Computer simulation analytic theory

Deserno, M., Holm, C., Blaul, J., Ballauff, M., and Rehahn, M. The osmotic coefficient of rod-like poly electrolytes Computer simulation, analytical theory, and experiment. European Physical Journal E, 2001, 5, No. 1, p. 97-103. 

Simulations in this area also require access to high-level computational capabilities. By definition, simulations involve both large numbers of atoms and dynamic behavior both consume large numbers of CPU cycles. Because the field is so demanding on computer time, analytical theory is very important, even if it yields only approximate solutions. 

A successful method to obtain dynamical information from computer simulations of quantum systems has recently been proposed by Gubernatis and coworkers [167-169]. It uses concepts from probability theory and Bayesian logic to solve the analytic continuation problem in order to obtain real-time dynamical information from imaginary-time computer simulation data. The method has become known under the name maximum entropy (MaxEnt), and has a wide range of applications in other fields apart from physics. Here we review some of the main ideas of this method and an application [175] to the model fluid described in the previous section. 

In addition to various analytic or semi-analytic methods, which are based on the theory of the liquid state and which are not the subject of this chapter, almost the entire toolbox of molecular computer simulation methods has been applied to the theoretical study of aqueous interfaces. They have usually been adapted and modified from schemes developed in a different context. 

Complementing these very well established approaches for the study of any scientific field, namely experiments and analytical theory, very recently, computer simulations have become a powerful tool for the study of a great variety of processes occurring in nature in general [4-6], as well as surface chemical reactions in particular [7]. Within this context, the aim of this chapter is not only to offer a critical overview of recent progress in the area of computer simulations of surface reaction processes, but also to provide an outlook of promising trends in most of the treated topics. 

Experimentally, these functions are usually determined only indirectly via the scattering functions of the whole system or the scattering functions of marked chains (see, e.g., [34]). This is one of the advantages of computer simulations over to experiments. However, in order to make significant statements for experimental systems it is always very important to directly compare computer simulations with experimental investigations as well as analytic theories. 

No informative experimental data have been obtained on the precise shape of segment profiles of tethered chains. The only independent tests have come from computer simulations [26], which agree very well with the predictions of SCF theory. Analytical SCF theory has proven difficult to apply to non-flat geometries [141], and full SCF theory in non-Cartesian geometry has been applied only to relatively short chains [142], so that more detailed profile information on these important, nonplanar situations awaits further developments. 

Computer simulation and analytical methods have both been used, based on diffusion equation, partition function and scaling theory approaches. There are a number of parameters which are common to most of these theories some of these are also relevant to theories of polymer solutions, i.e. 

The comparison of computer models with experimental data, then, tests the accuracy of the model. Assuming good agreement, we can take our analysis one step further by comparing equations of state with computer simulations, we test the assumptions implicit in the theories that lead to the EOS. That is, we shed light on what parameters in the analytical expression give rise to observations in the computer simulation. We can assess which underlying assumptions in the EOS constrain its usability. 

In sharp contrast to the large number of experimental and computer simulation studies reported in literature, there have been relatively few analytical or model dependent studies on the dynamics of protein hydration layer. A simple phenomenological model, proposed earlier by Nandi and Bagchi [4] explains the observed slow relaxation in the hydration layer in terms of a dynamic equilibrium between the bound and the free states of water molecules within the layer. The slow time scale is the inverse of the rate of bound to free transition. In this model, the transition between the free and bound states occurs by rotation. Recently Mukherjee and Bagchi [14] have numerically solved the space dependent reaction-diffusion model to obtain the probability distribution and the time dependent mean-square displacement (MSD). The model predicts a transition from sub-diffusive to super-diffusive translational behaviour, before it attains a diffusive nature in the long time. However, a microscopic theory of hydration layer dynamics is yet to be fully developed. 

An increase of the standard deviation at r 3 due to small number of survived particles, demonstrates a limited possibility of the direct statistical simulations for a system with a variable number of particles. However, certain conclusions could be drawn even for such limited statistical information. Say, if for equal concentrations the analytical theory based on the superposition approximation seems to be quite adequate, for unequal concentrations its deviation from the computer simulations greatly increases in time. The superposition approximation gives the lower bound estimate of the actual kinetic curves tia( ) but if for d = 2 shown in Fig. 5.8 the deviation is considerable, for d, = 1 (Fig. 5.7) it is not observed, at least for the reaction depths considered. 

Another important test of the accuracy of the superposition approximation is the diffusion-controlled A + B — 0 reaction. For the first time it was computer-simulated by Toussaint and Wilczek [27]. They confirmed existence of new asymptotic reaction laws but did not test different approximations used in the diffusion-controlled theories. Their findings were used in [28] to discuss divergence in the linear and the superposition approximations. Since analytical calculations [28] were performed for other sets of parameters as used in [27], their comparison was only qualitative. It was Schnorer et al. [29] who first performed detailed study of the applicability of the superposition approximation. 

The analysis of the diffusion-controlled computer simulations confirms once more conclusions drawn above for the static reactions of immobile particles. In particular, the superposition approximation gives the best lower bound estimate of the kinetics reaction, n = n(i). Divergence of computer simulations and analytical theory being negligible for equal concentrations become essential for large depths and when one of reactants is in excess. The obtained results allow us to use the superposition approximation for testing the applicability of simple equations of the linear theory in those cases when computer simulations because of some reasons cannot be performed. Examples will be presented in Chapter 6. 

To compare computer simulations with an analytical theory, it is convenient to introduce a distinctive parameter - dimensionless saturation concentration Uq = n(co)vo, where n(oo) is stationary concentration of accumulated defects at their saturation (t —> oo). (It is assumed that n(t) = n (t) = n t), vo is volume of the d-dimensional sphere having the recombination radius r0 v0 = 7iro/d.) In the continuous model it is clear that the quantity Uo, if it exists, is a universal parameter dependent on d only but not on vq. Indeed, most of previous theoretical studies were aimed mainly to obtain Uo-... 

In our opinion, this book demonstrates clearly that the formalism of many-point particle densities based on the Kirkwood superposition approximation for decoupling the three-particle correlation functions is able to treat adequately all possible cases and reaction regimes studied in the book (including immobile/mobile reactants, correlated/random initial particle distributions, concentration decay/accumulation under permanent source, etc.). Results of most of analytical theories are checked by extensive computer simulations. (It should be reminded that many-particle effects under study were observed for the first time namely in computer simulations [22, 23].) Only few experimental evidences exist now for many-particle effects in bimolecular reactions, the two reliable examples are accumulation kinetics of immobile radiation defects at low temperatures in ionic solids (see [24] for experiments and [25] for their theoretical interpretation) and pseudo-first order reversible diffusion-controlled recombination of protons with excited dye molecules [26]. This is one of main reasons why we did not consider in detail some of very refined theories for the kinetics asymptotics as well as peculiarities of reactions on fractal structures ([27-29] and references therein). 

Evident progress in studies of liquids has been achieved up to now with the use of computer simulations and of the models based on analytical theory. These methods provide different information and are mutually complementary. The first method employs rather rigorous potential functions and yields usually a chaotic picture of the multiple-particle trajectories but has not been able to give, as far as we know, a satisfactory description of the wideband spectra. The analytical theory is based on a phenomenological consideration (which possibly gives more regular trajectories of the particles than arise in reality ) in terms of a potential well. It can be tractable only if the profile of such a well is rather... 

Although recent years have witnessed an impressive confluence of experiments and statistical theories, presently there is no comprehensive understanding of the interrelation between chemical sequences in synthetic copolymers and the conditions of synthesis. One has merely to glance at recent literature in polymer science and biophysics to realize that the problem of sequence-property relationship is by no means entirely solved. As always, in these circumstances, an alternative to analytical theories is computer simulations, which are designed to obtain a numerical answer without knowledge of an analytical solution. 

Both analytic theory and computer simulations are included, and we note that the latter play an especially important role in understanding cluster reactions. Simulations not only provide quantitative results, but they provide insight into the dominant causes of observed behavior, and they can provide likelihood estimates for assessing qualitatively distinct mechanisms that can be used to explain the same experimental data. Simulations can also lead to a greater understanding of dynamical processes occurring in clusters by calculating details which cannot be observed experimentally. 

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