Generalized-ensemble Monte Carlo methods

Hansmann, U.H.E. Okamoto, Y., Generalized-ensemble Monte Carlo method for systems with rough energy landscapes, Phys. Rev. E 1997, 56, 2228-2233... 

This review discusses a newly proposed class of tempering Monte Carlo methods and their application to the study of complex fluids. The methods are based on a combination of the expanded grand canonical ensemble formalism (or simple tempering) and the multidimensional parallel tempering technique. We first introduce the method in the framework of a general ensemble. We then discuss a few implementations for specific systems, including primitive models of electrolytes, vapor-liquid and liquid-liquid phase behavior for homopolymers, copolymers, and blends of flexible and semiflexible... 

The Monte Carlo method is easily carried out in any convenient ensemble since it simply requires the construction of a suitable Markov chain for the importance sampling. The simulations in the original paper by Metropolis et al. [1] were carried out in the canonical ensemble corresponding to a fixed number of molecules, volume and temperature, N, V, T). By contrast, molecular dynamics is naturally carried out in the microcanonical ensemble, fixed (N, V, E), since the energy is conserved by Newton s equations of motion. This implies that the temperature of an MD simulation is not known a priori but is obtained as an output of the calculation. This feature makes it difficult to locate phase transitions and, perhaps, gave the first motivation to generalize MD to other ensembles. 

This chapter makes a humble attempt to introduce some of the basic concepts of the Monte Carlo method for simulating ensembles of particles. Before we jump into the heart of the matter, we would like to direct the interested reader to a more general description of different Monte Carlo methods by Hammersley and Handscomb [1] and to the excellent textbooks on particle simulations by Allen and Tildesley [2], Binder and Heer-mann [3], and Frenkel and Smit [4]. 

The remaining aspect of a trajectory simulation is choosing the initial momenta and coordinates. These initial conditions are chosen so that the results from an ensemble of trajectories may be compared with experiment and/or theory, and used to make predictions about the chemical system s molecular dynamics. In this chapter Monte Carlo methods are described for sampling the appropriate distributions for initial values of the coordinates and momenta. Trajectories may be integrated in different coordinates and conjugate momenta, such as internal [7], Jacobi [8], and Cartesian. However, the Cartesian coordinate representation is most general for systems of any size and the Monte Carlo selection of Cartesian coordinates and momenta is described here for a variety of chemical processes. Many of... 

Molecular dynamics simulations have generally a great advantage of allowing the study of time-dependent phenomena. However, if thermodynamic and structural properties alone are of interest, Monte Carlo methods might be more useful. On the other hand, with the availability of ready-to-use computer simulation packages (e g.. Molecular Simulations Inc. 1999), the implementation of particular statistical ensembles in molecular dynamics simulations becomes nowadays much less problematic even for an end user without deep knowledge of statistical mechanics. 

The molecular dynamics method is conceptually simpler than the Monte Carlo method. Here again, we can compute various averages of the form (2.107) and hence the RDF as well. The method consists in a direct solution of the equations of motion of a sample of N (j= 10 ) particles. In principle, the method amounts to computing time averages rather than ensemble averages, and was first employed for simple liquids by Alder and Wainwright (1957) [see review by Alder and Hoover (1968)]. The problem of surface effects is dealt with as in the Monte Carlo method. The sequence of events is now not random, but follows the trajectory which is dictated to the system by the equations of motion. In this respect, this method is of a more general scope, since it permits the computation of equilibrium as well as transport properties of the system. 

The Metropolis method is the simplest importance sampling Monte Carlo method and for this reason it is a good starting point for the simulation of a complex system. However, it is also one of the least efficient methods and thus one will often have to face the question of how to improve the efficiency of the sampling. One of the most frequently used tricks is to employ a modified statistical ensemble within the simulation mn and to reweight the obtained statistics after the simulation. The simulation is performed in an artificial generalized ensemble. 

Conventional, generalized-ensemble Monte Carlo methods can also be employed, of course, but require sophisticated Monte Carlo updates to be efiScient [306,309],... 

Master equation methods are not tire only option for calculating tire kinetics of energy transfer and analytic approaches in general have certain drawbacks in not reflecting, for example, certain statistical aspects of coupled systems. Alternative approaches to tire calculation of energy migration dynamics in molecular ensembles are Monte Carlo calculations [18,19 and 20] and probability matrix iteration [21, 22], amongst otliers. 

Another method of simulating chemical reactions is to separate the reaction and particle displacement steps. This kind of algorithm has been considered in Refs. 90, 153-156. In particular. Smith and Triska [153] have initiated a new route to simulate chemical equilibria in bulk systems. Their method, being in fact a generalization of the Gibbs ensemble Monte Carlo technique [157], has also been used to study chemical reactions at solid surfaces [90]. However, due to space limitations of the chapter, we have decided not to present these results. 

The first Monte Carlo study of osmotic pressure was carried out by Panagiotopoulos et al. [16], and a much more detailed study was subsequently carried out using a modified method by Murad et al. [17]. The technique is based on a generalization of the Gibbs-ensemble Monte Carlo (GEMC) method applied to membrane equihbria. The Gibbs ensemble method has been described in detail in many recent reports so we will only summarize the extension of the method to membrane equilibria here [17]. In the case of two phases separated by semi-permeable membranes... 

---