Smoluchowski equation times

Given the Smoluchowski equation, time correlation fimctions can be calculated. For the sake of simplicity we use x to denote the whole set of coordinatesXi,x, , Xn appearing in the Smoluchowski equation. Let G x, x t) be the probability that the system which was in the state x at time r s 0 is in the state x at time t. Qearly such probability is obtained... 

The transport of dissolved species in a solvent occurs randomly through movement of the Brownian type. The particles of the dissolved substance and of the solvent continuously collide and thus move stochastically with various velocities in various directions. The relationship between the mobility of a particle, the observation time r and the mean shift (jc2) is given by the Einstein-Smoluchowski equation (in three-dimensional case)... 

All models described up to here belong to the class of equilibrium theories. They have the advantage of providing structural information on the material during the liquid-solid transition. Kinetic theories based on Smoluchowski s coagulation equation [45] have recently been applied more and more to describe the kinetics of gelation. The Smoluchowski equation describes the time evolution of the cluster size distribution N(k) ... 

In an early attempt, Mozumder (1968) used a prescribed diffusion approach to obtain the e-ion geminate recombination kinetics in the pure solvent. At any time t, the electron distribution function was assumed to be a gaussian corresponding to free diffusion, weighted by another function of t only. The latter function was found by substituting the entire distribution function in the Smoluchowski equation, for which an analytical solution was possible. The result may be expressed by... 

In the general case, whether the e-ion pair is isolated or not, the probability density P(r, t) that an electron will remain extant at time t is given by the Smoluchowski equation... 

Equation (2.6) is called the Fokker-Planck equation (FPE) or forward Kolmogorov equation, because it contains time derivative of final moment of time t > to. This equation is also known as Smoluchowski equation. The second equation (2.7) is called the backward Kolmogorov equation, because it contains the time derivative of the initial moment of time to < t. These names are associated with the fact that the first equation used Fokker (1914) [44] and Planck (1917) [45] for the description of Brownian motion, but Kolmogorov [46] was the first to give rigorous mathematical argumentation for Eq. (2.6) and he was first to derive Eq. (2.7). The derivation of the FPE may be found, for example, in textbooks [2,15,17,18],... 

For example, in the case of PS and applying the Smoluchowski equation [333], it is possible to estimate the precipitation time, fpr, of globules of radius R and translation diffusion coefficient D in solutions of polymer concentration cp (the number of chains per unit volume) [334]. Assuming a standard diffusion-limited aggregation process, two globules merge every time they collide in the course of Brownian motion. Thus, one can write Eq. 2 ... 

Time-resolved fluorescence experiments carried out with 1,2-benzanthracene quenched by CBr4 in propane-1,2-diol show a better fit with the Collins-Kimball equation than with the Smoluchowski equation. 

By comparing time-resolved and steady-state fluorescence parameters, Ross et alm> have shown that in oxytocin, a lactation and uterine contraction hormone in mammals, the internal disulfide bridge quenches the fluorescence of the single tyrosine by a static mechanism. The quenching complex was attributed to an interaction between one C — tyrosine rotamer and the disulfide bond. Swadesh et al.(()<>> have studied the dithiothreitol quenching of the six tyrosine residues in ribonuclease A. They carefully examined the steady-state criteria that are useful for distinguishing pure static from pure dynamic quenching by consideration of the Smoluchowski equation(70) for the diffusion-controlled bimolecular rate constant k0,... 

In this approach, the diffusion constant, Di, is related to the corresponding characteristic time, x, describing the distortions of the normal coordinate, Westlund et al. (85) used the framework of the general slow-motion theory to incorporate the classical vibrational dynamics of the ZFS tensor, governed by the Smoluchowski equation with a harmonic oscillator potential. They introduced an appropriate Liouville superoperator ... 

The central problem in the theory of geminate ion recombination is to describe the relative motion and reaction with each other of two oppositely charged particles initially separated by a distance ro- If we assume that the particles perform an ideal diffusive motion, the time evolution of the probability density, w(r,t), that the two species are separated by r at time t, may be described by the Smoluchowski equation [1,2]... 

In the preceding part of this section, we have concentrated on the electron escape probability, which is an important quantity in the geminate phase of recombination, and can be experimentally observed. However, modern experimental techniques also give us a possibility to observe the time-resolved kinetics of geminate recombination in some systems. Theoretically, the decay of the geminate ion pairs can be described by the pair survival probability, W t), defined by Eq. (4). One method of calculating W t) is to solve the Smoluchowski equation [Eq. (2)] for w r,t) and, then, to integrate the solution over the space variable. Another method [4] is to directly solve Eq. (7) under relevant conditions. 

When the motion of electrons and positive ions in a particular system may be described as ideal diffusion, the process of bulk recombination of these particles is described by the diffusion equation. The mathematical formalism of the bulk recombination theory is very similar to that used in the theory of geminate electron-ion recombination, which was described in Sec. 10.1.2 ( Diffusion-Controlled Geminate Ion Recombination ). Geminate recombination is described by the Smoluchowski equation for the probability density w(r,i) [cf. Eq. (2)], while the bulk recombination is described by the diffusion equation for the space and time-dependent concentration of electrons around a cation (or vice versa), c(r,i). 

Finally, several attempts have been made to solve the Debye—Smoluchowski equation in the time domain using approximate techniques based on uniformly small perturbations (Montroll [74], Abell and Mozumder... 

A complete solution of eqn. (151) for the time-dependent density distribution does not appear feasible, but Hong and Noolandi [323—325] have found the long-time behaviour, as well as the steady-state solution. The mathematics are very complex since the complications encountered in the analysis of the Debye—Smoluchowski equation (Appendix A) are compounded by the applied electric field. For small electric fields and long times, the survival probability is approximately... 

Morse and Feshbach [499] have discussed the variation approach to a description of equations of motion for diffusion. Their approach is straightforward and is generalised here to consider the cases where there is an energy of interaction, U, between the pair of particles, separated by a distance r at time t. It is relatively easy to extend this to a many-body situation. The usual Euler form of the equation of motion is the Debye— Smoluchowski equation, which has been discussed in much detail before, viz. 

Exercise. For the Smoluchowski equation in one dimension show that the conditional mean first-passage time r (y) obeys... 

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. 

The above-described pair problem is treated by the Smoluchowski equation [3, 19] - see Fig. 1.10. It operates with the probability densities (Fig. 1.11) and contains the recombination rate characterizing particle motion. Knowledge of the probability density to find a particle at a given point at time moment t gives us (by means of a trivial integration over reaction volume) the quantity of our primary interest - survival probability of a particle in the system with... 

Onsager inverted snowball theory (Com.) relation to Smoluchowski equation in, 35 relaxation time by, 34 rotational diffusion and, 36 Ozone in the atmosphere, 108 alkene reactions with, 108 Crigee intermediate from, 108 molozonide from, 108 ethylene reaction with, 109 acetaldehyde effect on, 113 formic anhydride from, 110 sulfur dioxide effect on, 113 sulfuric acid aerosols from, 114 infrared detection of, 108 tetramethylethylene (TME) reaction with, 117... 

The eigenmode expansion was also used to determine the time-dependent solution of the Smoluchowski equations for diverse bistable potentials.185... 

The set of cfs constitutes the distribution of aggregate sizes. The Smoluchowski equation (Eq. 3) defines its change with time. 

Table 1. Number, weight, and z-average degrees of polymerization for the random homopolymerization of bifunctional monomer calculated using the Smoluchowski equation (Eq. 8) and expressed in terms of time (t) and conversion degree (p)...

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