Isothermal ensemble

Statistical thermodynamics gives us the recipes to perform this average. The most appropriate Gibbsian ensemble for our problem is the canonical one (namely the isochoric-isothermal ensemble N,V,T). We remark, in passing, that other ensembles such as the grand canonical one have to be selected for other solvation problems). To determine the partition function necessary to compute the thermodynamic properties of the system, and in particular the solvation energy of M which we are now interested in, of a computer simulation is necessary [1],... 

Najafbadi and Yip (18) have investigated the stress-strain relationship in iron under uniaxial loading by means of a MC simulation in the isostress isothermal ensemble. At finite temperatures, a reversible b.c.c. to f.c.c. transformation occurs with hysteresis. They found that the transformation takes place by the Bain mechanism and is accompanied by sudden and uniform changes in local strain. The critical values of stress required to transform from the b.c.c. to the f.c.c. structure or vice versa are lower than those obtained from static calculations. Parrinello and Rahman (14) investigated the behavior of a single crystal of Ni under uniform uniaxial compressive and tensile loads and found that for uniaxial tensile loads less than a critical value, the f.c.c. Ni crystal expanded along the axis of stress reversibly. 

In the isobaric-isothermal ensemble [119-122], the probability distribution Pnpt(E, V T, V) for potential energy E and volume V at temperature T and pressure V is given by... 

This weight factor produces an isobaric-isothermal ensemble at constant temperature (T) and constant pressure (V), and this ensemble yields bell-shaped distributions in both E and V. 

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]. 

To perform the multibaric-multithermal MD simulation, we just solve the above equations of motion (4.29)-(4.34) for the regular isobaric-isothermal ensemble (with arbitrary reference temperature T = To and reference pressure V = Vo), where the enthalpy H is replaced by the multibaric-multithermal enthalpy 7mbt in (4.30) and (4.34) [63]. 

After an optimal weight factor Wmu(E, V) is obtained, a long production simulation is performed for data collection. We employ the reweighting techniques [8] for the results of the production run to calculate the isobaric-isothermal-ensemble averages. The probability distribution / npt(A, V T, V) of potential energy and volume in the isobaric-isothermal ensemble at the desired temperature T and pressure V is given by... 

We now present an example of MREM. We consider an isobaric-isothermal ensemble and exchange not only the temperature but also the pressure values of pairs of replicas during a MC or MD simulation [94]. Namely, suppose we have M replicas with M different values of temperature and pressure (Tm,Vm). We are setting Eo = E, V = V, and Am = Vm in (4.58). We exchange replicas i and j which are at (Tm,Vm) and (Tn,Vn), respectively. The transition probability of this replica-exchange process is then given by (4.48), where (4.60) now reads [3,80,96]... 

To outline the fundamental basis of the model, we follow the notation of Hill (10) and extend his derivation to a three component mixture. Component 1 is the solvent which in our case is water, component 2 is a solute or polyethylene glycol, and component 3 is another solute or dextran. We base the theory on an isobaric-isothermal ensemble first introduced by Stockmayer (14). This choice of ensemble is the most appropriate because it )delds expressions for the chemical potentials of the components with temperature, pressure, and solute molality or mole fraction as the natural independent variables, and these are the independent variables normally used in calculation, experiment, and industrial practice. 

The most useful connection between thermodynamics and statistical thermodynamics is that established for a system at a given temperature T, volume V, and the number of particles N. The corresponding ensemble is referred to as the isothermal ensemble or the canonical ensemble. To obtain the T, V, N ensemble from the E, V, N ensemble, we replace the boundaries between the isolated systems by diathermal (i.e., heat-conducing) boundaries. The latter permits the flow of heat between the systems in the ensemble. The volume and the number of particles are still maintained constant. 

The isobaric-isothermal ensemble and its close relative, the isotension-isothermal ensemble, are often used in Monte Carlo simulations. Finite-size... 

A straightforward, but tedious, route to obtain information of vapor-liquid and liquid-liquid coexistence lines for polymeric fluids is to perform multiple simulations in either the canonical or the isobaric-isothermal ensemble and to measure the chemical potential of all species. The simulation volumes or external pressures (and for multicomponent systems also the compositions) are then systematically changed to find the conditions that satisfy Gibbs phase coexistence rule. Since calculations of the chemical potentials are required, these techniques are often referred to as NVT- or NPT- methods. For the special case of polymeric fluids, these methods can be used very advantageously in combination with the incremental potential algorithm. Thus, phase equilibria can be obtained under conditions and for chain lengths where chemical potentials cannot be reliably obtained with unbiased or biased insertion methods, but can still be estimated using the incremental chemical potential ansatz [47-50]. 

The isobaric-isothermic ensemble is characterized by a fixed number of atoms (N), a fixed pressure (P), and a fixed temperature (T). This method is applicable to periodic systems only. The unit cell vectors are allowed to change, and the pressure is adjusted by adjusting the volume (the size and shape of the unit cell). Several methods are available to control pressure. Those of Berendsen et al. (1984) and Anderson (1980) only vary the size of the unit cell, whereas that of Parrinello and Rahman (1982) allows both the cell volume and its shape to change. NPT is the ensemble of choice when the correct pressure, volume, and densities are important in the simulation. This ensemble can also be used during equilibration to achieve the desired temperature and pressure before changing to the constant-volume or constant-energy ensemble when data collection starts. 

So far in this chapter, we have considered only systems in which T,V,N) are constant. In statistical mechanics, such a system is called the Canonical ensemble. Ensemble has two meanings. First, it refers to the collection of all the possible microstates, or snapshots, of the arrangements of the system. We have counted the number of arrangements W of particles on a lattice or configurations of a model polymer chain. The word ensemble describes the complete set of all such configurations. Ensemble is also sometimes used to refer to the constraints, as in the T, V,N) ensemble. The (T, V,N) ensemble is so prominent in statistical mechanics that it is called canonical. A system constrained by (T, p,N), is called the isobaric-isothermal ensemble. 

For the binary system silica-calcia, the calculations were carried out hundreds of degrees above the melting in order to compare our results with available experimental measurements. The accuracy of the model was checked using experimental data from Ref. [21 ]. The evolution of the RDF for the system Si02-Ca0 calculated at r= 2000 K for a molar composition of 1 1 and calculated from an isobaric-isothermal ensemble is shown in Figure 3.4.4. Our values are summarized in Table 3.4.5 and compared with experimental data from Ref. [21]. 

The probability derrsity p for a classical isobaric-isothermal ensemble is given by... 

Asstrming the Gibbs entropy of the isobaric-isothermal ensemble is equivalent to the thermodynamic entropy, the average energy (H) and average voltrme (E) are equivalent to their thermodynamic analogues, p = l/k T, and the macroscopic parameter P is equivalent to the pressure, we can write... 

As for classical systems, the isobaric-isothermal ensemble is used to characterize the macroscopic state of a closed, isothermal qirantum system with a variable volume and fixed pressme. The semiclassical (part classical, part quantum) density operator p for a qirantum isobaric-isothermal ensemble is given by... 

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