Phase singularity

C Congruent melting of the pure binary Cu2 xS phase (singular... 

In our experiments using an optical tweezers set-up, only slightly modified compared to the conventional system described above, we have shown that using phase singular fields we can not only trap absorbing particles but also set them into rotation. This demonstrates the transfer of the angular momentum from the light beam to the particles. 

M.E.J. Friese, H.He, N.R. Heckenberg and H. Rubinsztein-Dunlop Transfer of angular momentum to absorbing particles from a laser beam with a phase singularity in N.B. Abraham and Y.I. Kanin (eds) Laser Optics 95 Nonlinear Dynamics in Lasers Proc SPIE 2792 190 -195(1996)... 

Winfree, A.T. 1989. Electrical instability in cardiac muscle Phase singularities and rotors. J. Theor. Biol, 138 353-405. 

The phase description can explain expanding target patterns in reaction-diffusion systems. The same method, however, breaks down for rotating spiral waves because of a phase singularity involved. The Ginzburg-Landau equation is then invoked. 

Unfortunately, however, such an attempt leads to a serious contradiction, the reason for which will be clarified below. The crucial point is that the spiral waves involve the notion of phase singularity. It is expected that the two-dimensional solution of (6.5.1) corresponding to a rotating spiral would be such that (p changes by an integer multiple of T each time the center of rotation (supposed to be situated at r = 0) was circled ... 

Before going into analytical theory, we show some results of the computer simulation carried out for the Ginzburg-Landau equation in the form of (2.4.18). From the nice symmetry of this model, we expect that the center of steady rotation is in the state of vanishing R, i.e., (X, Y) = (0,0), and hence the phase 0 of W cannot be defined there. Let this phase singularity be situated at r = 0 using polar coordinates (r, 6). Further, the rotation number / is assumed to be 1 as before ... 

One may imagine a more general circumstance where a number of such phase singularities coexist in the system. Then the sum of the associated rotation numbers // must be conserved as long as none of them happen to be absorbed by the wall. A convenient initial distribution satisfying (6.6.1) is shown in Fig. 6.5 where X and y have constant slopes in directions making 90° to each other. Consequently, the zero-level contours of X and Y intersect vertically at r = 0. It is clear that... 

It is expected that rotating spiral waves obtained for the Ginzburg-Landau equation become unstable and turbulent if l + CiC2<0. This is because spiral waves in general behave asymptotically as plane waves far from the core, and under the above condition no plane waves can remain stable. However, we are not much interested in this kind of turbulence in the present section, but we are more interested in the sort of turbulence which would be caused by the very existence of the phase singularity in the core. 

Koga, S. (1982) Rotating spiral waves in reaction-diffusion systems - Phase singularities and multiarmed waves. Prog. Theor. Phys. 67, 164... 

He, H., Friese, M. E. J., Heckenberg, N. R., and Rubinstein-Dunlop, H. (1995). Direct observation of transfer of angular-momentmn to absorptive particles from a laser beam with phase singularities. Physical Review Letters, 75, 826-829. 

The phase singularity is dimensionally trivial in the simplest examples. In a limit-cycle reaction implicating only two chemical concentrations, in the absence of diffusion, it is a reaction steady-state, a state of zero amplitude of the oscillations going on all around it (at azimuthally staggered phase, thus averaging out at the center). If there are A > 2 reactants, the singularity in the A-dimensional concentration space has codimension 2 it is a set of... 

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