Partition function open system

In the preceding section we have set up the canonical ensemble partition function (independent variables N, V, T). This is a necessary step whether one decides to use the canonical ensemble itself or some other ensemble such as the grand canonical ensemble (p, V, T), the constant pressure canonical ensemble (N, P, T), the generalized ensemble of Hill33 (p, P, T), or some form of constant pressure ensemble like those described by Hill34 in which either a system of the ensemble is open with respect to some but not all of the chemical components or the system is open with respect to all components but the total number of atoms is specified as constant for each system of the ensemble. We now consider briefly the selection of the most convenient formalism for the present problem. 

Consider the case where we have an open system consisting of a single protein molecule in solution. This system could consist, for example, of a biological molecule in a bath of water held at constant temperature. If this molecule adopts a number of conformational states, its NVT and NPT partition functions are the familiar quantities ... 

We note the at the outset that equilibrium between two or more phases, considered as coexisting open systems with no rigid partitions, requires minimally the uniformity of temperature and of pressure throughout the entire system. This makes it apposite to deal with the Gibbs free energy as the function of state for such a system. We also restrict ourselves to mechanical work the generalization to other types of work is dealt with in Exercise 2.1.5. 

The required statistical foundations have been outlined in chapter 1.3. Basically, two approaches are open, the canonical and the grand canonical. In the former, the adsorbate is considered as a closed system, characterized by the number of adsorbate molecules N. the area A and the temperature T. By model considerations, the canonical partition function Q(N.A.T) is formulated from... 

This reduction is not yet possible with [1.5.36] because of restriction [1.5.35b], but we can get rid of this by considering the system to be open, that is by changing to the grand canonical partition function ... 

The theory begins with a derivation of the semigrand partition function, T, which is defined for a system at constant P and T that is open with respect to components 2, 3 and 4 but not with respect to 1. The semigrand partition function is given in terms of the canonical partition function, Q, by... 

In equation (1.30), we have defined the T, V, fi partition function for a one-component system. In a straightforward manner we may generalize the definition for a multicomponent system. Let N=Ni,...,Nc be the vector representing the composition of the system, where AT is the number of molecules of species i. The corresponding vector fi = fiu. .., fic includes the chemical potential of each of the species. For an open system with respect to all components we have the generalization of (1.30)... 

An important case is a system open with respect to some of the species but closed to the others. For instance, in a two-component system of A and B we can define two partial grand partition functions as follows ... 

See, for example, Hill (1956). We shall not need this expansion in p of the pair correlation function. However, it should be noted that this expansion is derived in an open system, i.e., using the grand partition function. In a closed system, we always have an additional term of the order of N 1. See Appendix G for details. 

The derivation of the relationship between the thermodynamic quantities and KBIs consists of two parts. First, we use the normalization conditions for the singlet and the pair distribution functions in the T, V, p ensemble. This provides relationships between the KBIs and the fluctuations in the number of the particles in the open system. Next, by differentiation of the grand partition function, we obtain relationships between thermodynamic quantities and fluctuations in the number of particles. Finally, by eliminating the fluctuations in the number of particles, we obtain the required relations between thermodynamic quantities and the KBIs. 

In an open system (O), the singlet and the pair distribution functions are obtained from the grand partition function... 

This system, consisting of z identical open subsystems, is best represented by a grand canonical ensemble. From the grand ensemble partition function 3... 

As in the standard path integral method for finite temperature systems [19], the above pseudo partition function can be regarded as a configurational integral of classical polymers. However, in the variational path integral, the classical isomorphic systems consist of open-chain polymers. Furthermore, distributions of end-point coordinates at = 0 and M are affected by the trial wavefunction and respectively. 

The intriguing role the smeared potential in special and the smearing effect in general play in optimization of the total energy and partition function of a quantum system opens the possibility analyzing the smearing phenomenon of the quantum fluctuation in a more fundamental way (Putz, 2009). 

The summation in (L.27) cannot be carried out to obtain a closed form of the partition function. This is easily achieved, however, by transforming to an open system with respect to the solute molecules, i.e. we define the grand partition function by... 

To calculate the concentration or the number of molecules present in an open system at defined ju, V, T we must consider the grand canonical ensemble which is simply an ensemble in which each subsystem is itself a canonical ensemble. The grand partition function Z is simply the sum of the Boltzmann factors for each of the j canonical ensembles, weighted according to e / ... 

Finally, we introduce the so-called grand partition function, which is characterized by the thermodynamic variables T, F, and [jl, i.e., for an open system at constant T, V, and fi. The formal definition is... 

We recover the same partition function as that obtained by way of Z. The above equation indicates that the clathrate hydrate thus described is a partially open system with respect to the guest species. The thermodynamic potential for (V, T, A(v, /tig) ensemble is given as... 

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... 

In this particular example it is possible to compute Q(Au) and evaluate the canonical partition function of the system. A more elegant solution is obtained for the grand PF using the same matrix method as was used in previous sections. Thus, we open the system with respect to the particles and write... 

For the open system, one needs the grand canonical partition function or... 

The only subjectivity one can exercise in a strict partition of the molecule is to decide whether to adhere to the constraints imposed by physics or to ignore them and thereby abandon physics subject to one s particular needs. The constraints imposed by quantum mechanics on the definition of an atom are best stated in the form of two questions (a) Does the state function, the function that contains all of the infomiation that can be known about a system, predict a unique partitioning of the molecule into atoms and (b) Does quantum mechanics provide a complete description of the atoms so defined Affirmative answers to these two questions in effect require that quantum mechanics be extended to a uniquely defined set of open systems, termed proper open systems . 

The Bonding Evolution Theory, briefly presented in Appendix B, provides a description of the bonding features of a system, along with their evolution accompanying a reaction path. It relies on the variation of the ELF topological profile as a function of nuclear coordinates. The ELF makes a partition of the molecular space into open sets having a... 

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