Random reaction time simulation

The IRT model has been developed in detail in a series of papers of Green, Pimblott and coworkers and has been validated by comparison with full random flight simulations [47,49,51]. The IRT treatment of the radiation chemistry relies upon the generation of random reaction times from initial coordinate positions from pair reaction time distribution functions. A simulation, such as a random flight calculation, starts with the initial spatial distribution of the reactants. The separations between all the pairs of particles are evaluated... 

The independent reaction time (1RT) model was introduced as a shortcut Monte Carlo simulation of pairwise reaction times without explicit reference to diffusive trajectories (Clifford et al, 1982b). At first, the initial positions of the reactive species (any number and kind) are simulated by convolving from a given (usually gaussian) distribution using random numbers. These are examined for immediate reaction—that is, whether any interparticle separation is within the respective reaction radius. If so, such particles are removed and the reactions are recorded as static reactions. 

Analytical treatment of the diffusion-reaction problem in a many-body system composed of Coulombically interacting particles poses a very complex problem. Except for some approximate treatments, most theoretical treatments of the multipair effects have been performed by computer simulations. In the most direct approach, random trajectories and reactions of several ion pairs were followed by a Monte Carlo simulation [18]. In another approach [19], the approximate Independent Reaction Times (IRT) technique was used, in which an actual reaction time in a cluster of ions was assumed to be the smallest one selected from the set of reaction times associated with each independent ion pair. 

Lastly, we would like to mention here results of the two kinds of large-scale computer simulations of diffusion-controlled bimolecular reactions [33, 48], In the former paper [48] reactions were simulated using random walks on a d-dimensional (1 to 4) hypercubic lattice with the imposed periodic boundary conditions. In the particular case of the A + B - 0 reaction, D = Dq and nA(0) = nB(0), the critical exponents 0.26 0.01 0.50 0.02 and 0.89 0.02 were obtained for d = 1 to 3 respectively. The theoretical value of a = 0.75 expected for d = 3 was not achieved due to cluster size effects. The result for d = 4, a = 1.02 0.02, confirms that this is a marginal dimension. However, in the case of the A + B — B reaction with DB = 0, the asymptotic longtime behaviour, equation (2.1.106), was not achieved at all - even at very long reaction times of 105 Monte Carlo steps, which were sufficient for all other kinds of bimolecular reactions simulated. It was concluded that in practice this theoretically derived asymptotics is hardly accessible. 

The time evolution of the electronic wave function can be obtained in the adiabatic or in the diabatic basis set. At each time step, one evaluates the transition probabilities between electronic states and decides whether to hop to another siu-face. When hopping occurs, nuclear velocities have to be adjusted to keep the total energy constant. After hopping, the forces are calculated from the potential of the newly populated electronic state. To decide whether or not to hop, a Monte Carlo technique is used Once the transition probability is obtained, a random number in the range (0,1) is generated and compared with the transition probability. If the munber is less than the probability, a hop occurs otherwise, the nuclear motion continues on the same surface as before. At the end of the simulation, one can analyze populations, distribution of nuclear geometries, reaction times, and other observables as an average over all the trajectories. 

The simulation of lignin liquefaction combined a stochastic interpretation of depolymerization kinetics with models for catalyst deactivation and polymer diffusion. The stochastic model was based on discrete mathematics, which allowed the transformations of a system between its discrete states to be chronicled by comparing random numbers to transition probabilities. The transition probability was dependent on both the time interval of reaction and a global reaction rate constant. McDermott s ( analysis of the random reaction trajectory of the linear polymer shown in Figure 6 permits illustration. 

With models for catalyst decay and effectiveness now in hand, the simulation of lignin liquefaction could be achieved given the initial lignin structure (as described earlier) and model compound reaction pathways and kinetics, both thermal and catalytic. Construction of a random polymer, as outlined earlier, began the simulation. This structural information combined with the simulated process conditions to allow calculation of the reaction rate constants, selectivities and associated transition probabilities. The largest rate constant then specified the upper limit of the reaction time step size. 

The time step chosen is variable and changes throughout the simulation. It can be infinitely small or infinitely large and depends both on the random probabihty and the overall calculated rate. The variable-time method is mathematically an exact approach and there are no concerns about accuracy due to time step size. Systems which contain fast events have very small time steps and are thus dominated by the time scale scales of the fastest processes. Faster rates of reaction lead to higher probabilities that these steps are chosen. This leads to problems for systems with disparate rates since the simulation will spend nearly all of its time simulating the faster rates without ever simulating the slower processes. This is esp>ecially a problem for systems where diffusion is fast and reaction is slow and systems which contain fast processes which are nearly equihbrated together with slow processes. 

Fig. 167. Output of the simulation program SIMxNNy44/45. The program produces the distribution, the first peak (Fp), the median reaction time (Med), the internal meancycEN, and the externally observable (med-lin)/ET.Only the peaks of the xl ly distribution are simulated, the intermediate values are produced by a small random factor which has to be added ( z). A comma means 50ms on the x-axis. The times of Con, Lin etc. are given in millisecond, too...

It has been reported in the literature [48] that Smoluchowski s rate constant overestimates the rate of scavenging for a single target that can be hit multiple times (for example DNA). In their work, the authors found that Smoluchowski s rate constant overestimated the scavenging yield in comparison to Monte Carlo random flights simulation (which makes no assumptions on the rate of scavenging as they are explicitly treated). The authors have found that a modiflcation to Smoluchowski s rate constant is required in order to properly take the correlation of reaction times into account however, the independent pairs approximation is still made. 

The IRT model [1,2, 12, 15] is essentially a Monte Carlo algorithm which assumes the independence of reaction times (i.e. each reaction is independent of other such reactions and that the covariance of these reaction times is zero). Unlike the random flights simulation, the diffusive trajectories are not tracked but instead encounter times are generated by sampling from an appropriate probability density function conditioned on the initial separation of the pair. The first encounter takes place at the minimum of the key times generated min(h t2, fs...) and all subsequent reactions occur based on the minimum of surviving reaction times. Unlike random flights... 

In the random flights simulation, scavengers were treated explicitly, such that the e has to diffuse towards a stationary S to react. The IRT algorithm made use of Smoluchowski s time dependent rate constant, in which reaction times were generated from the probability distribution... 

Pig. 9.6 Survival probability calculated using the mean reaction time and compared with random flights simulations using an outer reflective boundary. The micelle radius was 30 A and the encounter radius was set to 4 A for all reactions, a Using the acmal mutual diffusion equation without scaling and b using Eq. (9.30) to correct for the mutual diffusion. Here MC refers to random flights simulation... 

---