Intrinsic random field

In order to further test this model of intrinsic random fields in the crossover regime to SG order, neutron diffraction experiments on the diluted Heisenberg antiferromagnetic system Eu Srj.j Te are underway (see its preliminary magnetic phase diagram in fig. 7b). As described in sec. 6.1, the borderline between SG and AFM behavior in Eu4.Sr,, Te has been determined up to now (Borgermann... 

The term used in the stochastic literature for these models is intrinsic random fields (IRFs) they appear under that name in textbooks of geostatistics (e.g. Cressie, 1991 Chiles and Delfiner, 1999). This chapter also introduces the intrinsic warps (IWs) intended to replace my principal and partial warps of 1989 for data analysis of landmark-like configurations under symmetries of scale. These IWs could well implement the movable potentially homologous characters that may someday underlie computer-assisted algorithms for stable ordinations or ecophenotypy surveillance much better than existing techniques could likely do. 

FIGURE 5.3 Twelve realizations of the same intrinsic random field on a 10 x 10 grid. Registration is to the lower left, upper left and upper right eomers. At this level of eomplexity, every grid appears to have reportable features, but all are meaningless by expUeit eonstruetion. 

Mardia, K.V., Bookstein, F.L. and Kent, J.T. (2006) Intrinsic random fields and image deformations. Journal of Mathematical Imaging and Vision, 26 59-71. 

H. Tanaka, A simple physical model of liquid-glass transition Intrinsic fluctuating interactions and random fields hidden in glass-forming liquids. J. Phys. Condens. Matter 10, L207-L214 (1998). 

Cottrell,F.R., Merrill,E. W Smith,K. A. Intrinsic viscosity and axial extension ratio of random-coiling macromolecules in a hydrodynamic flow field. J. Polymer Sci. Pt. A-2 8,289-294 (1970). 

Next, Fig. 5 shows a typical center of mass trajectory for the case of a full chaotic internal phase space. The eyecatching new feature is that the motion is no more restricted to some bounded volume of phase space. The trajectory of the CM motion of the hydrogen atom in the plane perpendicular to the magnetic field now closely resembles the random motion of a Brownian particle. In fact, the underlying equation of motion at Eq. (35) for the CM motion is a Langevin-type equation without friction. The corresponding stochastic Lan-gevin force is replaced by our intrinsic chaotic force — e B x r). A main characteristic of random Brownian motion is the diffusion law, i.e. the linear dependence of the travelled mean-square distance on time. We have plotted in Fig. 6 for our case of a chaotic force for 500 CM trajectories the mean-square distance as a function of time. Within statistical accuracy the plot shows a linear dependence. The mean square distance

of the CM after time t, therefore, obeys the diffusion equation... 

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