Field-theoretic Monte Carlo

As an alternative approach to samphng the fluctuating field theory, Diichs et al. [42,80] have proposed a Monte Carlo method. Since the weight exp[- ] 

The Monte Carlo simulation includes two different types of moves trial moves of W and co. The moves in co are straightforward. Implementing the moves in W is more involved. Every time that W is changed, the new saddle point i[7 [M j must be evaluated. In an incompressible blend, the set of self consistent equations 

Compared to the Complex Langevin method, the Monte Carlo method has the advantage of being well foimded theoretically. However, it can become very inefficient when spreads over a wide range and the reweighting factor oscillates strongly, hi practice, it relies on the fact that the integral is indeed dominated by one (or several) saddle points. 

Can we expect this to be the case here To estimate the range of the reweighting factor, we briefly re-inspect the Hubbard-Stratonovich transformation of the total density + b that leads to the fluctuating field 17. For simplicity we consider a one-component system. In a compressible polymer solution or blend, the contribution of the repulsive interaction energy to the partition function can be written as 

Thus U should to be distributed around the saddle point with a width proportional to K. The method should work best for very compressible solutions with small k. In contrast, in an incompressible blend, the contribution (Eq. 107) is replaced by a delta function constraint 8 a + 05 - ) The fluctuating field representation of this constraint is 

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We have demonstrated this for one specific case of an incompressible blend and suspect that it may be a featiue of incompressible blends in general. The observation that the fluctuations in W and o) are not correlated with each other is presumably related to the fact that the (vanishing) density fluctuations do not influence the composition fluctuations. If that is true, we can conclude that field-theoretic Monte Carlo can be used to study fluctuations in polymer mixtures in the limits of high and low compressibiUties. Whether it can also be applied at intermediate compressibilities will have to be explored in the future. 

We should note that the Monte-Carlo simulation with tw = 0 effectively samples the EP Hamiltonian. This version of field-theoretic Monte Carlo is equivalent to the real Langevin method (EPD), and can be used as an alternative. Monte Carlo methods are more versatile than Langevin methods, because an almost unlimited number of moves can be invented and implemented. In our applications, the W and tw-moves simply consisted of random increments of the local field values, within ranges that were chosen such that the Metropolis acceptance rate was roughly 35%. In principle, much more sophisticated moves are conceivable, e.g., collective moves or combined EPD/Monte Carlo moves (hybrid moves [84]). On the other hand, EPD is clearly superior to Monte Carlo when dynamic properties are studied. This will be discussed in the next section. 

In order to study this effect, Diichs et al. have performed field-theoretic Monte Carlo simulations of the system of Fig. 5 [80,83,104], in two dimensions. (For the reasons explained in Sect. 4.4.2, most of these simulations were carried out in the EP approximation). Some characteristic snapshots were already shown in Fig. 4. Here we show another series of snapshots at = 12.5 for increasing homopolymer concentrations (Fig. 8). For all these points, the self-consistent field theory would predict an ordered lamellar phase. In the... 

What is next Several examples were given of modem experimental electrochemical techniques used to characterize electrode-electrolyte interactions. However, we did not mention theoretical methods used for the same purpose. Computer simulations of the dynamic processes occurring in the double layer are found abundantly in the literature of electrochemistry. Examples of topics explored in this area are investigation of lateral adsorbate-adsorbate interactions by the formulation of lattice-gas models and their solution by analytical and numerical techniques (Monte Carlo simulations) [Fig. 6.107(a)] determination of potential-energy curves for metal-ion and lateral-lateral interaction by quantum-chemical studies [Fig. 6.107(b)] and calculation of the electrostatic field and potential drop across an electric double layer by molecular dynamic simulations [Fig. 6.107(c)]. 

By contrast, few such calculations have as yet been made for diffusional problems. Much more significantly, the experimental observables of rate coefficient or survival (recombination) probability can be measured very much less accurately than can energy levels. A detailed comparison of experimental observations and theoretical predictions must be restricted by the experimental accuracy attainable. This very limitation probably explains why no unambiguous experimental assignment of a many-body effect has yet been made in the field of reaction kinetics in solution, even over picosecond timescale. Necessarily, there are good reasons to anticipate their occurrence. At this stage, all that can be done is to estimate the importance of such effects and include them in an analysis of experimental results. Perhaps a comparison of theoretical calculations and Monte Carlo or molecular dynamics simulations would be the best that could be hoped for at this moment (rather like, though less satisfactory than, the current position in the development of statistical mechanical theories of liquids). Nevertheless, there remains a clear need for careful experiments, which may reveal such effects as discussed in the remainder of much of this volume. 

Real catalytic reactions upon solid surfaces are of great complexity and this is why they are inherently very difficult to deal with. The detailed understanding of such reactions is very important in applied research, but rarely has such a detailed understanding been achieved neither from experiment nor from theory. Theoretically there are three basic approaches kinetic equations of the mean-field type, computer simulations (Monte Carlo, MC) and cellular automata CA, or stochastic models (master equations). 

K. Coutinho, S. Canuto, Sequential Monte Carlo/quantum mechanics study of the dipole polarizability of atomic liquids The argon case, in Atoms, Molecules and Clusters in Electric Fields. Theoretical Approaches to the Calculation of Electric Polarizabilities, ed. by G. Maroulis (Imperial College Press, London, 2006), pp. 405 120... 

Hahn [47] developed a hybrid simulation based on BD and Monte Carlo methods. Incorporation of the statistical techniques of Monte Carlo methods relaxes the constraint that time steps must be sufficiently short such that external force fields can be considered constant, and the BD improves upon the Monte Carlo methods by allowing dynamic information to be collected. Hahn applied the model to the investigation of theoretical deposition by simulating a... 

This anomaly stems from the nonrandomness of the reactant distributions in low dimensions. Although in a classical reaction system the distribution of the reactants stays uniformly random, in a fractal-like reaction system the distribution tends to become less random. Similar changes take place in other reactions and other spaces. Such findings are well established today, and they have been observed experimentally and theoretically. Also, results from Monte Carlo simulations (a powerful tool in this field) are in very good agreement with these findings. 

The atomic radii may be further refined to improve the agreement between experimental and theoretical solvation free energies. Work on this direction has been done by Luque and Orozco (see [66] and references cited therein) while Barone et al. [67] defined a set of rules to estimate atomic radii. Further discussion on this point can be found in the review by Tomasi and co-workers [15], It must be noted that the parameterization of atomic radii on the basis of a good experiment-theory agreement of solvation energies is problematic because of the difficulty to separate electrostatic and non-electrostatic terms. The comparison of continuum calculations with statistical simulations provides another way to check the validity of cavity definition. A comparison between continuum and classical Monte Carlo simulations was reported by Costa-Cabral et al. [68] in the early 1980s and more recently, molecular dynamics simulations using combined quantum mechanics and molecular mechanics (QM/MM) force-fields have been carried out to analyze the case of water molecule in liquid water [69],... 

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