Canonical ensemble, potential energy surfaces

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... 

It is very interesting to note that at first order the effect of the fictitious mass /i is a renormalization of atomic masses. If we consider a canonical ensemble at temperature T, this effect should not alter the static properties of the system which depend only on the potential energy surface o(R-)- However, the temperature of the system is usually defined from the kinetic energy of the ions ... 

Equations [305]-[310] are strictly valid only for thermodynamic species, which are ordinarily associated with stationary points on the potential energy surface V(R), where R denotes the full set of solute coordinates. However, we also use the SMx solvation models to calculate potentials of mean force, which are called W(R,T). The gradient of W(R,T) gives the force on the solute molecule averaged over a canonical ensemble of solvent molecules and is a generalization of the one-dimensional radial potential of mean force that appears in Debye-Hiickel theory. Thus, we write... 

In the canonical ensemble approach, the adsorbed phase is treated as a separate phase with a known volume and containing a fixed number of molecules at constant temperature. This approach is somewhat unrealistic even though the results obtained can be successfully correlated with experimental data [8]. If we consider that the gas—solid interactions induce a smooth gradient in the density of the gas as the surface is approached, another formahsm is necessary. The solution is obtained by adopting the grand canonical ensemble, in which the fixed variables are the volume, temperature and chemical potential. The unknown quantities would be, e.g., the number of molecules, energy, and pressure. The chemical potential of the adsorbed phase, once the equifibrium condition has been achieved, is equal to the chemical potential of the gas phase, which is determined from the density at a point far from the surface. The amount adsorbed, can be defined as the difference between the total number of molecules in the system, N, and N, the number of molecules in a hypothetical system of equal volume but with no gas—solid interactions. 

The grand canonical ensemble describes a system of constant volume, but capable of exchanging both energy and particles with its environment. Simulations of open systems under these conditions are particularly useful in the study of adsorption equilibria, surface segregation effects, and nanoscopically confined fluids and polymers. Under these conditions, the temperature and the chemical potentials jti,- of the freely exchanged species are specified, while the system energy and composition are variable. This ensemble is also called the jx VT ensemble. In the case of a one-component system it is described by the equilibrium probability density... 

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