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Extra resources for Physics Reports vol.377
Both questions are addressed in Fig. 24, where Ä(N ) is computed in the FPU- model at a relatively low temperature. The convergence 50 S. Lepri et al. 2 10 10 2 10 1 1 10 2 3 10 10 N 4 10 2 10 3 10 4 N Fig. 22. Thermal conductivity of the FPU-ÿ model versus lattice length N for T+ = 0:11; T− = 0:09, and = 1. The inset shows the e ective growth rate e versus N . , respectively. 0 3 4 5 6 ln N 7 8 9 Fig. 23. Thermal conductivity of the FPU-ÿ model, T+ = 150; T− = 15, ÿxed boundary condition. The data are taken from Ref.
Lepri et al. / Physics Reports 377 (2003) 1 – 80 we ÿnd that Ne = m √ m N : (101) At this point, it becomes crucial to specify the boundary conditions. Let us ÿrst consider the case of free ones: the square amplitude of an extended eigenmode in a lattice of size N is of the order 1=N . This implies that the contribution to the heat ux of one of such modes is (T+ − T− )=N and the heat ux in Eq. (94) can be estimated as jfree ( ; N ) ˙ (T+ − T− ) 1 √ : N m m As a result, the conductivity diverges as m √ N : Äfree ˙ m (102) (103) This scaling was ÿrst derived in Ref.
For weak coupling, the conductivity has been evaluated by numerically computing the eigenvectors and averaging the Matsuda-Ishii formula (94) over 1000 realizations of the disorder. The asymptotic regime = −1=2 is approached very slowly (see the circles): one should consider N values much greater than 103 . Similar results are found at stronger coupling by directly simulating chains that interact with stochastic baths. The data (diamonds in Fig. 16) suggest that a relatively strong coupling reduces the amplitude of ÿnite-size corrections.