is discussed next. 

The Marcus theory, as described above, is a transition state theory (TST, see Section 14.3) by which the rate of an electron transfer process (in both the adiabatic and nonadiabatic limits) is assumed to be determined by the probability to reach a subset of solvent configurations defined by a certain value of the reaction coordinate. The rate expressions (16.50) for adiabatic, and (16.59) or (16.51) for nonadiabatic electron transfer were obtained by making the TST assumptions that (1) the probability to reach transition state configuration(s) is thermal, and (2) once the reaction coordinate reaches its transition state value, the electron transfer reaction proceeds to completion. Both assumptions rely on the supposition that the overall reaction is slow relative to the thermal relaxation of the nuclear environment. We have seen in Sections 14.4.2 and 14.4.4 that the breakdown of this picture leads to dynamic solvent effects, that in the Markovian limit can be characterized by a friction coefficient y The rate is proportional to y in the low friction, y 0, limit where assumption (1) breaks down, and varies like y when y oo and assumption (2) does. What stands in common to these situations is that in these opposing limits the solvent affects dynamically the reaction rate. Solvent effects in TST appear only through its effect on the free energy surface of the reactant subspace. 

The connection between the stochastic cell model and the usual deterministic model of reaction-diffusion systems were given by Kurtz for the homogeneous case in the same spirit. To give the relationship among a Markovian jump process and the solution p(r, i) of the deterministic model a law of large numbers and a central limit theorem hold (Arnold Theodoso-pulu, 1980 Arnold 1980). 

Equation [52] is also a Markovian stochastic process with zero mean and variance Af. The quantity X"(0,-Af) is correlated with X " (0,Af) through a bivariate Gaussian distribution. In the zero limit of the friction coefficient, this set of equations corresponds to the trajectories obtained with the Verlet algorithm. ... 

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