Configuration integral factorization

To obtain thermodynamic perturbation or integration formulas for changing q, one must go back and forth between expressions of the configuration integral in Cartesian coordinates and in suitably chosen generalized coordinates [51]. This introduces Jacobian factors... 

For a classical system of N point particles enclosed in a volume V,at a temperature T, the canonical partition function can be decomposed in two factors. The first one (Qt) comes from the integration over the space of momenta of the kinetic term of the classical Hamiltonian, which represents the free motion of noninteracting particles. The second one, which introduces the interactions between the particles and involves integration over the positions, is the configuration integral. This way, equation (30)... 

Given an approximation to V( 0s ) acceptable for the purposes at hand, one can proceed to compute equilibrium, i.e., statistically mechanically averaged, values

for properties P( 0s ) of interest using standard procedures which weight each conformation of the carbohydrate molecule by the Boltzmann factor of V( 0s ) normalized by the configuration integral given in eqn. (9). 

F(J ) is the pair interaction potential, T the temperature and k the Boltzmann factor. The integral to the right is over the volume v of the container it is often referred to as the configuration integral, Q2(T). For large enough volumes, the integral may be replaced by the volume. With the help of Hill s effective potentials, Eq. 2.28, this expression may be written as a sum of free and bound pair state sums, respectively [183, 184],... 

However, these results do not immediately carry over to the problems of interest here where (while PBCs are the norm) the ensembles are frequently open or constant pressure, and the systems do not fit in to the lattice model framework. Even in the apparently simple case of crystalline solids in NVT, the free translation of the center of mass introduces /-dependent phase space factors in the configurational integral which manifest themselves as additional finite-size corrections to the free energy these may not yet be fully understood [58, 97]. If one adopts the traditional stance, then, one is typically faced with having to make extrapolations of the free-energy densities in each of the two phases, without a secure understanding of the underlying form (jf . ..) of the corrections involved. 

Here a(cx) is a rate-determining factor for the approach of D to De(c ) and may be a function of clt as indicated. One may expect that a becomes larger as the polymer segments more rapidly change configurations. Integration of Eq. (12), with cx fixed, gives... 

In Chapter 2, we saw that the configuration integral is the key quantity to be calculated if one seeks to compute thermal properties of classical (confined) fluids. However, it is immediately apparent that this is a formidable task because it reejuires a calculation of Z, which turns out to involve a 3N-dimensional integration of a horrendously complex integrand, namely the Boltzmann factor exp [-C7 (r ) /k T] [ see Eq. (2.112)]. To evaluate Z we either need additional simplifjfing assumptions (such as, for example, mean-field approximations to be introduced in Chapter 4) or numerical approaches [such as, for instance, Monte Carlo computer simulations (see Chapters 5 and 6), or integral-equation techniques (see Chapter 7)]. 

The configuration correction factor F can be obtained by analytical or numerical integration of Equations (6.108a) and (6.108b) as a function of two dimensionless ratios, P and R, defined as... 

Fortunately, this can be simplified by factoring the total energy U into terms, each of which represents the interaction energy of a certain number of molecules or clusters. We can consider interactions of molecules or clusters in groups of two, three, four, etc., and rewrite the configuration integral as the sum of these terms. 

An alternate method of considering configurational entropy factors is through the Fourier integral representation of the microcanonical distribution described earlier. We will show how that formalism can be applied to a two-component, period-two DNA molecule as well as the one-component DNA molecule. 

The last of Eqs. (51) defines a thermodynamic fundamental equation for G = G(N, P, 7) in the Gibbs energy representation. Note that passing from one ensemble to the other amounts to a Legendre transformation in macroscopic thermodynamics [39]. Vq is just an arbitrary volume used to keep the partition function dimensionless. Its choice is not important, as it just adds an arbitrary constant to the free energy. The NPT partition function can also be factorized into the ideal gas and excess contributions. The configurational integral in this case is ... 

Some authors also include the N and Vq factors in the configurational integral. A simulation in the isothermal isobaric ensemble should be conducted with the volume allowed to change, but the way these changes are implemented must provide for the proper probability density distribution given by Eq. (52). 

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