Expanded ensembles

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... 

Lyubartsev, A. P., Jacobsson, S. P., Sundholm, G., Laaksonen, A. Water/ octanol systems. A expanded ensemble molecular dynamics simulation smdy of log P parameters. J. Phys. Chem. B 2001, 105, 7775-7782. 

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... 

Extended sampling strategies utilizing this kind of representation appear in the literature with a variety of titles expanded ensemble [23], simulated tempering [24], and temperature scaling [25]. 

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],... 

To do so, one must determine a set of weights that moderate the relative contributions of each histogram. There is more than a passing similarity here to the expanded ensemble framework... 

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... 

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. 

To achieve uniform sampling of the different canonical ensembles within an expanded ensemble simulation, the weights should ideally obey... 

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. 

Several factors affect the performance of HPTMC. First, factors that affect the performance of the underling expanded ensemble simulation clearly influence the performance of HPTMC. With regard to HPTMC itself, the frequency and success rate of configuration swaps are the most important factors. A simple rule-of-thumb that we have adopted in these applications is to make the frequency of successful swaps of the same order of magnitude as the frequency of successful particle insertions/removals. Note, however, that simulations of different complex fluids are likely to require some fine-tuning to arrive at optimal parallel tempering algorithms for complex fluids. 

Lyubartsev AP, Jacobsson SP, Sundholm G, Laaksonen A. Solubility of organic compounds in water/octanaol systems. An expanded ensemble molecular dynamics simulation study of log P parameters. J Phys Chem B 2001 105 7775-7782. 

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. 

Each state of the expanded ensemble is equilibrated as usual (e.g., through molecular rearrangements). However, a new type of trial move is now introduced trial transitions between neighboring states. Such trial moves are subject to a Metropolis-like acceptance criterion ... 

The efficiency of expanded ensemble simulations can be further increased by implementing a configurational-bias scheme for the insertion and deletion processes associated with the segmental coupling-decoupling of the tagged chain [76]. 

The main advantage of the expanded ensemble approach is that it provides a natural framework for inserting and removing chain molecules from a system. By facilitating such insertion-deletion moves, chemical potential equilibration can be effectively attained in grand canonical and Gibbs... 

Figure 3. Illustration of the transitions between states in a Expanded Ensemble simulation of the chemical potential. The tagged chain of variable length (anisotropic coupling) is depicted in black (solid circles).

Applications of the expanded-ensemble approach have been reported for the partitioning of polymers between a slit pore and the bulk [52,77,79], and for fluid-fluid equilibria [78-80]. Figure 4 shows results of expanded Gibbs ensemble simulations of phase coexistence in a binary system of Lennard-Jones chains dissolved in their own monomer [80] (cross-inter-... 

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. 

If a fiVT ensemble simulation can be turned into a ( quasi ) NPT ensemble-type simulation (e.g., a pseudo- FT ensemble), the inverse transformation (a pseudo-NPT ensemble) is also possible. The key relationship for a pseudo-NPT ensemble technique is Eq. (5.1) [78]. Such a reverse strategy can be practical only if molecular insertion and deletion moves can be performed efficiently for the system under study (e.g., by expanded ensemble moves for polymeric fluids). Replacing volume moves by particle insertions can be advantageous for polymeric and other materials that require simulation of a large system (due to the sluggishness of volume moves for mechanical equilibration of the system) such an advantage has been clearly demonstrated for a test system of dense, athermal chains [78]. 

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]. 

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