Additional remarks

The above derivation is based on an expansion (A3) in the displacements $x_n$. This excludes some nonlinearities in the force which can lead to additional nonlinear terms in Eq. (A11). The most general form of Eq. (A11) is

$$f_1 \simeq {\cal A} x_1 + {\cal B} \vert x_1\vert^2 x_{1} + \ {\cal C} x_1\vert f_1\vert^2 + {\cal D} x_{-1} f_1^2$$

$$+ {\cal E} \vert x_1\vert^2 f_1 + {\cal F}x_1^2 f_{-1} + {\cal G} \vert f_1\vert^2 f_1 \quad .$$ (30)

However, for small forces $f_1$ and small amplitudes $x_1$, the results derived above are not affected. The regime of nonlinear reponse $\vert f_1\vert\sim \vert x_1\vert^{3}$, as well as the linear response regime $\vert f_1\vert\sim \vert x_1\vert$ still exist. If $\vert f_1\vert\sim \vert x_1\vert$, the nonlinear terms in $f_1$ renormalize the third order term, which in this regime is negligable. If $\vert f_1\vert\sim \vert x_1\vert^{3}$, the nonlinear terms in$f_1$ are of even higher order and can be neglected.
 

2000-03-29