Ensemble dependence

A specific example will be sufficient to fix these ideas. Consider calculations in an isothermal-isobaric ensemble (Shing and Chung, 1987 Smith, 1999)  

Simulation calculations on finite systems will entail some error associated with the submacroscopic size considered (Lebowitz and Percus, 1961a b 1963). For example, periodic boundary conditions will influence molecular correlations to some extent (Pratt and Haan, 1981a b). Support of a claim of accuracy would typically involve some practical investigation of the thermodynamic limit. A claim of preference for calculations in one ensemble over another is typically made first on the basis of convenience rather than on the basis of accuracy defined in some absolute way. Thus, advantages of practical accuracy for ensemble-specialized alternatives to Eq. (3.18) are not proven typically, and they are not necessary fundamentally. 

But notice that V, in this example, introduces a fluctuating global variable into a formula otherwise involving the more local quantity thus, introduction 

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Lebowitz, J. L. Percus, J. K. Verlet, L., Ensemble dependence of fluctuations with application to machine computations, Phys. Rev. 1967,153, 250... 

The statistical behavior of interest is encapsulated in the equilibrium probability density function P )( q c). This PDF is determined by an appropriate ensemble-dependent, dimensionless [6] configurational energy 6( q, c). The relationship takes the generic form... 

Each structural model must be chosen so as to match a particular set of observations. That is, the symmetry of the model of a molecule or of a molecular ensemble depends on the conditions of the relevant physical (or chemical) measurement, and may vary for the same system according to time scale of observation and instrumental sensitivity. Hence, whether the model of a chemical system is chiral or achiral depends on the conditions of observation. In what... 

Since percolation is a property of macroscopic many-particle systems, it can be analyzed in terms of statistical mechanics. The basic idea of statistical mechanics is the relaxation of the perturbed system to the equilibrium state. In general the distribution function p(p,q t) of a statistical ensemble depends on the generalized coordinates q, momentum p, and time t. However, in the equilibrium state it does not depend explicitly on time [226-230] and obeys the equation... 

The fluctuation relations for the various viscosity coefficients involve time correlation functions. Some of them are ensemble dependent and others are ensemble independent. Only correlation functions the components of which couple with the angular velocity of the director or its conjugate torque density... 

Figure 11 The shear viscosity for liquid decane at T = 480 K and P = 170 atm calculated three different ways NEMD-NVT, NEMD-NPX and by means of the Green-Kubo formula (Eqs. [198]). The Green-Kubo value is actually at y = 0 but is placed on the y axis as a guide to the eye. The linear regime is reached when y 0.005 ps". Note the ensemble-dependent results in the shear thinning regime.

The main aim of this contribution is to demonstrate the severe ensemble dependence of near-critical thermodynamic properties of the RPM and AHS fluid, and to examine the effects of ion pairing - or rather the lack thereof - on the criticality in ionic fluids. We will deal with these topics in turn, and then present a summary and discussion in Sec. 5. 

The kinetics of Ca -transport, as studied by spectrophotometric techniques, show a fast and a slow phase the latter, lying in the range of seconds, can be identified with the translocation of Ca across the membrane ". Synchronous triggering of the ensemble of Ca -ATPase molecules within a oriented multilayer of membranes can be achieved by flash photolysis of caged ATP. The time-scale of the effective synchronization of the ensemble depends on the duration of the UV-light flash required to produce a sufficient quantity of ATP and is ultimately limited to the millisecond range due to the kinetics of the dark-reaction of the photolytic process. 

The one-field model gives a description of the approximate grand canonical ensembles introduced in Chapter 11, Section 5. These ensembles, called equilibrium ensembles , depend on only two fugacities, which determine respectively the average number of polymers and the average number of monomers (constituting the polymers). In the continuous case, a connected partition function J(/, a) can be associated with this ensemble it is defined by... 

Thus it is necessary to use the appropriate ensemble-dependent expressions in calculating the heat capacity, compressibility or thermal expansion. 

It has been argued that the unzipping transition for the quenched averaged RANI case is second order [41]. However for real DNA, it is not the quenched averages that matter. There is strong ensemble dependence and sample to sample variation. This has been exploited to identify point mutants by a comparison of the unzipping force in a fixed distance ensemble [42]. An experimental determination of the unzipping phase boundary for a real DNA has been reported in Ref. [43]... 

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. 

An ensemble is defined by the density of states p included in it. In principle all states that satisfy the imposed external conditions are considered to be members of the ensemble. There are three types of ensembles the microcanonical, canonical and grand canonical. The precise definition of the density p for each ensemble depends on the classical or quantum mechanical nature of the system. 

Electrochemistry at nano-ITIES arrays (or ensembles depending on their periodicity) was pioneered by Dryfe et as summarized in an excellent... 

The entropy S of the microcanonical ensemble depends only on the variables E, V, and N. Thus, the differential dS can be written... 

Expressions similar to those given above may be derived easily from partition functions in other ensembles.The choice of ensemble is very important in calculations of hydration entropy, enthalpy, and heat capacity, as discussed below. Many other quantities, including all free energies, are ensemble invariant, with the choice of ensemble affecting only system size dependence. For simplicity, the discussion here is therefore limited to the canonical ensemble except in such cases where a true ensemble dependence exists. 

Results of entropy and energy calculations are strongly ensemble-dependent, a fact not always appreciated in the literature. In particular, it can be shown readily that the difference between the constant-pressure solvation energy AE)p and the constant volume solvation energy AE)y is given by... 

In their study of krypton hydration, Durell and Wallqvist also reported a calculation of the enthalpy of hydration evaluated by the direct method of Eq. [31]." Both constant volume and constant pressure enthalpies were determined by varying the volume of the krypton solution. Their results are displayed in Table 1. The enthalpy of hydration in the constant volume case (—6.3 1.3 kcal/mol) is significantly more exothermic than in the constant pressure case (—3.4 1.3 kcal/mol). The latter number agrees very well with the experimental value of —3.3 kcal/mol, also obtained at constant pressure. The calculated enthalpies of solvation were decomposed into solute-water and water-water (solvent reorganization) terms. The solute-water contribution is comparable and favorable (—5.4 kcal/mol) in both the constant volume and constant pressure calculations. The solvent reorganization term, in contrast, shows a large ensemble dependence. In the constant-pressure case, the solvent reorganization term has a value of 2.0 1.3 kcal/mol. The overall favorable enthalpy of hydration of krypton at constant pressure therefore results from the solute-water attractions rather than from a... 

Hie properties of NP ensembles depend on their aggregation number. Therefore, it is important to predict and control the number of NPs organized in ID polymer-like stmrtures. For example, significant effort has been applied to achieving... 

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