Stochastic Treatments

The methods discussed so far are essentially limited to isolated ion-pairs or, in the admittedly crude approximation, to cases when a multiple ion-pair spur can be considered to be a collection of single ion-pairs. Additionally, it is difficult to include an external field, as that will destroy the spherical symmetry of the problem. Stochastic treatments can incorporate both multiple ion-pairs and the effects of an external field. 

The methodology of stochastic treatment of e-ion recombination kinetics is basically the same as for neutrals, except that the appropriate electrostatic field term must be included (see Sect. 7.3.1). This means the coulombic field in the dielectric for an isolated pair and, in the multiple ion-pair case, the field due to all unrecombined charges on each electron and ion. All the three methods of stochastic analysis—random flight Monte Carlo (MC), independent reaction time (IRT), and the master equation (ME)—have been used (Pimblott and Green, 1995). 

MC simulation for multiple ion-pair case is straightforward in principle. A recombination, if necessary with a given probability, is assumed to have taken place when an e-ion pair is within the reaction radius. Simulation is continued until either only one pair is left or the uncombined pairs are so far apart from each other that they may be considered as isolated. At that point, isolated pair equations are used to give the ultimate kinetics and free-ion yield. 

For an isolated ion-pair with rc = 29 nm, r0 (initial separation) = 6.0 nm, D = 2.5 x 10-5 cm2s 1, and R (reaction radius) = 1.0 nm, all appropriate to n-hexane, random flight MC simulation reproduces accurately the kinetics of 

Green and Pimblott (1991) criticize the truncated distributions of Mozumder (1971) and of Dodelet and Freeman (1975) used to calculate the free-ion yield in a multiple ion-pair case. In place of the truncated distribution used by the earlier authors, Green and Pimblott introduce the marginal distribution for all ordered pairs, which is statistically the correct one (see Sect. 9.3 for a description of this distribution). 

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On the other hand, the stochastic treatment in terms of equation (2.2.2) is applied to the macroscopic values a(t) characterizing a system. As is generally-accepted, thermal fluctuations of macroscopic quantities being very small are not of great interest. The only exception when fluctuations appear to be important (which leads to an untrivial result) is the case when the solution of a set of equations F (c, ..., cs) = 0 finds itself near the bifurcation point [26, 34, 90], This is why we are going to consider now the spatially-extended systems [26, 67, 68],... 

In other words, K(t) is afunctional of the joint correlation function of similar particles. In this respect, a set of equations (8.2.12) and (8.2.13) is similar to the stochastic treatment of the Lotka-Volterra model (equations (2.2.68) and (2.2.69)) considered in Section 2.3.1 using the similar time-dependent reaction rate (2.2.67). 

In order to develop a simple theoretical model of the plasma under consideration, we must now give a microphysical interpretation of the stability criterion. Unfortunately, very little work has been done on the subject. A detailed stochastic treatment of several simple models has shown that stability will be ensured if the time scales associated with the fluctuations in the system are much shorter than the time scales associated with the outside world45,50) a. This condition is similar to that for local thermodynamic equilibrium45,52. ... 

The impact of bath molecules on the atoms of the molecule of interest cannot be treated as impulsive because the strong binding forces of chemical bonds places a significant fraction of the vibrational sjjectrum of the molecule above the collisional bandwidth, broadly defined as the reciprocal of the duration of a collision. Thus collisional vibrational relaxation and excitation are inefficient relative to rotational relaxation. Binary collision theory is well develojjed at the microcanonical level because of the its importance in chemical reactions. The relationship to the friction is of interest, " primarily because stochastic treatments have the potential of bridging the gas-phase limit of resolved binary collisions and the liquid phase where collective phenomena of the solvent can preclude interpretation in terms of binary collisions. 

The altitude effecf and the continental effecf (Rozanski et al. 1993) refer to the tendencies for and 5D of precipitation to become progressively lower towards continental interiors and higher elevations. The altitude effect includes important control by temperature. Both may reflect a stochastic tendency for air masses moving away from moisture sources to encounter conditions (frontal/convective) that cause moisture extraction and so isotopic fractionation for °0 and D H it is possible that modeling of and 5D of precipitation could involve stochastic treatments. As reviewed by Rozanski et al. (1993), the continentality depletion for 5 0 of precipitation reaches 8 per mil over 4500 km into Europe from the Atlantic coast. This continental effect is more pronounced in winter than in summer, perhaps because evapotranspiration returns most summertime precipitation to the atmosphere, whereas runoff is more important in wintertime. 

When detailed information about the nature of the bath is lacking or unimportant for the problem at hand, it can be convenient to treat the bath stochastically, that is, as a source of random fluctuations of the subsystem Hamiltonian [30,35,38,48]. In a stochastic treatment the bath operators in V are replaced with classical random variables. 

It was logical to extend the results of the thermostatic theory to temporal changes. Stochastic thermostatics adopts an intermediate level between statistical and phenomenological thermodynamics. Analogously, in principle the stochastic treatment of thermodynamics processes has an intermediate character between nonequilibrium statistical mechanics and phenomenolog-... 

Modeling approaeh Lagrangian/Eulerian treatment, deterministic/stochastic treatment... 

See Ramkrishna (1981). See Fox and Fan (1988) for a stochastic treatment of breakage and coalescence using the master equation approach. 

O. Cohendet, Ph. Combe, M. Sirugue, M. Sirugue-Collin, A stochastic treatment of the dynamics of an integer spin, J. Phys. A 21 (1988) 2875. 

In stochastic treatments the coordinates of the bath atoms are projected out, thereby circumventing the description of the recombination processes in terms of cluster equilibria and kinetics [27]. 

In the case of low and moderate diffusion coefficients the stochastic treatment will show that the diffusion can have interesting effects on the reaction for instance it can make easier the passage from a homogeneous metastable state to a homogeneous stcible state via inhomogeneous configurations/ which explains their spontaneous emergence. 

R. Nassar, L.T. Fan, J.R. Too, L.S. Fan, A Stochastic Treatment of Unimolecular reactions in an Unsteady State Continuous Flow System, Chemical Engineering... 

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