Entropy topological

These fascinating bicontinuous or sponge phases have attracted considerable theoretical interest. Percolation theory [112] is an important component of such models as it can be used to describe conductivity and other physical properties of microemulsions. Topological analysis [113] and geometric models [114] are useful, as are thermodynamic analyses [115-118] balancing curvature elasticity and entropy. Similar elastic modulus considerations enter into models of the properties and stability of droplet phases [119-121] and phase behavior of microemulsions in general [97, 122]. 

The earliest and simplest approach in this direction starts from Langevin equations with solutions comprising a spectrum of relaxation modes [1-4], Special features are the incorporation of entropic forces (Rouse model, [6]) which relax fluctuations of reduced entropy, and of hydrodynamic interactions (Zimm model, [7]) which couple segmental motions via long-range backflow fields in polymer solutions, and the inclusion of topological constraints or entanglements (reptation or tube model, [8-10]) which are mutually imposed within a dense ensemble of chains. 

Fig. 2.12. Enthalpy, entropy, and free energy differences for the ethane —> ethane zero-sum alchemical transformation in water. The molecular dynamics simulations are similar to those described in Fig. (2.7). 120 windows (thin lines) and 32 windows (thick lines) of uneven widths were utilized to switch between the alternate topologies, with, respectively, 20 and lOOps of equilibration and 100 and 500 ps of data collection, making a total of 14.4 and 19.2 ns. The enthalpy (dashed lines) and entropy (dotted lines) difference amount to, respectively, —0.1 and +1.1 kcalmol-1, and —0.5 and +4.1 calmol-1 K For comparison purposes, the free energy difference is equal to, respectively, +0.02 and —0.07kcalmol I, significantly closer to the target value. Inset Convergence of the different thermodynamic quantities...

The entropy change upon elongation is a matter of conformations, so purely topological, and is not influenced by temperature. 

The first two terms on the right of Eq. (7.30) will be recognized as the independent strand contribution to the entropy. The topological or entanglement contribution is then... 

The concatenation is performed without constraint on the successive symbols so that the motion on the repeller corresponds to a Bernoulli random process. The regions around the fundamental periodic orbits are successively visited in a random fashion without memory of the previous fundamental periodic orbit visited. As a consequence, the periodic orbits proliferate exponentially with their period, as described by (2.17). The topological entropy per symbol is equal to htop = In M. 

As a result of the branched chain architecture, TASP molecules exhibit some unique conformational properties)5 12-14 47 75 76 148 For example, the folding to a compact state proceeds via two distinct steps the onset of secondary structure in the attached peptide blocks followed by their template-directed self-assembly to a three-dimensional packing topology. Due to its characteristic branched chain connectivity, the conformational space accessible in the unfolded state is considerably reduced compared to a linear chain of similar size (excluded volume effect), resulting in a smaller chain entropy. Thus, folded TASP molecules are expected to show higher thermodynamic stability compared to unbranched polypeptides of comparable size. 

Thus, it should be stressed that the mathematical topological theory investigates, as a rule, the problems of classification of knots and links, the construction of topological invariants, definitions of topological classes, etc. whereas the fundamental physical problem in the theory of topological properties of polymer chains is the determination of the entropy, S = In Z with the fixed topological state of chains. Both these problems are very difficult, but important. 

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