Partition function independent variables

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. 

The Helmholtz free energy, A, which is the thermodynamic potential, the natural independent variables of which are those of the canonical ensemble, can be expressed in terms of the partition function ... 

It is also important to keep in mind the independent (state) variables that were specified in deriving q. That is, the partition function was derived with the number of molecules N, the volume of the system V, and the temperature T specified as the independent variables. Thus, when taking the derivative with respect to temperature, as will be needed later, it is good to keep in mind that q = q(N, V, T). The partition function for the entire system of identical molecules, with independent variables N, V, and T, is denoted by a capital Q. If the molecules are indistinguishable, as would normally be the case when calculating thermochemical properties for a given species, then the system partition function is related to the molecular partition function by... 

The set of independent variables (N, V, T) defines the canonical partition function. Systems defined by the number of molecules, the total energy, and the volume (N, E, V) lead to the microcanonical partition function, and systems specified by N, the temperature, and a pressure, namely (N, p, T), lead to the isothermal-isobaric partition function, denoted A. 

In addition to being a function of T, the partition function is also a function of V, on which the quantum description of matter tells us that the molecular energy levels, , depend. Because, for single-component systems, all intensive state variables can be written as functions of two state variables, we can think of q(T, V) as a state function of the system. The partition function can be used as one of the independent variables to describe a single-component system, and with one other state function, such as T, it will completely define the system. All other properties of the system (in particular, the thermodynamic functions U, H, S, A, and G) can then be obtained from q and one other state function. 

For a system of n independent space coordinates, there are 2 n such variables which comprise the 2n-dimensional phase space . The partition function is... 

Name of Ensemble Independent Variables Types of Contact with Next System Partition Function Fundamental Thermodynamic Equation... 

In view of relations (8) and (9), we have for a given value of b two independent variables, for instance bc and Nce. An appropriate statistical mechanical procedure is now to extremize the partition function (5) in turn with respect to bc and Nce. Details of the procedure are described in the first of our papers on the present topic (23). If the polymer concentration in solution is not very high, all z - external interactions of a segment in an extended bundle are directed towards solvent molecules, and characterization of such segments in terms of a ve parameter is not necessary. Since our treatment is not significantly affected by specific values of v-type parameters, an assumption made in (23) and repeated here is ve = 0. 

Finally, we note, that while in thermodynamics independent variables are changed via Legendre Transform, partition functions are changed via Laplace Transforms. In the thermodynamics limit, only the maximum term contributes to the integrals, and the two transformations become manifestly identical E Ethernio, V Vthermo and A Athermo- Thus, lu the thermodyuamic limit,... 

We begin with the canonical partition function for a three component system which is given by Equation 1 and where the independent variables are temperature, volume and mole numbers. 

From a series of transformations of Equation 1 we obtain a new partition function (T) whose independent variables are temperature, pressure, solvent mole number, and the chemical potentials of the solutes (components 2 and 3). These transformations consist of first creating a partition function with pressure rather than volume as an independent variable, and then using this result to create yet another partition function in which we have switched independent variables from solute mole numbers to solute chemical potentials. These operations are analogous to the Legendre transforms commonly employed in thermodynamics. 

The independent areas s have then to be integrated, but it must be noted that, when the preceding rules are applied, the integrals do not always converge for small areas. In order to cure these divergences a cut-off s0 is introduced, and s0 acts as a lower bound for the integration variables. We denote the presence of a cut-off by using an index + we thus write + J"(S) for the partition function, and, in a similar way +Je(... , S) for the restricted partition functions and +i (..., 5) for their Fourier transforms. 

The first of these difficulties can be avoided for symmetrical polymer mixtures (Na = Nb = N) by working in the semigrandcanonical ensemble of the polymer mixture [107] rather than keeping the volume fractions < )A, B and hence the numbers of chains nA, nB individually fixed, as one would do in experiment and in the canonical ensemble of statistical thermodynamics, we keep the chemical potential difference Ap = pA — pB between the two types of monomers fixed as the given independent variable. While the total volume fraction 1 — < )v taken by monomers is held constant, the volume fractions < )A, B of each species fluctuate and are not known beforehand, but rather are an output of the simulation. Thus in addition to the moves necessary to equilibrate the coil configuration (Fig. 16, upper part), one allows for moves where an A-chain is taken out of the system and replaced by a B-chain or vice versa. Note that for the symmetrical polymer mixture the term representing the contributions of the chemical potentials pA, pB to the grand-canonical partition function Z... 

ZfN) (not to be confused with Z, the number of entanglements on a polymer) is the partition function of a LP with N beads and is indicative of the total number of conformations that it may adopt. As shown in Sections 7.3 and 7.4, static properties of LPs are unaffected by the composition of the CLB hence is assumed to be independent of ( )c. The partition function Zf N) does not enter into calculations explicitly, beyond an additive constant, which cancels out when we calculate MN, c) is the probability with which an LP adopts a conformation in which its ends overlap. The second equality in Equation (7.25) offers a more tractable computational route to calculate P Rc)-Computation of F Rc) thus requires knowledge of (a) X N, ( )c), the probability with which an LP adopts a conformation indistinguishable from lhat of a CP, which occurs when its ends essentially overlap, and (b) P Rc), the probability density function of CPs characterized using the size Rc as the macrostate variable. These quantities are expected to change with the composition of the CLB. 

There are four main ensembles in statistical thermodynamics for which the independent variables are NVE (microcanonical), NVT (canonical), NpT (Gibbs or isothermal isobaric), and VT (grand canonical). The characteristic fnnetions provided in Equations 1.2 and 1.3 can be expressed in terms of a series of partition functions such that (Hill 1956)... 

Of course, N and Nb in (5.75) are not independent variables, i.e., we cannot prepare a system with any chosen values of N and Nb, as in the case of a real mixture of two components. Hence, we refer to the A-cules and the -cules as quasicomponents. One may envisage a device which prevents the conversion of molecules between A and B, Such a device may be called an inhibitor (or an anticatalyst) for the conversion reaction A B. A system in the presence of this inhibitor is referred to as being frozen in with respect to the conversion A B, Clearly, the partition function of our system in the frozen-in state is (5.75) and not (5.74). 

Since the mixed solvent is a particle reservoir of both components, we introduce the activity aa of each type of solvent as independent variables (functions of the solvent composition), and move to the grand partition function ... 

The vdWP theory with a set of the independent variables necessarily requires semipermeable membrane to separate water from the guest fluid, hence the pressure Pg of the fluid phase is different in general from the pressure p of the hydrate phase. The appropriate thermodynamic potential is derived in Section II.B. In the light of the phase rule and the experimental conditions, it is desirable to alternate the ensemble to meet the thermodynamic potential F [30,31]. The partition function S is converted into the generalized partition function T as... 

Generally, in the system of balance equations we suppose a priori that the variables values are restricted to certain intervals, thus to a multidimensional interval in the whole variables space. The enthalpy function H is assumed to be infinitely differentiable (imagine a polynomial expression). Then, as can be shown formally, the solutions (8.1.2 and 3) and that of (8.1.5) with (8.1.4) are infinitely differentiable functions of the (chosen) independent variables , in number 2 + 2 + 2+ l= 7 for 3 dependent variables (our solutions). The special partition of variables into independent and dependent as chosen is not critical. The set of equations (8.1.1) can be written as... 

The quantity Q T, F, N) defined in Eq. (1.3.6) is called the T, F, A/ partition function (PF) or the canonical or isothermal-isochoric PF. In the following subsections we shall encounter other partition functions pertaining to systems characterized by the independent variables T, P, N or T, F, /i. The canonical PF is by far the one most used in theoretical work, although other PFs are also useful in some specific applications. 

The general structure of the theoretical tool of ST should now be quite clear. For each set of independent variables we define a partition function. This partition function is related to a thermodynamic quantity through one of the fundamental relationships. On the other hand, each of the summands in the PF is proportional to the probability of realizing the specific value of the variable on which the summation is carried out. Having the probability distribution, for each set of independent variables, one can write down various averages over that distribution function. The calculation of such averages consists of the main outcome of ST. 

In this chapter we have deseribed a canonieal ensemble. In fact, an ensemble is chosen according to the set of independent variables selected, i.e. volume, temperature and amount of matter of the canonical ensemble. For this ensemble, we defined a canonical partition function by relation [5.8]. 

In the same way as we chose in Chapters 1 and 2 the characteristic function and a characteristic matrix for each set of independent variables Ep, we can link to this set an ensemble, as in section 5.2, snch that the relation between the characteristic function and the partition function linked to this ensemble, i.e. in the same form as in relation [5.42]. 

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