Coarse graining approximation

Quantum calculations are the starting point for another objective of theoretical and computational chemical science, multiscale calculations. The overall objective is to understand and predict large-scale phenomena, such as deformation in solids or transport in porous media, beginning with fundamental calculation of electronic structure and interactions, then using the results of that calculation as input to the next level of a more coarse-grained approximation. 

The Gaussian subchain model and its possible generalisations allows one to calculate, in a coarse-grained approximation, the different characteristics of a macromolecule and systems of macromolecules, playing a fundamental role in the theory of equilibrium and non-equilibrium properties of polymers. The model does not describe the local structure of the macromolecule in detail, but describes correctly the properties on a large-length scale. 

To describe the system in the coarse-grained approximation, the position of each macromolecule can be defined, as before, by specifying certain points along the macromolecule, spaced at distances that are equal, but not too small as before, we shall refer to these points as particles. If one takes N + 1... 

Considering the entangled systems in the coarse-grained approximation, we forget about segments the theory contains the effective elastic forces between the fictious adjacent particles, and the stress tensor (equation (6.7)) can be expressed through the variables and u"k in the form... 

Because of the layered structure, defects in the cholesteric can be likened in many respects to those in smectic A. Both of them exhibit focal conic textures and both allow for the existence of screw and edge dislocations. To discuss these similarities we employ a coarse-grained approximation in which the cholesteric distortions are considered to be small and to vary slowly over a pitch. In this approximation the free energy of distortion may be expressed in terms of layer displacement u parallel to the twist axis ... 

Thus, in the coarse-grained approximation the energy density becomes... 

When intertheoretic relationships are studied in flesh-and-blood science, one can see the different kinds of resources needed to establish the links. In fact, the relations between theories are usually much more subtle and varied than what the traditional perspective supposes they involve limits, coarse-graining, approximations and other mathematical techniques far more complex than the simple logical links involved in reduction. Moreover, they are not mere tools to which we turn in response to our perceptual or technological limitations. 

In the coarse graining approximation, a DNA N base pairs long is divided into A /M blocks, each block containing M base pairs. Each block is treated as a single unit with its properties derived from an average over its components. Some fine graining is restored in blocks that border on bonded and unbonded regions. 

Fig. 8. Comparison of coarse graining approximation for same random sequence using block sizes of M = 5, 10. 25, 50. N = 5000, = 49.5, = 99.5, = 0.0194.

---