Macroscopic properties, canonical

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). 

For most practical applications, we consider not isolated systems but systems in which the temperature is fixed. This is the canonical system consistent with the fixed macroscopic properties (N, V, T), where T is the absolute temperature. 

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. 

We shall treat more compUcated cases, such as systems with a larger number of identical or different sites, and also cases of more than one type of ligand. But the general rules of constructing the canonical PF, and hence the GPF, are the same. The partition functions, either Q or have two important properties that make the tool of statistical thermodynamics so useful. One is that, for macroscopic systems, each of the partition functions is related to a thermodynamic potential. For the particular PFs mentioned above, these are... 

The canonical partition function is linked, firstly to the molecular canonical functions, and secondly to the thermodynamic functions that define the phase on the macroscopic level (U, F, G, S, etc.). These two types of relation mean that the canonical partition function forms the link between the microscopic definition of the phase and its macroscopic thermodynamic properties. 

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... 

What is now the advantage of the microcanonical temperature compared to its canonical counterpart Since ( ) is directly derived from the fundamental system quantities S and E, its curve E) should contain all information about the system behavior if it undergoes a macroscopic change such as a cooperative transition. It will turn out that transitions occur if /3( ) responds least sensitively to changes in system energy. By means of this least-sensitivity principle, it is possible to identify transitions uniquely, even in small systems. This also means that a transition point can be uniquely assigned a single transition temperature - this is typically not possible in the canonical formalism, where the transition point depends on the fluctuation extremum of the chosen order parameter. In contrast to the canonical counterpart, the microcanonical temperature is a system property and not an external parameter that can be controlled. 

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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