Renormalized dispersion relations of $\beta$-Fermi-Pasta-Ulam chains in equilibrium and nonequilibrium states

We study the nonlinear dispersive characteristics in $\beta$-Fermi-Pasta-Ulam (FPU) chains in both thermal equilibrium and nonequilibrium steady state. By applying a multiple scale analysis to the FPU chain, we analyze the contribution of the trivial and nontrivial resonance to the renormalization of the dispersion relation. Our results show that the contribution of the nontrivial resonance remains significant to the renormalization, in particular, in strongly nonlinear regimes. We contrast our results with the dispersion relations obtained from the Zwanzig-Mori formalism and random phase approximation to further illustrate the role of resonances. Surprisingly, these theoretical dispersion relations can be generalized to describe dispersive characteristics well at the nonequilibrium steady state of the FPU chain with driving-damping in real space. Through numerical simulation, we confirm that the theoretical renormalized dispersion relations are valid for a wide range of nonlinearities in thermal equilibrium as well as in nonequilibrium steady state. We further show that the dispersive characteristics persist in nonequilibrium steady state driven-damped in Fourier space.

Figure 1

The renormalization factors as functions of the energy $H$. The chain was simulated using the parameters of $N=256$ and $\beta =100$. Plotted is the renormalization factor ${\eta }_{\mathrm{meas}}$ [Eq. (27)] (red circles). For comparison, also plotted are the renormalization factors ${\eta }_L,{\eta }_N$, and ${\eta }_M$ from Eqs. (23), (24), and (25) with black triangles, green hexagons, and blue squares, respectively. It can be seen that ${\eta }_{\mathrm{meas}},{\eta }_L$, and ${\eta }_N$ almost overlap with each other. (Inset) Near-$k$ independence of the renormalization factors ${\tilde{\eta }}_{\mathrm{meas}}\left(k\right)$. The red circles correspond to ${\tilde{\eta }}_{\mathrm{meas}}\left(k\right)$ for each mode $k$. The solid red line corresponds to the mean value ${\eta }_{\mathrm{meas}}$ [Eq. (27)].

Figure 2

Comparison of renormalized dispersion relations for the $x$-space driven-damped $\beta$-FPU chain. The chain is simulated with the nonlinear strengths (a) $\beta =100$ and (b) $\beta =0$. The other parameters of the chain are $N=256,{\sigma }_L=1,{\sigma }_R=10$, and ${\gamma }_L={\gamma }_R=1$. Plotted is the logarithmic modulus $ln|{\widehat{Q}}_k{\left(\omega \right)|}^2$ by the WFS analysis with its magnitude color coded. The measured dispersion relation ${\Omega }_k^{\mathrm{meas}}$ is the location of the frequency peak of $ln|{\widehat{Q}}_k{\left(\omega \right)|}^2$ for each mode $k$. For comparison, also shown are ${\omega }_k^L$ [Eq. (18)], ${\omega }_k^M$ [Eq. (20)], ${\omega }_k^N$ [Eq. (19)] and the linear dispersion relation ${\omega }_k^{\left(0\right)}$ [Eq. (3)]. It can be seen in the left panel (a) that ${\omega }_k^L$ and ${\omega }_k^N$ almost overlap with each other and with ${\Omega }_k^{\mathrm{meas}}$. It can also be observed in the right panel (b) that all the dispersion relations overlap with the linear dispersion relation.

Figure 3

The renormalization factor as a function of driving strength ${\sigma }_R$. The chain was simulated with $N=256,\beta =100,{\sigma }_L=1$, and ${\gamma }_L={\gamma }_R=1$. The renormalization factor ${\eta }_{\mathrm{meas}},{\eta }_L,{\eta }_N$, and ${\eta }_M$ are depicted with red curve with circles, black curve with triangles, green curve with hexagons, and blue curve with squares, respectively. It can be observed that ${\eta }_{\mathrm{meas}},{\eta }_L$, and ${\eta }_N$ nearly overlap one another. (Inset) The renormalization factor $\eta \left(k\right)$ versus wave number $k$ for the system corresponding to that in Fig. 2. The parameter is ${\sigma }_R=10$. The red circle corresponds to ${\tilde{\eta }}_{\mathrm{meas}}\left(k\right)$ in Eq. (28) and the black triangle corresponds to ${\tilde{\eta }}_L\left(k\right)$ in Eq. (26). The solid red line and the dashed black line correspond to the mean values ${\eta }_{\mathrm{meas}}$ [Eq. (27)] and ${\eta }_L$ [Eq. (23)], respectively.

Figure 4

Comparison of renormalized dispersion relations for the $k$-space driven-damped $\beta$-FPU chain. The left panel (a) displays the renormalized dispersion relations of the $\beta$-FPU chain for the weakly damped case. The parameters are $N=256,\beta =100,\sigma =2,\gamma =0.1,n_{dv}=16$, and $n_{dp}=16$. The right panel (b) displays the dispersion relations for the strongly damped $\beta$-FPU chain with the parameters $N=256,\beta =100,\sigma =2,\gamma =10,n_{dv}=16$, and $n_{dp}=32$ (note that more damping modes are used here than in (a), in addition to the stronger damp coefficient, for this strongly damped case). The spatiotemporal spectrum $ln|{\widehat{Q}}_k{\left(\omega \right)|}^2$ is plotted with its value color-coded; its corresponding peak ${\Omega }_k^{\mathrm{meas}}$ is the measured dispersion relation. Also plotted are the theoretical predictions for the renormalizations ${\omega }_k^L$ [Eq. (18)], ${\omega }_k^M$ [Eq. (20))], ${\omega }_k^N$ [Eq. (19)], and linear dispersion relation ${\omega }_k^{\left(0\right)}$ [Eq. (3)]. Note that ${\omega }_k^L,{\omega }_k^N$ nearly overlap with ${\Omega }_k^{\mathrm{meas}}$ in the left panel (a). The right panel (b) displays the center of the dispersive band ${\Omega }_k^c$ [Eq. (32)] with the dash-dot cyan (light gray) curve.

Figure 5

The kinetic energy $\langle |P_k{|}^2\rangle$ for modes $k$ in the inertial range for (a) the weakly damped and (b) the strongly damped FPU chain whose spatiotemporal spectrum is shown in Figs. 4 and 4, respectively.

Figure 6

(a) The renormalized dispersion relations for the purely quartic chain in thermal equilibrium. The chain was simulated with $N=256,\beta =100$, and $H=1000$. Plotted is the measured dispersion relation ${\Omega }_k^{\mathrm{meas}}$ [solid blue (dark gray) curve]. For comparison, also plotted are the theoretical predictions for the renormalized dispersion relations ${\omega }_k^L$ [Eq. (18)] (solid white curve), ${\omega }_k^M$ [Eq. (20)] (dashed blue curve), and ${\omega }_k^N$ [Eq. (19)] (dash-dot black curve). Note that ${\omega }_k^L$ and ${\omega }_k^N$ nearly overlap with each other. (b) The renormalization factors as a function of the energy $H$ for purely quartic chains. The chain was simulated using the parameters of $N=256$ and $\beta =100$. Plotted is the renormalization factor ${\eta }_{\mathrm{meas}}$ [Eq. (27)] (red circles). For comparison, also plotted are the renormalization factors ${\eta }_L,{\eta }_N$, and ${\eta }_M$ from Eqs. (23), (24), and (25) with black triangles, green hexagons, and blue squares, respectively. Note that ${\eta }_{\mathrm{meas}},{\eta }_L$, and ${\eta }_N$ nearly overlap with each other.

Figure 7

(a) The energy flux $f_{j,j-1}$ [Eq. (C1)] and (b) the kinetic energy $\langle p_j^2\rangle$ for various driving strengths ${\sigma }_R$ corresponding to the systems in Fig. 3. The simulation for the $\beta$-FPU chain is performed with parameters of $N=256,\beta =100,{\gamma }_L={\gamma }_R=1$, and ${\sigma }_L=1$ . The blue line with squares, magenta line with triangles, green line with hexagons and red line with circles correspond to (a) the flux profile and (b) the kinetic energy profile for the system with driving strengths ${\sigma }_R$ of 10, $7.07$, 5 and $1.25$, respectively.