Self-consistency approach to frequency renormalization

We now turn to the discussion of how the trivial resonances give rise to the dispersion renormalization. This question was examined in [12] before. There, it was shown that the renormalization of the linear dispersion of the $\beta$-FPU chain arises due to the collective effect of the nonlinearity. In particular, the trivial resonant interactions of type $(2\rightarrow 2)$, i.e., the solutions of Eq. (41), enhance the linear dispersion (the renormalized dispersion relation takes the form $\tilde{\omega}_k=\eta\omega _k$ with $\eta>1$), and effectively weaken the nonlinear interactions. Here, we further address this issue and present a self-consistency argument to arrive at an approximation for the renormalization factor $\eta$. As it was mentioned above, the contribution of the non-resonant terms have a vanishing long time effect to the statistical properties of the system, therefore, in our self-consistent approach, we ignore these non-resonant terms. By removing the non-resonant terms and using the canonical transformation

$$\tilde{a}_k=\frac{P_k-\imath \eta_{sc}\omega _k Q_k}{\sqrt{2\eta_{sc}\omega _k}},$$ (45)

where $\eta _{sc}$ is a factor to be determined, we arrive at a simplified effective Hamiltonian from Eq. (32) for the finite $\beta$-FPU system

$$H_{\rm {eff}} = \sum_{k=1}^{N-1}\frac{\omega _k}{2}\left(\eta_{sc}+\frac{1}{\eta_{sc}}\right)\vert\tilde{a}_k\vert^2$$ (46)

$$+ \sum_{k,l,m,s=1}^{N-1}T^{kl}_{ms}\Delta^{kl}_{ms}\tilde{a}_k^*\tilde{a}_l^*\tilde{a}_m\tilde{a}_s.$$

The ``off-diagonal'' quadratic terms $\tilde{a}_k\tilde{a}_{N-k}$ from Eq. (32) are not present in Eq. (46), since $\tilde{a}_k$ are chosen so that $\langle \tilde{a}_k\tilde{a}_{N-k}\rangle =0$ (see Section II). The contribution of the trivial resonances in $H_{\rm {eff}}$ is

$$H_4^{\rm {tr}}=4\sum_{k,l=1}^{N-1}T^{kl}_{kl}\vert\tilde{a}_l\vert^2\vert\tilde{a}_k\vert^2,$$ (47)

which can be ``linearized'' in the sense that averaging the coefficient in front of $\vert\tilde{a}_k\vert^2$ in $H_4^{\rm {tr}}$ gives rise to a quadratic form

$$H_2^{\rm {tr}}\equiv\sum_{k=1}^{N-1}\left(4\sum_{l=1}^{N-1}T^{kl}_{kl}\langle \vert\tilde{a}_l\vert^2\rangle \right)\vert\tilde{a}_k\vert^2.$$

Note that the subscript $2$ in $H_2^{\rm {tr}}$ emphasizes the fact that $H_2^{\rm {tr}}$ now can be viewed as a Hamiltonian for the free waves with the familiar effective linear dispersion $\Omega_k=4\sum_{l=1}^{N-1}T^{kl}_{kl}\langle \vert\tilde{a}_l\vert^2\rangle$ [12,5]. This linearization is essentially a mean-field approximation, since the long-time average of trivial resonances in Eq. (47) is approximated by the interaction of waves $\tilde{a}_k$ with background waves $\langle \vert\tilde{a}_l\vert\rangle$. The self consistency condition, which determines $\eta _{sc}$, can be imposed as follows: the quadratic part of the Hamiltonian (46), combined with the ``linearized'' quadratic part, $H_2^{\rm {tr}}$, of the quartic $H_4^{\rm {tr}}$, should be equal to an effective quadratic Hamiltonian $\tilde{H}_2=\sum_{k=1}^{N-1}\tilde{\omega}_k\vert\tilde{a}_k\vert^2$ for the renormalized waves, i.e.,

$$\sum_{k=1}^{N-1}\frac{\omega _k}{2}\left(\eta_{sc}+\frac{1}{\eta_{sc}}\right)\vert\tilde{a}_k\vert^2$$

$$+\sum_{k=1}^{N-1}\left(4\sum_{l=1}^{N-1}T^{kl}_{kl}\langle \vert\... ...vert\tilde{a}_k\vert^2=\sum_{k=1}^{N-1}\tilde{\omega}_k\vert\tilde{a}_k\vert^2,$$

where $\tilde{\omega}_k$ is the renormalized linear dispersion, which is used in the definition of our renormalized wave, Eq. (45), and $\tilde{\omega}_k=\eta_{sc}\omega _k$. Equating the coefficients of $\omega _k\vert\tilde{a}_k\vert^2$ on both sides for every wave number $k$ yields

$$\frac{1}{2}\left(\eta_{sc}+\frac{1}{\eta_{sc}}\right)+4\sum_{l=1}... ...\beta}{8N\eta_{sc}^2}\omega _l\langle \vert\tilde{a}_l\vert^2\rangle=\eta_{sc},$$

where use is made of Eq. (33). After algebraic simplification, we have the following equation for $\eta _{sc}$

$$\eta_{sc}^3-\eta_{sc}=\frac{3\beta}{N}\sum_{l=1}^{N-1}\omega _l\langle\vert\tilde{a}_l\vert^2\rangle.$$ (48)

Using the property (21) of the renormalized normal variables $\tilde{a}_k$, we find the following dependence of $\langle \vert\tilde{a}_k\vert^2\rangle$ on $\eta _{sc}$,

$$\langle \vert\tilde{a}_l\vert^2\rangle =\frac{1}{2\eta_{sc}\omega... ...l\vert^2\rangle +\eta_{sc}^2\omega _l^2\langle \vert Q_l\vert^2\rangle \right).$$ (49)

Combining Eqs. (49) and (50) leads to

$$\eta_{sc}^4-A\eta_{sc}^2-B=0,$$ (50)

where

$$A = 1+\frac{3\beta}{2N}\sum_{l=1}^{N-1}\omega _l^2\langle\vert Q_l\vert^2\rangle=1+\frac{3\beta}{N}\langle U\rangle ,$$

$$B = \frac{3\beta}{2N}\sum_{l=1}^{N-1}\langle\vert P_l\vert^2\rangle=\frac{3\beta}{N}\langle K\rangle .$$

The only physically relevant solution of Eq. (51) is

$$\eta_{sc}=\sqrt{\frac{A+\sqrt{A^2+4B}}{2}}.$$ (51)

The constants $A$ and $B$ can be easily derived using the Gibbs measure.

Next, we compare the renormalization factor $\eta$ [Eq. (25)] with its approximation $\eta _{sc}$ [Eq. (52)] from the self-consistency argument. In Appendix A, we study in detail the behavior of both $\eta$ and $\eta _{sc}$ in the two limiting cases, i.e., when nonlinearity is small ( $\beta\rightarrow 0$ with fixed total energy $E$), and when nonlinearity is large ( $\beta\rightarrow\infty$ with fixed total energy $E$). As is shown in Appendix A, for the case of small nonlinearity, both $\eta$ and $\eta _{sc}$ have the same asymptotic behavior in the first order of the small parameter $\beta$,

$$\eta = 1+\frac{3E}{2N}\beta+O(\beta^2),$$ (52)

$$\eta_{sc} = 1+\frac{3E}{2N}\beta+O(\beta^2).$$

Moreover, in the case of strong nonlinearity $\beta\rightarrow\infty$, both $\eta$ and $\eta _{sc}$ scale as $\beta^\frac{1}{4}$, i.e.,

$$\eta\sim\eta_{sc}\sim\beta^{\frac{1}{4}}$$ (53)

(see Appendix A for details). Note that, in [12], we numerically obtained the scaling $\eta\sim\beta^{0.2}$, which differs from the exact analytical result (54) due to statistical errors in the numerical estimate of the power.

Figure 5: The renormalization factor as a function of the nonlinearity strength $\beta$ for small values of $\beta$. The renormalization factor $\eta$ [Eq. (25)] is shown with the solid line. The approximation $\eta _{sc}$ [Eq. (52)](via the self-consistency argument) is depicted with diamonds connected with the dashed line. The small-$\beta$ limit [Eq. (53)] is shown with the solid circles connected with the dotted line. Note that, abscissa is of logarithmic scale. Inset: The renormalization factor as a function of the nonlinearity strength $\beta$ for large values of $\beta$. The renormalization factor $\eta$ [Eq. (25)] is shown with the solid line. $\eta _{sc}$ [Eq.(52)] is depicted with diamonds connected with the dashed line. The large-$\beta$ scaling [Eq. (54)] is shown with the dashed-dotted line. Note that, the plot is of log-log scale with base 10.

In Fig. 5, we plot the renormalization factor $\eta$ and its approximation $\eta _{sc}$ for the case of small nonlinearity $\beta$ for the system with $N=256$ particles and total energy $E=100$. The solid line shows $\eta$ computed via Eq. (25), the diamonds with the dashed line represent the approximation via Eq. (52), and the solid circles with the dotted line correspond to the small-$\beta$ limit (53). In Fig. 5 (inset), we plot the renormalization factor $\eta$ and its approximation $\eta _{sc}$ for the case of large nonlinearity $\beta$ for the system with $N=256$ particles and total energy $E=100$. The solid line shows $\eta$ computed via Eq. (25), the diamonds with the dashed line represent the approximation via Eq. (52), and the dashed-dotted line correspond to the large-$\beta$ scaling (54). Figure 5 shows good agreement between the renormalization factor $\eta$ and its approximation $\eta _{sc}$ from the self consistency argument for a wide range of nonlinearity, from $\beta\sim 10^{-3}$ to $\beta\sim 10^4$. This agreement demonstrates, that (i) the effect of the linear dispersion renormalization, indeed, arises mainly from the trivial four-wave resonant interactions, and (ii) our self-consistency, mean-field argument is not restricted to small nonlinearity.