Ensemble mixed isostress isostrain

Therefore, in the classic limit, thermal averages in the grand mixed isostress isostrain ensemble may be cast as... 

Figure 5.20 Shear modulus C44 as a function of shear strain asxo- (O) MC simulations in grand mixed isostress isostrain ensemble (—) representation of small-strain approximation C44 (asxo) = 00 + 02 (asxo) (see Eqs. (5.113) and (5.114)].

To obtain a more concise picture of thermodynamic stability of different film morphologies, we plot grand mixed isostress isostrain ensemble, we calculate (j> directly from Eq. (5.119) using the molecular expression for Tyy [see Eq. (5.85)], which does not contain any fluid substrate contribution between the fluid substrate... 

Figure 5.22 (a) Normal compressional stress (see Appendix E.3 for molecular expressions) as a function of substrate separation from GCEMC simulations (O) (o.Sxo = 0.0). Solid lines aie intended to guide the eye. (b) As (a) but for [see Eq. 5.119]. Intersections between the latter and the vertical lines demarcate (meta- or thermodynamically) stable states in the grand mixed isostress isostrain ensemble for = 0.0 (see text). 

Figure 5.25 (a) as a fimction of shear strain qSxO for mono- (O), bi- (A), and trilayer (-I-) morphologies calculated in grand mixed isostress isostrain ensemble... 

Table 5.5 Overview of fluctuation-related response coeflScients in the mixed isostress isostrain ensemble at constant T, N, T, and s o (see text for details of the derivations).

The discussion of the Joule-Thonison effect in the previous section clearly showed that it is advantageous in theoretical treatments of confined fluids to tackle a given physical problem by a combination of different methods. This was illustrated in Section 5.7 whore wo employed a virial expansion of the equation of state, a van der Waals type of equation of state, and MC simulations in the specialized mixed isostress isostrain ensemble to investigate various aspects of the impact of confinement on the Joule-Thomson effect. The mean-field approach was particularly useful because it could predict certain trends on the basis of analytic equations. However, the mean-field treatment developed in Sections 4.2.2 and 5.7.5 is hampered by the assump-... 

This being the primary goal of the subsequent discussion we would also like to emphasize two other, perhaps more practical, aspects. On account of the rigidity of the underl3dng lattice it seems inconceivable to develop mixed isostress isostrain ensembles suitable for lattice MC simulations. On the other hand, lattice simulations are computationally much less demanding because molecules can occupy only discrete positions in. space. Hence, the number of configurations possible on a lattice is greatly reduced compared with simulations of continuous model systems. 

To this end we performed MC simulations in the mixed isostress isostrain ensemble introduced in Section 5.7 where a thermodynamic state is specified... 

The demonstration of equivalence departs from the observation that partition functions in any statistical physical ensemble comply with the general form given in Eq. (B.52) with individual summands as given in Eq. (B.54). Thus, according to th( discussion in Appendix B.4, wo may approximate the grand mixed isostress isostrain ensemble partition functiou x b (2.70a) b> the maximum terms in the sums appearing on the right side of that equation... 

According to the above discussion the distribution of microstates in the current mixed isostress isostrain ensemble is governed by the probability density given in Eq. (5.193). The similarity between the present probability density and the one relevant in the closely related ensemble discussed in Section 5.2.4 suggests we should design an adapted Metropolis algorithm closely related to the one described in that section. In fact, from the detailed discussion in Section 5.2.4, it turns out that we just need to replace the substrate separation Sj ly the area. 4. More specifically, we need to replace Elqs. (5.46)-(5.48)... 

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