Fluctuations topological

Gompper G, Kroll D (1998) Membranes with fluctuating topology Monte Carlo simulations. Phys Rev Lett 81(ll) 2284-2287... 

SmA phases, and SmA and SmC phases, meet tlie line of discontinuous transitions between tire N and SmC phase. The latter transition is first order due to fluctuations of SmC order, which are continuously degenerate, being concentrated on two rings in reciprocal space ratlier tlian two points in tire case of tire N-SmA transition [18,19 and 20], Because tire NAC point corresponds to the meeting of lines of continuous and discontinuous transitions it is an example of a Lifshitz point (a precise definition of tliis critical point is provided in [18,19 and 20]). The NAC point and associated transitions between tire tliree phases are described by tire Chen-Lubensky model [97], which is able to account for tire topology of tire experimental phase diagram. In tire vicinity of tire NAC point, universal behaviour is predicted and observed experimentally [20]. 

VII. Computer Simulations of the Basic Landau-Ginzburg Model 711 A. Topological fluctuations 711... 

The second difference is related to the structure of the lamellar phase. The Euler characteristic has been assumed zero in the whole lamellar phase by Gompper and Kraus [47], whereas we show that it fluctuates strongly in the lamellar phase between the transition line and the topological disorder fine. The notion of the topological disorder line has not appeared in their paper. We think that the topological disorder line is much closer to the transition... 

The topological fluctuations can also be induced by confinement. It has been found that confinement between parallel walls exhibits topological fluctuations even if they are absent in the bulk system. 

For diffuse and delocahzed interfaces one can still define a mathematical surface which in some way describes the film, for example by 0(r) = 0. A problem arises if one wants to compare the structure of microemulsion and of ordered phases within one formalism. The problem is caused by the topological fluctuations. As was shown, the Euler characteristic averaged over the surfaces, (x(0(r) = 0)), is different from the Euler characteristics of the average surface, x((0(r)) = 0), in the ordered phases. This difference is large in the lamellar phase, especially close to the transition to the microemulsion. x((0(r)) =0) is a natural quantity for the description of the structure of the ordered phases. For microemulsion, however, (0(r)) = 0 everywhere, and the only meaningful quantity is (x(0(r) = 0))-... 

X. Michalet, D. Bensimon, B. Fourcade. Fluctuating vesicles of nonspherical topology. Phys Rev Lett 72 168-171, 1994. 

R. Holyst, W. Gozdz. Fluctuating Euler characteristics, topological disorder line, and passages in lamellar phases. J Chem Phys 706 4773- 780, 1997. 

R. Holyst, P. Oswald. Confinement induced topological fluctuations in a system with internal surfaces. Phys Rev Lett 79 1499-1502, 1997. 

There is a strong dependence on the exact lattice structure intermediate topologies typically have large measure fluctuations. 

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. 

Here w is a weighting factor. Asr (s) is the absorption factor (in this case for symmetrical absorption), Hz (5) and In consider the non-ideal character of the two-phase topology (cf. p. 124, Fig. 8.10) by consideration of a smooth phase transition zone and density fluctuations inside the phases. 

From (2.70), it follows that the free energy cannot be divided simply into two terms, associated with the interactions of type a and type b. There are also coupling terms, which would vanish only if fluctuations in AUa and AUb were uncorrelated. One might expect that such a decoupling could be accomplished by carrying out the transformations that involve interactions of type a and type 6 separately. In Sect. 2,8.4, we have already discussed such a case for electrostatic and van der Waals interactions in the context of single-topology alchemical transformations. Even then, however, correlations between these two types of interactions are not... 

Classical molecular theories of rubber elasticity (7, 8) lead to an elastic equation of state which predicts the reduced stress to be constant over the entire range of uniaxial deformation. To explain this deviation between the classical theories and reality. Flory (9) and Ronca and Allegra (10) have separately proposed a new model based on the hypothesis that in a real network, the fluctuations of a junction about its mean position may may be significantly impeded by interactions with chains emanating from spatially, but not topologically, neighboring junctions. Thus, the junctions in a real network are more constrained than those in a phantom network. The elastic force is taken to be the sum of two contributions (9) ... 

The equilibrium shear modulus of two similar polyurethane elastomers is shown to depend on both the concentration of elastically active chains, vc, and topological interactions between such chains (trapped entanglements). The elastomers were carefully prepared in different ways from the same amounts of toluene-2,4-diisocyanate, a polypropylene oxide) (PPO) triol, a dihydroxy-terminated PPO, and a monohydroxy PPO in small amount. Provided the network junctions do not fluctuate significantly, the modulus of both elastomers can be expressed as c( 1 + ve/vc)RT, the average value of vth>c being 0.61. The quantity vc equals TeG ax/RT, where TeG ax is the contribution of the topological interactions to the modulus. Both vc and Te were calculated from the sol fraction and the initial formulation. Discussed briefly is the dependence of the ultimate tensile properties on extension rate. 

Figure 7. Topological fluctuations of the lamellar phase at different points of the phase diagram, (a) Single fusion between the lamellae by a passage (this configuration is close to the topological disorder line), (b) Configuration close to the transition to the disordered microemulsion phase the Euler characteristic is large and negative.

It was found that normal zero-point oscillations lie on top of large gluon fluctuations - instantons and anti-instantons with random positions and sizes. The left column - action density and the right column - topological charge density. Here instantons are peaks and anti-instantons are holes. 

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