Macroscopic properties, canonical ensemble

In MC methods the ultimate objective is to evaluate macroscopic properties from information about molecular positions generated over phase space. To evaluate average macroscopic properties, p, in the canonical ensemble from statistical mechanics, the following expression is used ... 

The grand canonical ensemble is appropriate for adsorption systems, in which the adsorbed phase is in equilibrium with the gas at some specified temperature. The use of a computer simulation allows us to calculate average macroscopic properties directly without having to explicitly calculate the partition function. The grand canonical Monte Carlo (GCMC) method as applied in this work has been described in detail earlier (55). The aspects involving binary fluid mixtures have been described previously in our Xe-Ar work (30). 

The Kirkwood—Buff (KB) theory of solution (often called fluctuation theory) employs the grand canonical ensemble to relate macroscopic properties, such as the derivatives of the chemical potentials with respect to concentrations, the isothermal compressibility, and the partial molar volnmes, to microscopic properties in the form of spatial integrals involving the radial distribution function. This theory allows one to obtain information regarding some microscopic characteristics of mnlti-component mixtures from measurable macroscopic thermodynamic quantities. However, despite its attractiveness, the KB theory was rarely used in the first three decades after its publication for two main reasons (1) the lack of precise data (in particular regarding the composition dependence of the chemical potentials) and (2) the difficulty to interpret the results obtained. Only after Ben-Naim indicated how to calculate numerically the Kirkwood—Buff integrals (KBIs) for binary systems was this theory used more frequently. 

The KB theory of solution [15] (often called fluctuation theory of solution) employed the grand canonical ensemble to relate macroscopic properties, such as the derivatives of the chemical potentials with respect to concentrations, the isothermal compressibility and the partial molar volumes to microscopic properties in the form of spatial integrals involving the radial distribution function. 

In statistical thermodynamics, a system with interacting particles is depicted with the canonical ensemble that describes a collection of a large number of macroscopic systems under identical conditions (for instance, N particles in a volume V at temperature T). In each system, laws that describe interactions between molecules are identical. They differ by the coordinates of each particular molecule corresponding to a microstate. The static picture of the canonical ensemble is equivalent to the development of a system over time [10,14]. In other words, the measurement of a macroscopic property reflects a succession of microstates. Thus, the measured property corresponds to a time-averaged mean value and thermodynamic equilibrium corresponds to the most probable macroscopic state. 

The probability law (2.2.1) characterizes the so called canonical ensemble ). The macroscopic equilibrium value. P of a property P whose value is Pr when the system is in the state r, is given by... 

The ideal gas law and the thermodynamic properties of an ideal gas are completely derived from the canonical ensemble partition function. This is a remarkable illustration of how statistical mechanics explains macroscopic observables in terms of microscopic properties. 

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