Lattice sizes

It is important to note that we assume the random fracture approximation (RPA) is applicable. This assumption has certain implications, the most important of which is that it bypasses the real evolutionary details of the highly complex process of the lattice bond stress distribution a) creating bond rupture events, which influence other bond rupture events, redistribution of 0(microvoid formation, propagation, coalescence, etc., and finally, macroscopic failure. We have made real lattice fracture calculations by computer simulations but typically, the lattice size is not large enough to be within percolation criteria before the calculations become excessive. However, the fractal nature of the distributed damage clusters is always evident and the RPA, while providing an easy solution to an extremely complex process, remains physically realistic. 

The solution for the discretized model of the continuous functional is obtained with a certain accuracy which depends on the value of the lattice spacing h and the number of points N. The accuracy of our results is checked by calculating the free energy and the surface area of (r) = 0 for a few different sizes of the lattice. The calculation of the free energy is done with sufficient accuracy for N = 129, which results in over 2 million points per unit cell. The calculation of the surface area of (r) = 0 is sufficiently accurate even for a smaller lattice size. 

The properties of the periodic surfaces studied in the previous sections do not depend on the discretization procedure in the hmit of small distance between the lattice points. Also, the symmetry of the lattice does not seem to influence the minimization, at least in the limit of large N and small h. In the computer simulations the quantities which vary on the scale larger than the lattice size should have a well-defined value for large N. However, in reality we work with a lattice of a finite size, usually small, and the lattice spacing is rather large. Therefore we find that typical simulations of the same model may give diffferent quantitative results although quahtatively one obtains the same results. Here we compare in detail two different discretization... 

Using successively an inverse Cluster Variation Method and an IMC algorithm, we determined a set of nine interactions for each alloy (for the IMC procedure, we used a lattice size of 4 24 ). For each alloy, the output from the inverse procedure has been used as an input interaction set in a direct MC simulation, in order to calculate a... 

Figure 3 Two-dimensional X-Y order-parameter and spin maps for temperature 1675 K [lattice sizes 24x36x36 in unit of fee cubes].

Fig, 3,20 Number of cyclic states cis a function of lattice size for a few representative one-dimensional elementary rules see text. 

Dynamical behavior depends on lattice size N only for A near A transient length, for example, grows exponentially with N for A = Ac. 

As A increases beyond A,., transient lengths decrease until, at A = 3/4, they no longer show any dopendeiu-.e on lattice size. 

Numerical Observations Figure 3.42 shows a schematic plot of H versus A for A = 8 Af = 5 two dimensional CA. The lattice size is 64 x 64 with periodic boundary conditions. In the figure, the evolution of the single-site entropy is traced for four different transition events. In each case, for a given A, a rule table consistent with that A is randomly chosen and the system is made to evolve for 500 steps to allow transients to die out before H is measured. 

The key result in this section will be the derivation of linear recurrence relations for 7)v,p in terms of 7j,p, for j < n [jen 88a]. We begin by introducing an invariance matrix, whose powers correspond to the lattice sizes on which f> is defined. 

This same technique can be used to find the number of limit cycles of period p on lattice size n for any two-neighbor CA rule. 

While this bound is not a particularly strong one and convergence is generally faster in practice [goles90], it does clearly point out the important fact that transient times are linearly bounded by the lattice size n. Notice that this is not true of more general classes of matrices, even those of the preceding section that are both symmetric and integer-valued. Equation 5.140 shows that the transient time depends on both A and 26 — Al if both A and b are arbitrary (save, perhaps, for A s symmetry), there is of course no particular reason to expect t to be linearly... 

Figures 6.4 shows some of the variety of possible shapes of P f) for elementary rules shown in the figures are the power spectra for rules Rll, R56, R150 and R200. The plots were generated for lattice size N = 2048, ignoring the first 15 transient steps and averaging a total of 20 runs. Also, since there are only N data points but 2N real Fourier components, half of the components are redundant. Thus, only the first half of the components are shown (see [H89b] or [H87] for a complete set of power spectra).

In Regime I, for example, there are many possible domain sizes and, consequently, Q D) 0 for many different ) s moreover, the maximum domain size increases with the total lattice size [kaneko89]. Regimes II and III, on the other hand, allow only a relatively few domain sizes so that Q D) 0 for only a few domain sizes D. In addition, there appears to be a cutoff size Dc (which does not depend on lattice size) such that Q(D > Dc) = 0. In the case cf fully developed turbulence in Regime IV, Q D) oc expf—constant. D), which is indicative of the random generation of domains. 

Figure 8.4. Graphical separation of lattice size and lattice distortion effects according to Warren-Averbach... 

Think of instrumental broadening, finite lattice size. 

Figure 31. The domain growth during the phase separation process reflected by the shift of the first zero in the pair correlation function (a) and by the surface area reduction (b). Although the surface area and first zero of the pair-correlation functions are equivalent lengthscales, the time dependence of the surface area is less affected by the finite lattice size affects.

Figure 9. (a) Heat current versus temperature Tl (at fixed Tr = 0.2) for different coupling constants, feint, with lattice size N = 50. The system parameters are Vl = 5, Vr = 1, fci = 1, fcfl = 0.2. (b) Same as (a) but for different system size N. kint = 0.05. Notice that when Tl < 0.1 the heat current increases with decreasing the external temperature difference. 

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