Ensemble method

In this chapter, the foundations of equilibrium statistical mechanics are introduced and applied to ideal and weakly interacting systems. The coimection between statistical mechanics and thennodynamics is made by introducing ensemble methods. The role of mechanics, both quantum and classical, is described. In particular, the concept and use of the density of states is utilized. Applications are made to ideal quantum and classical gases, ideal gas of diatomic molecules, photons and the black body radiation, phonons in a hannonic solid, conduction electrons in metals and the Bose—Einstein condensation. Introductory aspects of the density... 

The Gibbs ensemble method has been outstandingly successfiil in simulating complex fluids and mixtures. 

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... 

Several techniques are available to calculate dJf /dQ. Ciccotti and coworkers [17, 19-27] have developed a technique, called blue-moon ensemble method or the method of constraints, in which a simulation is performed with fixed at some value. This can be realized by applying an external force, the constraint force, which prevents from changing. From the statistics of this constraint force it is possible to... 

For buried solvent molecules, open ensemble methods should be helpful, although extension of the existing methods to allow for solute flexibility is needed.18... 

Particle size measurement, 18 132-156 data representation in, 18 133-138 distribution averages in, 18 134-136 ensemble methods for, 18 151-154... 

Wittig, C., Nadler, I., Reisler, H., Noble, M., Catanzarite, J., and Radhakrishnan, G. (1985). Nascent product excitations in unimolecular reactions The separate statistical ensembles method, J. Chem. Phys. 83, 5581-5588. 

Figure 16 shows the gas-liquid phase diagrams calculated with DHH+ DHHDS [65, 81], compared to the ones calculated from MSA and from BB+ODS. Also include the data of Loth et al. [102] and those obtained from the Gibbs ensemble method by Panagiotopoulos [103]. Note that the reported values obtained from DHH+DHHDS are only estimates, because as said above, this approximation, even if accurate, exhibits little thermodynamic inconsistencies. As... 

Indirect Methods Test particle method Grand canonical ensemble method Biased sampling methods Thermodynamic integration... 

We now introduce the idea of the multicanonical technique into the isobaric-isothermal ensemble method and refer to this generalized-ensemble algonthnmsthemultibaric-multithermalalgorithm(MlJBAT i) [62,63,123-125]. 

Wongkoblap et al.307 study Lennard-Jones fluids in finite pores, and compare their results with Grand canonical ensemble simulations of infinite pores. Slit pores of 3 finite layers of hexagonally arranged carbon atoms were constructed. They compare the efficiency of Gibbs ensemble simulations (where only the pore is modelled) with Canonical ensemble simulations where the pore is situated in a cubic cell with the bulk fluid, and find that while the results are mostly the same, the Gibbs ensemble method is more efficient. However, the meniscus is only able to be modelled in the canonical ensemble. 

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. 

In the first part of this chapter we will give a short overview of Monte Carlo simulations for classical lattice models in Sect. 2.1 and will then review the extended and optimized ensemble methods in Sect. 3. We will focus the discussion on a recently developed algorithm to iteratively achieve an optimal ensemble, with the fastest equilibration and shortest autocorrelation times. 

The tunneling problem at first order phase transitions and for disordered systems, where tunneling times often diverge exponentially can be overcome using extended ensemble methods, which are the main topic of the next chapter. 

Extended ensemble methods, such as the multicanonical ensemble, Wang-Landau sampling or parallel tempering can also be generalized to quantum systems [35,36], as we will show in the next two sections. 

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