Grand-canonical and isothermal-isobaric ensembles

We are now in a position to introduce other ensembles. Two interesting ones for practical applications are the grand-canonical ensemble and the isothermal-isobaric ensemble. 

A member system of the grand-canonical ensemble has constant volume and temperature, and it can exchange particles with the environment. At equilibrium, net diffusion of particles stops when the chemical potential of the particles inside the system is the same as outside. The grand-canonical ensemble is also called the (xUr ensemble (Fig. 6.3). This ensemble is useful for calculating phase equilibria of different systems (see Further reading). 

The probabUity of an open, isothermal system at equilibrium, with constant p.TT, having Nk number of particles and energy Ei can be shown to be 

Note that H is the grand-canonical partition function 

We can associate the grand-canonical and the canonical partition functions as follows  

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CANONICAL, GRAND CANONICAL, AND ISOTHERMAL-ISOBARIC ENSEMBLES [2, 3]... 

Methods for simulation of the liquid-vapour coexistence are well developed and were reviewed by Panagiotopoulos263 however in some cases these have been shown to be sensitive to the potential energy surface and factors such as many-body interactions, and therefore new results continue to be obtained to investigate these issues and to study new systems (see for example,110 for mercury,264 for methane, and265 for water). The Gibbs ensemble approaches and grand canonical and isothermal-isobaric MC simulations with histogram... 

In Ihc canonical, microcanonical and isothermal-isobaric ensembles the number of particles is constant but in a grand canonical simulation the composition can change (i.e. the number of particles can increase or decrease). The equilibrium states of each of these ensembles are cha racterised as follows ... 

Similar results can be obtained for the other three ensembles of Fig. 5.1 canonical (CE), grand canonical (GCE), and isothermal-isobaric ensembles (HE). We collect all the results and add a few lines about a fifth ensemble, the generalized ensemble (GE). 

There are several conditions in which molecular dynamics simulations can take place. These include the microcanonical ensemble (NV ), canonical ensemble (NVT), isothermal-isobaric ensemble (NPT), and grand canonical ensemble (p-VT), where N is the number of particles, V is the volume, E is the energy, P is the pressure, T is the temperature, and x is the chemical potential. In each of these ensembles, the thermodynamic variables held constant are designated by the appropriate letters. 

The MD simulations can be perfomed in maity different ensembles, such as grand canonical (pVT), microcanonical (NVE), canonical (NVT) and isothermal-isobaric (NPT). The constant temperature and pressure can be controlled by adding an appropriate thermostat (e g., Berendsen, Nose, Nose-Hoover, and Nose-Poincare) and barostat (e.g., Andersen, Hoover, and Berendsen), respectively. Applying MD into polymer composites allows us to investigate into the effects of fillers on polymer stracture and dynamics in the vicinity of polymer-filler interfaee and also to probe the effects of polymer-filler interactions on the materials properties. 

The basic idea is to develop expressions for common thermodynamic quantities in terms of fluctuations in a system open to all species. The key lies in the fact that the fluctuating quantities characteristic of the grand canonical ensemble can then be transformed into expressions, which provide properties representative of the isothermal isobaric ensemble. Using an equivalence of ensembles argument, one can then consider these fluctuations to represent properties of small microscopic local regions of the solution of interest. This approach can be used to understand many properties of isobaric, isochoric, or osmotic systems in terms of particle number (and energy) fluctuations. 

Simulation protocols begin by specifying the statistical ensemble used for the system. The four most commonly used statistical ensembles in this area of research are the microcanonical (E, V, N), the canonical (T, V, N), the isothermal-isobaric (T, P, N), and the grand canonical (T, V, p.) ensembles, where E is total energy, N is the total number of molecules, T is the temperature, P the pressure, V the volume, and p the chemical potential of the system. Depending on the chosen ensemble, the three listed thermodynamic quantities are required to be conserved throughout the simulation. All these ensembles are implemented with both MC and MD methods. 

G[T P] = G = —RTIn A, where A is the isothermal-isobaric partition function and U[T,n ] = —RTInE, where S is the grand canonical ensemble partition function. When a system involves several species, but only one can pass through a membrane to a reservoir, L/(7 jux] = — PTlnT, where T is the semigrand ensemble partition function. The last chapter of the book is on semigrand partition functions. 

There are many excellent reviews on the standard molecular dynamics method dealing with calculations in the microcanonical ensemble as well as on the Monte Carlo method involving calculations in the canonical, isothermal isobaric, and grand canonical ensemble (< ). In the present article, we shall limit ourselves exclusively to those developments that have taken place since the work of Andersen (4). In the molecular dynamics method, the developments are the constant-pressure, constant-temperature, constant-temperature-constant-pressure, variable shape simulation cell MD, and isostress calculations in the Monte Carlo method, it is the variable shape simulation cell calculation. 

In the last section we have assumed that we perform our simulation for a fixed number, N, of particles at constant temperature, T, and volume, V, the canonical ensemble. A major advantage of the Monte Carlo technique is that it can be easily adapted to the calculation of averages in other thermodynamic ensembles. Most real experiments are performed in the isobaric-isothermal (constant- ) ensemble, some in the grand-canonical (constant-pFT) ensemble, and even fewer in the canonical ensemble, the standard Monte Carlo ensemble, and near to none in the microcanonical (constant-NFE) ensemble, the standard ensemble for molecular-dynamics simulations. 

A. Canonical, Isobaric-Isothermal, and Grand Canonical Ensembles... 

The discussion so far has focused on the use of canonical ensemble averages like (2), in which the variables (N, V, T) are held fixed. That is the most common application of MC work to fluids. However, averages in any ensemble can be put into form (1) and evaluated using similar techniques. In practice computations have been carried out in the grand canonical ensemble [with (fi, V, T) fixed] and in the isothermal isobaric [with (N, P, 7) fixed]. In the former the number of particles N is a fluctuating quantity that must be sampled during the MC experiment, while in the latter this is true of the volume V. 

Abstract Fluctuation Theory of Solutions or Fluctuation Solution Theory (FST) combines aspects of statistical mechanics and solution thermodynamics, with an emphasis on the grand canonical ensemble of the former. To understand the most common applications of FST one needs to relate fluctuations observed for a grand canonical system, on which FST is based, to properties of an isothermal-isobaric system, which is the most common type of system studied experimentally. Alternatively, one can invert the whole process to provide experimental information concerning particle number (density) fluctuations, or the local composition, from the available thermodynamic data. In this chapter, we provide the basic background material required to formulate and apply FST to a variety of applications. The major aims of this section are (i) to provide a brief introduction or recap of the relevant thermodynamics and statistical thermodynamics behind the formulation and primary uses of the Fluctuation Theory of Solutions (ii) to establish a consistent notation which helps to emphasize the similarities between apparently different applications of FST and (iii) to provide the working expressions for some of the potential applications of FST. 

There are four main ensembles in statistical thermodynamics for which the independent variables are NVE (microcanonical), NVT (canonical), NpT (Gibbs or isothermal isobaric), and VT (grand canonical). The characteristic fnnetions provided in Equations 1.2 and 1.3 can be expressed in terms of a series of partition functions such that (Hill 1956)... 

Simulation of adsorption has been performed in various ensembles canonical, grand canonical, isobaric-isothermal, and Gibbs ensemble. The choice of the ensemble depends on the nature of the investigated system and the aim of the simulations. In the case of adsorption on heterogeneous surfaces, usually the grand canonical Monte Carlo simulation method (GCMC) has been used. 

In PI work on homogeneous and isotropic fluids one utilizes the three basic statistical ensembles canonical, isothermal-isobaric, and grand canonical. In the canonical ensemble the partition function is given by the trace (Tr) of the density matrix operator... 

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