Gibbsian statistical mechanics

The potential distribution theorem has been around for a long time [13-17], but not as long as the edifice of Gibbsian statistical mechanics where traditional partition functions were first encountered. We refer to other sources [10] for detailed derivations of this PDT, suitably general for the present purposes. 

Our perspective is that the PDT should be recognized as directly analogous to the partition functions which express the Gibbsian ensemble formulation of statistical mechanics. From this perspective, the PDT is a formula for a thermodynamic potential in terms of a partition function. Merely the identification of a... 

This argument has descended onto the equation of state as a principal determinant of peculiar temperature dependences of hydrophobic effects. The statistical thermodynamic model discussed above, however, started with probabilities and fluctuations. But equations of state and fluctuations are connected by the most basic of results of Gibbsian statistical mechanics, e.g. Eq. (2.24), p. 27. Ad hoc models, such as the more simplistic lattice gas models, can be adjusted to agree with solubility at a thermodynamic state point, but if they don t agree with the equation of state of liquid water more broadly they can t be expected to describe molecular fluctuations of liquid water consistently and realistically. Thus, models of that sort are unlikely to be consistent with the picture explored here. 

Before we introduce the concept of an open system, it is useful to discuss the specific heat of the subsystem itself. The Hamiltonian "Xg in Equation 11.7 does qualify for describing the prototype thermodynamics of a smah quantum system, like a harmonic oscillator. What we have to do is to imagine Jig to be weakly coupled to a classical heat bath, with which the system undergoes exchange of energy. The consequent energy fluctuations provide the temperature of the system. All this can be put into statistical mechanical perspective in terms of the Gibbsian partition function... 

The entropy is calculated from the number of conformations available to the molecules. This sounds simple but we will see that there are many difficulties which have to be solved first. In this chapter we restrict ourselves very much to fundamental theoretical problems based on molecular pictures, rather than on attempts to resolve them phenomenologically. This requires the generalization of Gibbsian statistical mechanics to cover the existence of permanent constraints, mathematical realization of chains, crosslinks and entanglements, and discussion of applications to real systems. 

Note that the Gibbsian terms have a whilst the permanent constraints have a Y[ over the replicas, stating that the potential can be everywhere, whilst the crosslinks are permanent. The statistical mechanics of this system have been studied extensively and a detailed discussion of the analysis is beyond of the scope of this text and we only summarize the outline and the main results. For simplicity consider the interaction term chosen as a pseudopotential jji 5[if(s) -- if(s )]dsds to account for excluded volume forces, v is again the excluded volume parameter. This approximation of the potential is very limited and can only be justified in various concentration regimes. 14,179, iso This leads to the generalized partition function... 

Being a contemporary of the nineteenth century, Gibbs could obviously not have had any concept of quantum mechanics and its role in laying a sound foundation of modern statistical thermodynamics. In this modern formulation, the classic Gibbsian version of statistical thermodjmamics does, however, emerge as a limiting case as our discussion in Section 2.5 reveals. 

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