Symmetry breaking transition

So far we have considered the formation of tubules in systems of fixed molecular chirality. It is also possible that tubules might form out of membranes that undergo a chiral symmetry-breaking transition, in which they spontaneously break reflection symmetry and select a handedness, even if they are composed of achiral molecules. This symmetry breaking has been seen in bent-core liquid crystals which spontaneously form a liquid conglomerate composed of macroscopic chiral domains of either handedness.194 This topic is extensively discussed in Walba s chapter elsewhere in this volume. Some indications of this effect have also been seen in experiments on self-assembled aggregates.195,196... 

Here, the final three terms are a Ginzburg-Landau expansion in powers of i j. The coefficient t varies as a function of temperature and other control variables. When it decreases below a critical threshold, the system undergoes a chiral symmetry-breaking transition at which i becomes nonzero. The membrane then generates effective chiral coefficients kHp = k n>i f and kLS = which favor membrane curvature and tilt modulations, respec-... 

The irreversibility inherent in the equations of evolution of the state variables of a macroscopic system, and the maintenance of a critical distance from equilibrium, are two essential ingredients for this behavior. The former confers the property of asymptotic stability, thanks to which certain modes of behavior can be reached and maintained against perturbations. And the latter allows the system to reveal the potentialities hidden in the nonlinearity of its kinetics, by undergoing a series of symmetry breaking transitions across bifurcation points. 

Larter, R. Strickholm, A. Ortoleva, P. Cell, "A Detailed Model of the Electrical Symmetry-Breaking Transition in Early Fucus Egg Development" (submitted for publication). 

Induced transition from Ej) to ,). The one- and two-photon routes interfere and, as llllnplied in Section 3.3.3, allow for symmetry-breaking transitions. 

Although not immediately obvious, this control scenario relies entirely upon quantum interference effects. To see this note that in the absence of an e0 (0 pulse, excitation from D) or L) to level /), for example, occurs via one photon excitation with e, (0, i = 1, 2. In this case, as noted above, there is no chiral control. By contrast, with nonzero e0 (0 there is an additional (interfering) route to [ /), i.e., a two-photon route using e7 (t) excitation to level ),j i, followed by an e0 (0-induced transition from Ej) to Ei). The one- and two-photon routes interfere, thus causing symmetry breaking transitions. 

When electron-phonon interactions are taken into account, the regular conducting chains are found to be unstable they undergo a low-temperature phase transition which results in both a modulation of the chains and the opening of an energy gap at the Fermi level. This is the so-called Peierls transition, at T = TP, a second-order symmetry-breaking transition that is characteristic of the quasi-one-dimensional conductors [2,3]. 

Jog C. S., Sankarasubramanian R. and Abinandanan T. A., Symmetry-Breaking Transitions in Equilibrium Shapes of Coherent Precipitates, J. Mech. Phys. Solids, 48, 2363 (2000). 

Condensed matter phases and structures are commonly reached via symmetry breaking transitions. In such systems, when the continuous symmetry is broken, temporary domain-t5q)e patterns are formed. The domain structures eventually coarsen, and disappear in the long-time limit, leaving a uniform broken-symmetry state. This state possesses so-called long-range order (LRO), in which the spatially dependent order parameter correlation function does not decay to zero in the limit of large distances. 

Symmetric-ridge reconstruction, 240 Symmetry of bands, 145, 442f, 483 Symmetry-breaking transition, on surface, 234 Symmetry forbidden reactions, 39 Symmetry notation for bands, 145 Symmetry points, in Briltouin Zone, 73... 

At the united atom limit (J2 = 0), the potential surface Woo has a single well, but at distances near the equilibrium bond length at R 2) double minima become prominent [3]. At large R these evolve into a pair of isolated wells in the separated atom limit. The critical point at which the symmetry breaking transition from single to double wells occurs is determined from the conditions = 0 and = 0, both evaluated at z = 0. At that point Rg = (27/16) / = 1.299038 Pc = (27/32)1/2 = 0.918559 Wg = -32/27 = -1.185185. Figure 1 shows the variation with R of the coordinates ram, and bm that correspond to the minima of Woo-... 

Many developmental phenomena depend on a sequence of patterns, for example from simple extending tip growth to branching. RD theory provides a means for understanding the kinetic constraints involved in such symmetry-breaking transitions. The development of RD theory for growing domains, in conjunction with experimental tests, illuminates how chemical kinetics shape the plants around us, from ferns to spruce trees. 

Figure 19.6 A symmetry-breaking transition or bifurcation in the presence of a small bias that favors one of the bifurcating branches. It can be analyzed through the general equation (19.3.17) and the probability of the system making a transition to the favored branch is given by equation (19.3.18)...

Phase transitions in condensed phases are characterized by symmetry changes, i.e. by transformations in orientational and translational ordering in the system. Many soft materials form a disordered (isotropic) phase at high temperatures but adopt ordered structures, with different degrees of translational and orientational order, at low temperatures. The transition from the isotropic phase to ordered phase is said to be a symmetry breaking transition, because the symmetry of the isotropic phase (with full rotational and translational symmetry) is broken at low temperatures. Examples of symmetry breaking transitions include the isotropic-nematic phase transition in hquid crystals (Section 5.5.2) and the isotropic-lamellar phase transition observed for amphiphiles (Section 4.10.2) or block copolymers (Section 2.11). 

The studies presented so far were performed for angular amplitudes of the oscillating field between 40° and 60°. If this angle is increased further and for sufficiently large Sp, a symmetry breaking transition occurs and the swimmer does not move any longer along the z-axis [57]. 

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