Aggregation-field effect model

Most traditional models focus on looking for equilibrium solutions among some set of (pre-defined) aggregate variables. The LEs are effectively mean-field equations, in which certain variables (i.e. attrition rate) are assumed to represent an entire force, the equilibrium state is explicitly solved for and declared the battle outcome. In contrast, ABMs focus on understanding the kinds of emergent patterns that might arise while the overall system is out of (or far from) equilibrium. 

More than 20 years ago, Matsushita et al. observed macroscopic patterns of electrodeposit at a liquid/air interface [46,47]. Since the morphology of the deposit was quite similar to those generated by a computer model known as diffusion-limited aggregation (D LA) [48], this finding has attracted a lot of attention from the point of view of morphogenesis in Laplacian fields. Normally, thin cells with quasi 2D geometries are used in experiments, instead of the use of liquid/air or liquid/liquid interfaces, in order to reduce the effect of convection. 

In the past, the equivalence between the size distribution generated by the Smoluchowski equation and simple statistical methods [9, 12, 40-42] was a source of some confusion. The Spouge proof and the numerical results obtained for the kinetics models with more complex aggregation physics, e.g., with a presence of substitution effects [43,44], revealed the non-equivalence of kinetics and statistical models of polymerization processes. More elaborated statistical models, however, with the complete analysis made repeatedly at small time intervals have been shown to produce polymer size distributions equivalent to those generated kinetically [45]. Recently, Faliagas [46] has demonstrated that the kinetics and statistical models which are both the mean-field models can be considered as special cases of a general stochastic Markov process. 

Section IV is devoted to excitons in a disordered lattice. In the first subsection, restricted to the 2D radiant exciton, we study how the coherent emission is hampered by such disorder as thermal fluctuation, static disorder, or surface annihilation by surface-molecule photodimerization. A sharp transition is shown to take place between coherent emission at low temperature (or weak extended disorder) and incoherent emission of small excitonic coherence domains at high temperature (strong extended disorder). Whereas a mean-field theory correctly deals with the long-range forces involved in emission, these approximations are reviewed and tested on a simple model case the nondipolar triplet naphthalene exciton. The very strong disorder then makes the inclusion of aggregates in the theory compulsory. From all this study, our conclusion is that an effective-medium theory needs an effective interaction as well as an effective potential, as shown by the comparison of our theoretical results with exact numerical calculations, with very satisfactory agreement at all concentrations. Lastly, the 3D case of a dipolar exciton with disorder is discussed qualitatively. 

One of the main factors influencing the activation barrier in fast electron-transfer reactions is the change in the polarization of the immediate space surrounding the activated complex in solution. The more-well-known salt effects as well as the relatively new field of micellar effects can be used as mechanistic probes in this context. Since micelles have a hydrophobic as well as a hydrophilic part, this creates two different kinds of interfaces where electron transfer can occur if one of either the oxidant or reductant is contained or associated with these molecular aggregates. A futuristic approach could be that studies of this kind may serve as models for enzymatic reactions with complex bioaggregates such as membranes. 

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