Markovian diffusion

Assuming Markovian diffusion on a Bravais single-crystalline lattice, the intermediate scattering function is given by... 

Straub J E and Berne B J 1986 Energy diffusion in many-dimensionai Markovian systems the oonsequenoes of oompetition between inter- and intramoieouiar vibrationai energy transfer J. Chem. Phys. 85 2999-3006... 

Straub J E and Berne B J 1986 Energy diffusion in many dimensional Markovian systems the consequences of the competition between inter- and intra-molecular vibrational energy transfer J. Chem. Phys. 85 2999 Straub J E, Borkovec M and Berne B J 1987 Numerical simulation of rate constants for a two degree of freedom system in the weak collision limit J. Chem. Phys. 86 4296... 

Keilson-Storer kernel 17-19 Fourier transform 18 Gaussian distribution 18 impact theory 102. /-diffusion model 199 non-adiabatic relaxation 19-23 parameter T 22, 48 Q-branch band shape 116-22 Keilson-Storer model definition of kernel 201 general kinetic equation 118 one-dimensional 15 weak collision limit 108 kinetic equations 128 appendix 273-4 Markovian simplification 96 Kubo, spectral narrowing 152... 

In order to examine the nature of the friction coefficient it is useful to consider the various time, space, and mass scales that are important for the dynamics of a B particle. Two important parameters that determine the nature of the Brownian motion are rm = (m/M) /2, that depends on the ratio of the bath and B particle masses, and rp = p/(3M/4ttct3), the ratio of the fluid mass density to the mass density of the B particle. The characteristic time scale for B particle momentum decay is xB = Af/ , from which the characteristic length lB = (kBT/M)i lxB can be defined. In derivations of Langevin descriptions, variations of length scales large compared to microscopic length but small compared to iB are considered. The simplest Markovian behavior is obtained when both rm << 1 and rp 1, while non-Markovian descriptions of the dynamics are needed when rm << 1 and rp > 1 [47]. The other important times in the problem are xv = ct2/v, the time it takes momentum to diffuse over the B particle radius ct, and Tp = cr/Df, the time it takes the B particle to diffuse over its radius. 

Mixing models based on the CD model have discrete jumps in the composition vector, and thus cannot be represented by a diffusion process (i.e., in terms of and B ). Instead, they require a generalization of the theory of Markovian random processes that encompasses jump processes136 (Gardiner 1990). The corresponding governing equation... 

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... 

In this chapter, the motion of solute and solvent molecules is considered in rather more detail. Previously, it has been emphasised that this motion approximates to diffusion only over times which are long compared with the velocity relaxation time (see Chap. 8, Sect. 2.1). At times comparable with or a little longer than the velocity relaxation time, the diffusion equation does not provide a satisfactory description of molecular motion. An alternative approach must be sought. This introduces considerable complications to a theoretical analysis of very fast reactions in solution. To develop an understanding of chemical reactions occurring over very short time intervals, several points need to be discussed. Which reactions might be of interest and over what time scale What is known of the molecular motion of solute and solvent molecules Why does the Markovian (hydrodynamic) continuum analysis fail and what needs to be done to develop a better theory These points will be considered in further detail in this chapter. 

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]. 

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. 

So far we studied the first passage of Markov processes such as described by the Smoluchowski equation (1.9). On a finer time scale, diffusion is described by the Kramers equation (VIII.7.4) for the joint probability of the position X and the velocity V. One may still ask for the time at which X reaches for the first time a given value R, but X by itself is not Markovian. That causes two complications, which make it necessary to specify the first-passage problem in more detail than for diffusion. 

In this Chapter we introduce a stochastic ansatz which can be used to model systems with surface reactions. These systems may include mono-and bimolecular steps, like particle adsorption, desorption, reaction and diffusion. We take advantage of the Markovian behaviour of these systems using master equations for their description. The resulting infinite set of equations is truncated at a certain level in a small lattice region we solve the exact lattice equations and connect their solution to continuous functions which represent the behaviour of the system for large distances from a reference point. The stochastic ansatz is used to model different surface reaction systems, such as the oxidation of CO molecules on a metal (Pt) surface, or the formation of NH3. 

In this Section we introduce a stochastic alternative model for surface reactions. As an application we will focus on the formation of NH3 which is described below, equations (9.1.72) to (9.1.76). It is expected that these stochastic systems are well-suited for the description via master equations using the Markovian behaviour of the systems under study. In such a representation an infinite set of master equations for the distribution functions describing the state of the surface and of pairs of surface sites (and so on) arises. As it was told earlier, this set cannot be solved analytically and must be truncated at a certain level. The resulting equations can be solved exactly in a small region and can be connected to a mean-field solution for large distances from a reference point. This procedure is well-suited for the description of surface reaction systems which includes such elementary steps as adsorption, diffusion, reaction and desorption.The numerical part needs only a very small amount of computer time compared to MC or CA simulations. 

Let us study now a stochastic model for the particular a+ib2 -> 0 reaction with energetic interactions between the particles. The system includes adsorption, desorption, reaction and diffusion steps which depend on energetic interactions. The temporal evolution of the system is described by master equations using the Markovian behaviour of the system. We study the system behaviour at different values for the energetic parameters and at varying diffusion and desorption rates. The location and the character of the phase transition points will be discussed in detail. 

In order to complete the above analysis, one needs to solve the full non-Markovian Langevin equation (NMLE) with the frequency-dependent friction for highly viscous liquids to obtain the rate. This requires extensive numerical solution because now the barrier crossing dynamics and the diffusion cannot be treated separately. However, one may still write phenomenologically the rate as [172],... 

The second part (sections H and I) is devoted to a detailed discussion of the dynamics of unimolecular reactions in the presence and the absence of a potential barrier. Section H presents a critical examination of the Kramers approach. It is stressed that the expressions of the reaction rates in the low-, intermediate-, and high-friction limits are subjected to restrictive conditions, namely, the high barrier case and the quasi-stationary regime. The dynamics related to one-dimensional diffusion in a bistable potential is analyzed, and the exactness of the time dependence of the reaction rate is emphasized. The essential results of the non-Markovian theory extending the Kramers conclusions are also discussed. The final section investigates in detail the time evolution of an unimolecular reaction in the absence of a potential barrier. The formal treatment makes evident a two-time-scale description of the dynamics. 

The first non-Markovian approach to chemical reactions in solutions, developed by Smoluchowski [1], was designed for contact irreversible reactions controlled by diffusion. Contrary to conventional (Markovian) chemical kinetics in the Smoluchowskii theory, the reaction constant of the bimolecular reaction, k(t), becomes a time-dependent quantity instead of being tmly constant. This feature was preserved in the Collins-Kimball extension of the contact theory, valid not only for diffusional but for kinetic reactions as well [2]. 

In the kinetic limit (ko 3> ko) there is no difference between the initial, ko, and final, k ko, values of the time dependent rate constant that is, k(t) const and the theory becomes Markovian. In the opposite limit (ko << ko) the reaction is controlled by diffusion and the final value of the rate constant k = ko is less than the initial one. Hence, the time dependence of k(t) is well pronounced and at long times takes the following well-known form [56] ... 

The modified rate equation (MRE) approximation [177] was also constructed to describe the non-Markovian character of diffusion-controlled reversible reactions. The forward reaction rate k(t) is the same as in contact DET, Eq. (3.21), but with ka substituted for ko. As for the backward reaction rate, it was modified to be proportional to k(t) ... 

According to the Smoluchowski theory of diffusion-controlled bimolecular reactions in solutions, this constant decreases with time from its kinetic value, k0 to a stationary (Markovian) value, which is kD under diffusional control. In the contact approximation it is given by Eq. (3.21), but for remote annihilation with the rate Wrr(r) its behavior is qualitatively the same as far as k(t) is defined by Eq. (3.34)... 

Hence the Markovian distribution is common for all non-Markovian ones but only in the limit X[> —> oo. This distribution completely ignores the nonstationary annihilation and therefore does not depend on the exciton concentration and lifetime. The difference between /o(r) and fm(r) becomes more pronounced when xD is reduced (Fig. 3.99). Under diffusion control both of them have a well-pronounced maximum near the effective ionization radius. However, the nearcontact contribution of the nonstationary annihilation increases with shortening xD, on account of the main maximum. Finally (as xD > 0), the UT distribution tends to become exponential, as W/(r), while the Markovian one remains unchanged. 

Figure 9.8 Structured Markovian model. Diffusion is expressed by means of h+, h, and compartments 1 to is. Erlang-type elimination is represented by means of ho and compartments is to m. The drug is given in compartment is and cleared from compartment m.

In the case where the correlation function <3> (f) has the form of Eq. (148), with p fitting the condition 2 < p < 3, the generalized diffusion equation is irreducibly non-Markovian, thereby precluding any procedure to establish a Markov condition, which would be foreign to its nature. The source of this fundamental difficulty is that the density method converts the infinite memory of a non-Poisson renewal process into a different type of memory. The former type of memory is compatible with the occurrence of critical events resetting to zero the systems memory. The second type of memory, on the contrary, implies that the single trajectories, if they exist, are determined by their initial conditions. 

R. A. Dabrowski, H. Dehling, Jump Diffusion Approximation for a Markovian Transport Model, in Asymptotic Methods in Probability and Statistics,... 

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