Kink trajectory

The ground state tunneling splitting for two symmetrically placed diabatic terms can be found in the same manner, as described in sections 2.4 and 4.2. Since the kink trajectory crosses the barrier once, we shall obtain... 

Fig. 3.16 Space-time pattern of fc — 2, r — 1 rule R18 kink-nites (i.e.. neighboring a = 1 site.s) are indicated in solid black. Notice the stochastic-like kink trajectories, despite the strictly deterministic rule.

Each extremal trajectory includes n kink-antikink pairs, where n is an arbitrary integer, and the kink centers are placed at the moments 0 < tj < < tin < P forming the instanton gas (fig. 22). Its contribution to the overall path integral may be calculated in exactly the same manner as was done in the previous subsection, with the assumption that the instanton gas is dilute, i.e., the kinks are independent of each other. 

In the case of a symmetric (or Just slightly asymmetric) potential the instanton trajectory consists of kink and antikink, which are separated by infinite time and do not interact with each other. In other words, we may change the boundary conditions, namely, suppose that the time spans from — 00 to -t- 00 for a single kink, and then multiply the action in (5.72) by factor 2. 

Figure 7 Application of the dimer method to a two-dimensional test problem. Three different starting points are generated in the reactant region by taking extrema along a high temperature dynamical trajectory. From each one of these, the dimer isjirst translated only in the direction of the lowest mode, but once the dimer is out of the convex region a full optimization of the effective force is carried oat at each step (thus the kink in two of the paths). Each one of the three starting p>oints leads to a different saddle point in this case.

It is readily seen that this quantity is independent of the kink s position on the real axis, tx i.e., whenever the kink is started, after performing half of the vibration with imaginary period 2ntIm the coordinate remains real and the action satisfies (3.27). The contours with n = 1, 2, 3,. . . correspond to multiple barrier crossings. It is clear from the semiclassical picture that the barrier transparency is proportional to exp[-2W( )]. The trajectories for which the imaginary increment is 277rc/ > with integer n are associated with reflection from the barrier. 

Boundary-layer fronts at sea. These fronts or convergence lines develop in the cold air at sea when there are large bends or kinks in the shape of the upstream land or ice botmdary from which the cold air is flowing. The air on one side of the convergence line has had a different over-water trajectory than that on the other side. The existence of such convergence in the cold air may, under the right conditions, very well be a factor in the genesis of polar vortices, which may later be intensified irrto polar (arctic) lows. 

It was shown some time ago that in ionic surfactants (such as SDS), the trajectory of CMCs versus temperature intersects the Krafft phase boundary at the CMC Krafft point [7,44,45]. Just as the CMC itself is not a thermodynamic discontinuity [46], there is no kink or cusp in the Krafft boundary at this intersection. Nevertheless, this behavior is important because below the temperature of the CMC Krafft point micellar structure does not exist in equilibrium surfactant solutions. Metastable micellar solutions may, however, easily be formed below the Krafft boundary by cooling concentrated liquid phases [47]. Cooling liquid-crystal phases below the Krafft eutectic typically yields metastable liquid-crystal (not liquid) phases. 

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