Effect of lattice size

The renormalized WT theory also predicts that the knee width should scale with lattice size (Eq. (9)) whereas the three wave time scale should not (Eq. (10)). These properties are compatible with a thermodynamic limit. In figure 3 the logarithm of the fraction of modes to the total lattice size is plotted against time for seven ensembles of experiments with increasing lattice size $N=32, 64, 128, 256, 1024, 2048, 4096$, and fixed $\epsilon = 0.1$. These experiments were again initialized with half the number of modes of the predicted knee. Three features of this plot stand out most clearly. First, the spectral entropy follows a universal evolution [#!jdl:ueeoc!#,#!pp:swtsr!#] for lattice sizes larger than $N=128$. Secondly, the number of excited modes exponentially increases with time prior to three wave equilibrium. Finally, for $N=32$ (where the initial number of excited modes is approximately $7$) there is no equilibrium, but rather quasi-periodic behavior. In fact, a shadow of this behavior is present for $N=64$ and $128$ also. These are reminiscent of the integrability discovered in Fermi, Pasta and Ulam's original work [#!ef:snp!#], and indicate the breakdown of the WT theory scaling predictions for small lattice sizes.