Expanded ensemble method

In order to estimate the free energy many canonical simulations at different temperatures are necessary furthermore, it is often difEcult to define a suitable reference state with a known entropy Sq. Two alternatives can be followed to overcome these difficulties (i) expanded ensemble methods and (ii) multicanonical methods. 

The expanded-ensemble method has been shown to be capable of handling molecules with O(102) sites. The obvious disadvantages of the method are that preliminary, iterative simulations are sometimes needed and that, as the number of sites per molecule increases, the number of intermediate states must increase accordingly. 

Escobedo and de Pablo have proposed some of the most interesting extensions of the method. They have pointed out [49] that the simulation of polymeric systems is often more troubled by the requirements of pressure equilibration than by chemical potential equilibration—that volume changes are more problematic than particle insertions if configurational-bias or expanded-ensemble methods are applied to the latter. Consequently, they turned the GDI method around and conducted constant-volume phase-coexistence simulations in the temperature-chemical potential plane, with the pressure equality satisfied by construction of an appropriate Cla-peyron equation [i.e., they take the pressure as 0 of Eq. (3.3)]. They demonstrated the method [49] for vapor-liquid coexistence of square-well octamers, and have recently shown that the extension permits coexistence for lattice models to be examined in a very simple manner [71]. 

The EXEDOS method that we use combines an expanded ensemble formalism with a density-of-states [9,10] scheme for the determination of the potential of mean force. In the expanded ensemble method, besides the canonical variables (number of mesogens N, volume V, and temperature T) the state of the system is also labelled by the value of C coordinate we consider M intervals of width S of the reaction coordinate, then the system is said to be in state m ii m - 1 < /5 < m. For suflSciently narrow intervals, we can associate the midpoint Cm = (m — 1/2)5 as the representative value. 

Lyubartsev A P, MartsInovskI A A, Shevkunov S V and Vorontsov-Velyamlnov P N 1992 New approach to Monte Carlo calculation of the free-energy—method of expanded ensembles J. Chem. Phys. 96... 

Fenwick, M. K. Escobedo, F. A., Expanded ensemble and replica exchange methods for simulation of protein-like systems, J. Chem. Phys. 2003,119, 11998-12010... 

The ESPS method draws on and synthesizes a number of ideas in the extensive free-energy literature, including the importance of representations and space transformations between them [63, 68, 69], the utility of expanded ensembles in turning virtual transitions into real ones [23], and the general power of multicanonical methods to seek out macrostates with any desired property [27],... 

A. P. Lyubartsev, A. A. Martsinovski, S. V. Shevkunov and P. N. Vorontsov-Velyaminov (1992) New Approach to Monte-Carlo Calculation of the Free-Energy - Method of Expanded Ensembles. J. Chem. Phys. 96, p. 1776 E. Marinari and G. Parisi (1992) Simulated Tempering - A New Monte-Carlo Scheme. Europhysics Lett. 19, p. 451... 

The goal of the method is to perform a random walk in space. Consider a system consisting of N particles interconnected to form a molecule, and having volume V and temperature T. The end-to-end distance of the molecule ( ) can be discretized into distinct states each state is characterized by its end-to-end distance, in some specified range of interest [, +j and represent a lower and an upper bound, respectively. The partition function 17 of this expanded ensemble is given by... 

Chang, J. Sandler, S.I. Determination of liquid-solid transition using histogram reweighting method and expanded ensemble simulations. J. Chem. Phys. 2003, 118, 8930. 

At liquid-like densities, the configurational-bias ghost particle approach is reliable only for chain molecules of intermediate length [71]. This limitation can be partially overcome by applying other techniques, such as the method of expanded ensembles. 

This chapter is organized as follows. In section 1.1, we introduce our notation and present the details of the molecular and mesoscale simulations the expanded ensemble-density of states Monte Carlo method,and the evolution equation for the tensor order parameter [5]. The results of both approaches are presented and compared in section 1.2 for the cases of one or two nanoscopic colloids immersed in a confined liquid crystal. Here the emphasis is on the calculation of the effective interaction (i.e. potential of mean force) for the nanoparticles, and also in assessing the agreement between the defect structures found by the two approaches. In section 1.3 we apply the mesoscopic theory to a model LC-based sensor and analyze the domain coarsening process by monitoring the equal-time correlation function for the tensor order parameter, as a function of the concentration of adsorbed nanocolloids. We present our conclusions in Section 1.4. 

Chem. Phys., 96, 1776 (1992). New Approach to Monte Carlo Calculation of the Free Energy Method of Expanded Ensembles. 

Just as configurational bias moves, parallel tempering methods come in a number of flavors. We begin by discussing briefly Parallel Tempering in one dimension, and we then proceed to discuss its implementation in several dimensions or in combination with expanded ensemble ideas. 

In the particular case of polymeric molecules, it is also possible to combine multidimensional parallel tempering and expanded ensemble ideas into a powerful technique, which we call hyperparallel tempering (HPTMQ [47], In that method, a mutation along the number of sites of a tagged chain is also considered (see Section IV.B.l) Fig. 19 provides a schematic representation of the algorithm. Two of the axes refer to replicas having different chemical potentials or different temperatures. A third axis or dimension is used to denote the fact that, in each replica of the system. 

Reller H, Kirowa-Eisner E, Gileadi E (1984) Ensembles of microelectrodes Digital simulation by the two-dimensional expanding grid method. Cyclic voltammetry, iR effects and applications. J Electroanal Chem 161 247. 

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