Canonical approach

Due to its more complicated structure the canonical expansion is not well adapted for a discussion of general features like the properties of the theory under renormalization. Thus in the sequel we exclusively work with the grand canonical approach. 

Gaussian-like distribution of energy around the energy average. Other ensembles with non-Boltzmann distributions can enhance the sampling considerably for example, in the multi-canonical approach [97, 98], all the conformations are equiprobable in energy in Tsallis statistics [99], the distribution function includes Boltzmann, Lorentzian, and Levy distributions. 

Nowadays the study of a reaction mechanism may be done by performing a well determined sequence of computational steps we define this sequence as the canonical approach to the study of chemical reactions. At first, one has to define the geometry of reagents and products, then that of other locally stable intermediates, especially those acting as precursors of the true reaction process, and finally that of the transition state or states and of the reaction intermediates, if any. The determination of these geometries will of course be accompanied by the computation of the relative energies. All the points on the potential energy hypersurface we have mentioned are stationary points, defined by the condition ... 

On grounds of the solar abundances of [40], it has been demonstrated in [37] that the derived MES distribution of neutron irradiation agrees qualitatively with the exponential distributions assumed in the canonical model, even though some deviations are noticed with respect to the canonical weak and strong components.5 The MES provides an excellent fit to the abundances of the 35 nuclides included in the considered set of species, and in fact performs to a quite-similar overall quality as that of the exponential canonical model predictions of [40]. Even a better fit than in the canonical framework is obtained for the s-only nuclides (see [37] for details). The MES model is therefore expected to provide a decomposition of the solar abundances into their s- and r-components that is likely to be more reliable than the one derived from the canonical approach for the absence of the fundamental assumption of exponential distributions of neutron exposures. 

For Ni [6], Cu [11,12], Pt [13,14] and other fcc-crystallizing metals, one usually considers three canonical-approach sites atop, bridge and threefold, this latter being either of hollow (fee) or filled (hep) type, with the fee site being usually more favourable than the hep site to the chemisorption of single atoms or small molecules. 

There are mainly historical reasons why the canonical approach, as discussed in the previous section, has been used most frequently for the analysis of structural and phase transitions. This is because the exponential form of the canonical microstate probability (2.22) in suitable for the development of approximation methods and field-theoretical formulations which enable analytic calculations of thermodynamic quantities. 

Before computers and efficient simulation methods became available, the canonical approach was the only way for a theoretical discussion of thermodynamic phenomena. On the experimental side, it was appealing to use the heat bath temperature as an external control parameter. Because of this, in equilibrium, temperature seemed to be an easily accessible parameter to control the macrostate of the system, and transition points of thermally driven phase transitions are typically defined by transition temperatures. The canonical analysis of phase transitions has been extremely successful and it enabled the introduction of fundamental physical concepts such as universality. 

However, the uniqueness of the canonical approach can only be maintained as long as the investigated system fulfills the requirement of the thermodynamic limit, i.e., if finiteness, in particular surface effects, do not matter. Thus, the rapidly grown interest in the structural behavior of notoriously finite systems such as biomolecules confronted the thermodynamic analysis with the problem, to what extent a canonical analysis is still appropriate for the discussion of cooperative thermal behavior associated with structure formation processes such as protein folding or aggregation. Because of its simplicity, it is still popular to also apply the conventional canonical approach to such systems. However, the obvious violation of the thermodynamic limit condition leads to conceptual problems. Even the imagination of what temperature is, is strongly affected and requires a careful consideration. 

Garel, T., Monthus, C. and Orland, H. (2000). Copolymer at a Selective Interface and Two-dimensional Wetting a Grand Canonical Approach, Eur. Phys. J. B 17, pp. 121-130. 

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