Method Monte Carlo

The method has not taken on elsewhere. One paper [248] reports the use of PSPICE, but for simulating actual resistance in an electrolyte, modelled as a resistance network. This is quite a different application, and much more directly relevant. 

In a paper reporting the results of some simulations of diffusion of hydrogen into palladium [249], the authors describe their method of solution as the Treanor method. This is described in a few texts [250, 251] and goes back to a paper by Treanor in 1966 [252]. 

The method is one way to handle a stiff set of ode%, and is an extension of fourth-order explicit Runge-Kutta. The function to be solved is approximated over the next time interval by a combination of a linear function of the dependent variable and a quadratic function of time (assuming that it is strongly time-dependent) and this increases the accuracy and stability of the fourth-order Runge-Kutta method considerably. Today, however, we have other methods of dealing with stiff sets of odes, so this method might be said to have outlived its usefulness. 

Feldberg SW (1969) Digital simulation a general method for solving electrochemical diffusion-kinetic problems. In Bard AJ (ed) Electroanalytical chemistry, vol 3. Marcel Dekker, New York, pp 199-296 

Rudolph M, Reddy DR Feldberg SW (1994) A simulator for cyclic voltammetry responses. Anal Chem 66 589A-600A 

The Monte Carlo method is another stochastic procedure for the numerical determination of the change in concentrations of reactants, intermediates or products of a reaction as a function of time. This method can simulate the time evolution of any chemical system. 

The major advantage of the Monte Carlo method stems from the fact that its convergence is independent of the dimension of the integral. Let us consider an integral of dimension j 

In a reaction of rate constant k = W sec if we consider a timescale with units of 1 nsec and 10 events are accomplished in each cycle, the probability of a reaction occurring in each cycle will be p = 10 The description of the time evolution of the system can thus be given in terms of the dependence of the molar fraction as a function of the number of cycles or, using the previous relation and converting the initial molar fraction into the initial concentration, as the variation of concentration with time. 

If the reaction is second order, the probability of a molecule occupying a coordinate is independent of another occupying another coordinate. Consequently for each cycle m, two events must be made, one for each reactant molecule. If a molecule is found in both selections, then the reaction will occur. 

Particularly useful applications of the Monte Carlo method include modelling complex oscillatory reactions and studying enzyme catalysis [8,9]. As an example of the latter treatment, we will consider a system involving an initial reversible complex formation between the enzyme and the substrate, accompanied by a reversible step of inhibition of the catalyst 

Several codes are available for carrying out a direct Monte Carlo simulation of a reactor problem using detailed geometrical models and continuous energy (or very fine group) representation of nuclear data. These can be used to provide reference values and investigate the effects of approximations in deterministic methods. Some widely used Monte Carlo codes are MCNP [4.39], MORSE [4.66] and KENO [4.67], amongst others. 

The best of these deterministic methods, correctly applies, gives a perfectly adequate treatment of most reactor problems. However there are some important benefits from a more direct model of the physical problem. The current deterministic methods involve so many specialised approximations, the subject of may man years of specialists effort during their development, that their range of validity is sometimes well understood only by their 

If problems of this kind are to be avoided, potential users will have to be trained to an exceptionally high level in the modelling technique used in the old methods and the overhead in doing this will be high. The Monte Carlo method could be the remedy. Due to its inherent slowness relative to the deterministic methods it cannot be used at present for all routine calculations. However, it can provide reference values which are transparent to physicists and engineers. 

The VIM code was probably the first Monte Carlo code specially tailored to treat continuous energy neutron data in a manner suitable for general fast reactor problems. Mention can also be made of the MOCA Monte Carlo code which has been used for fast reactor control rod calculations and is included in the ERANOS scheme. 

One of the most promising directions of development is the combinatorial approach. In this approach, hybrid schemes (Monte Carlo codes coupled with deterministic methods) are used, the Monte Carlo method being applied only for treating sub-regions having complex geometry. Outside, these complex domains simpler and quicker deterministic methods are used. A detailed review of these combined methods is presented in [4.69]. 

Generally, at molecular level, the off-lattice MC simulation considers more detail than lattice MC simulation does, such as bond angle, bond length, and chain flexibility. However, the researchers pay more attention to the morphology than to the molecular details in the field of microphase separation of block copolymers. Furthermore, the lattice MC simulation runs much faster than the off-lattice MC simulation does. Therefore, many MC simulations employed the lattice model to explore the microphase separation of block copolymers. 

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The alternative simulation approaches are based on molecular dynamics calculations. This is conceptually simpler that the Monte Carlo method the equations of motion are solved for a system of A molecules, and periodic boundary conditions are again imposed. This method pennits both the equilibrium and transport properties of the system to be evaluated, essentially by numerically solvmg the equations of motion... 

Specific solute-solvent interactions involving the first solvation shell only can be treated in detail by discrete solvent models. The various approaches like point charge models, siipennoleciilar calculations, quantum theories of reactions in solution, and their implementations in Monte Carlo methods and molecular dynamics simulations like the Car-Parrinello method are discussed elsewhere in this encyclopedia. Here only some points will be briefly mentioned that seem of relevance for later sections. 

Mbiler-Krumbhaar H and Binder K 1973 Dynamic properties of the Monte-Carlo method in statistical mechanics J. Stat. Phys. 8 1-24... 

Binder K (ed) 1995 The Monte Carlo Method in Condensed Matter Physics vol 71 Topics in Applied Physics 2nd edn (Berlin Springer)... 

Newman M E J and Barkema G T 1999 Monte Carlo Methods in Statistical Physics (Oxford Clarendon)... 

Salsburg Z W, Jacobson J D, Fickett W and Wood W W 1959 Application of the Monte Carlo method to the lattice gas model. Two dimensional triangular lattice J. Chem. Phys. 30 65-72... 

McDonald I R and Singer K 1967 Calculation of thermodynamic properties of liquid argon from Lennard-Jones parameters by a Monte Carlo method Discuss. Faraday Soc. 43 40-9... 

Heermann D W and Burkitt A N 1995 Parallel algorithms for statistical physics problems The Monte Carlo Method In Condensed Matter Physios vol 71 Toplos In Applied Physios ed K Binder (Berlin Springer) pp 53-74... 

Bruce A D, Wilding N B and Ackland G J 1997 Free energies of crystalline solids a lattice-switch Monte-Carlo method Phys. Rev. Lett. 79 3002-5... 

Demidov A A 1999 Use of Monte-Carlo method in the problem of energy migration in molecular complexes Resonance Energy Transfer e6 D L Andrews and A A Demidov (New York Wiley) pp 435-65... 

Agranovich V M, Efremov N A and Kirsanov V V 1980 Computer simulation of kinetics of excitation bimolecular quenching by Monte-Carlo method Fiz. Tverd. Tela 22 2118-27... 

The basic scheme of this algorithm is similar to cell-to-cell mapping techniques [14] but differs substantially In one important aspect If applied to larger problems, a direct cell-to-cell approach quickly leads to tremendous computational effort. Only a proper exploitation of the multi-level structure of the subdivision algorithm (also for the eigenvalue problem) may allow for application to molecules of real chemical interest. But even this more sophisticated approach suffers from combinatorial explosion already for moderate size molecules. In a next stage of development [19] this restriction will be circumvented using certain hybrid Monte-Carlo methods. 

Ch. Schiitte, A. Fischer, W. Huisinga, P. Deuflhard. A Hybrid Monte-Carlo Method for Essential Molecular Dynamics. Preprint, Preprint SC 98-04, Konrad Zuse Zentrum, Berlin (1998)... 

Treatment of Multiple Ionizations by a Monte Carlo Method... 

Monte Carlo Methods for Pure Tsallis Statistics... 

Alternatively, one may use a phase space Monte Carlo method with uniform random trial moves and an acceptance probability... 

A similar algorithm has been used to sample the equilibrium distribution [p,(r )] in the conformational optimization of a tetrapeptide[5] and atomic clusters at low temperature.[6] It was found that when g > 1 the search of conformational space was greatly enhanced over standard Metropolis Monte Carlo methods. In this form, the velocity distribution can be thought to be Maxwellian. 

B. Mehlig, D. W. Heermann, and B. M. Forrest. Hybrid Monte Carlo method for condensed-matter systems. Phys. Rev. B, 45 679-685, 1992. 

In this chapter we shall discuss some of the general principles involved in the two most common simulation techniques used in molecular modelling the molecular dynamics and the Monte Carlo methods. We shall also discuss several concepts that are common to both of these methods. A more detailed discussion of the two simulation methods can be found in Chapters 7 and 8. 

Differences Between the Molecular Dynamics and Monte Carlo Methods... 

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