Monte field-theoretic

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... 

At d = 1 one has a completely stretched chain with ly = 1. At d = 2 the exact result v = 3/4) [13] is obtained. The upper critical dimension is d = 4, above which the polymer behaves as a random walker. The values of the universal exponents for SAWs on d - dimensional regular lattices have also been calculated by the methods of exact enumerations and Monte Carlo simulations. In particular, at the space dimension d = 3 in the frames of field-theoretical renormalization group approach one has (v = 0.5882 0.0011 [11]) and Monte Carlo simulation gives (i/ = 0.592 0.003 [12]), both values being in a good agreement. 

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... 

Single-Chain-in-Mean-Field Simulations and Grid-Based Monte Carlo Simulation of the Field-Theoretic Hamiltonian... 

The thermodynamic integration scheme can be appUed to different models including coarse-grained, partide-based models of amphiphihc systems and membranes [133, 134] (e.g., soft DPD-models [135-137], Lennard-Jones models [138,139], or solvent-free models [140-142] of membranes) as well as field-theoretic representations [28]. It can be implemented in Monte Carlo or molecular dynamic simulations, as well as SCMF simulations [40-42, 86], field-theoretic simulations [28], and external potential dynamics [27, 63, 64] or dynamic density functional theory [143, 144]. 

Similar attempts to include fluctuations beyond one-loop in SC FT have also been exercised in the context of neutral polymers using field theoretical simulations [55,90] or by bridging SCFT with Monte Carlo techniques [91]. However, these techniques have not been applied for the case of polyelectrolytes with counterions and added salt ions due to very high computational cost. 

Since the results of Monte Carlo (MC) simulations on PE structure are discussed in detail in chapter Thermodynamic and Rheological Properties of Polyelectrolyte Systems, only a brief summary of this simulation method and some results related to this chapter are discussed. In addition, a short overview of recent developments in field theoretical approaches is given. 

An novel approach is the combination between self-consistent field theoretical approaches and Monte Carlo simulations called theoretically informed coarse grain simulations [120]. It was developed in the field of block copolymers and could easily be extended to PE based copolymers and also thermodynamic calculations [120]. 

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. 

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