Energy localization in the ${\Phi }^4$ oscillator chain

We study energy localization in a finite one-dimensional ${\Phi }^4$ oscillator chain with initial energy in a single oscillator of the chain. We numerically calculate the effective number of degrees of freedom sharing the energy on the lattice as a function of time. We find that for energies smaller than a critical value, energy equipartition among the oscillators is reached in a relatively short time. On the other hand, above the critical energy, a decreasing number of particles sharing the energy is observed. We give an estimate of the effective number of degrees of freedom as a function of the energy. Our results suggest that localization is due to the appearance, above threshold, of a breather-like structure. Analytic arguments are given, based on the averaging theory and the analysis of a discrete nonlinear Schrödinger equation approximating the dynamics, to support and explain the numerical results.

Figure 1

$N_{\mathit{eff}}$ vs time for $m=7$, at different values of the energy $E$. Equipartition level at $N=500$ is $\nu N=350$ and $E_c=130$.

Figure 2

$N_{\mathit{eff}}$ vs time for $m=100$, at different values of the energy $E$. Equipartition level at $N=500$ is $\nu N=350$ and $E_c=266$.

Figure 3

Logarithm of the energy $E$ vs $1∕N_{\mathit{eff}}(T_m,E)$ for four different values of $m$. Note that the slope of all curves look the same but for the curve at $m=1.73$.

Figure 4

Logarithm of the ratio $E∕m^2$ vs $1∕N_{\mathit{eff}}(T_m,E)$ for the same four different values of $m$ of Fig. 3. Note that all curves are superposed but the curve at $m=1.73$.