Dynamic structure factors and Lyapunov modes in disordered chains

We investigate various dynamic structure factors for harmonic and anharmonic chains. For harmonic chains with mass disorder we find some unexpected features, such as a fine structure contributing to a central peak, which is present also in the spatial spectra of the eigenfunctions. These results are contrasted with structure factors of Lyapunov modes obtained for the disordered Lennard-Jones chain. For this nonlinear system the static and the dynamic Lyapunov structure factors show opposite trends in their temperature dependence.

Figure 1

(Color online) Dynamic structure factors of a harmonic chain of alternating masses with ratio of 8:1, temperature $\beta =100$, and periodic boundary conditions: (a) mass density [$S_m\left(k,\omega \right)$, weights $w_{\nu }=m_{\nu }$], (b) number density [$S_{\varrho }\left(k,\omega \right)$, weights $w_{\nu }=1$]. If not stated otherwise, the same color coding is used in the other figures.

Figure 2

(Color online) Mass density dynamic structure factor $S_m\left(k,\omega \right)$ of harmonic chains of alternating masses for a mass ratio of 2:1, inverse temperature $\beta =100$, and different boundary conditions: (a) periodic, (b) fixed, (c) open. Note the appearance of a central peak $\left(\omega =0\right)$ in (c).

Figure 3

(Color online) Density of states for a harmonic chain with binary isotopic disorder for three different mass ratios $m_0/m_1$: (a) 3:2, (b) 2:1, (c) 4:1. Data have been obtained from averaging over $100000$ random permutations of a chain of length $N=5000$, with different masses to occur with equal probability. The dotted red line shows for comparison the density of states of the corresponding periodic chain.

Figure 4

(Color online) Mass density structure factor $S_m\left(k,\omega \right)$ of a binary disordered harmonic chain at inverse temperature $\beta =1000$ for mass ratios (a) 3:2 (subcritical), (b) 4:1 (supercritical, cf. Figure 3) and (c) number density structure factor $S_{\varrho }\left(k,\omega \right)$ for mass ratio 4:1. The data have been averaged over a sample of ten realizations.

Figure 5

(Color online) Dynamic structure factor of a binary disordered harmonic chain of masses with ratio 4:1 and temperature $\beta =100$: (a) structure factor computed from the mass density, (b) structure factor computed from the number density.

Figure 6

(Color online) Central peak in the mass structure factor as a function of the wavenumber. Symbols correspond to cross sections of the data shown in Fig. 5a, lines are numerical fits with a Lorentzian.

Figure 7

(Color online) Linewidth and spectral weight of the central peak as a function of the wave number [cf. Figs. 5a, 6]. The straight lines correspond to least square fits for low wave numbers with slope two and zero, respectively.

Figure 8

(Color online) (a) Mass structure factor of a binary disordered harmonic chain of $N=1000$ particles at inverse temperature $\beta =1000$ with mass ratio 4:1 for a single disorder realization [cf. 4(b) for an average over 10 realizations]. (b) Detailed view of the fine structure for small wave numbers.

Figure 9

(Color online) Spatial power spectrum of the eigenmodes as function of the eigenfrequencies for a binary disordered chain of (a) $N=200$ and (b) $N=1000$ particles (periodic boundary conditions, mass ratio 4:1).

Figure 10

(Color online) Dynamic structure factors of a Lennard-Jones chain of $N=200$ particles with binary disordered masses with ratio 4:1, density $\rho =1$, and increasing temperatures: (a,c,e) dynamical structure factor of the mass density, (b,d,f) dynamical Lyapunov structure factor for mode number 196. Inverse temperatures: (a,b) $\beta =100$, (c,d) $\beta =10$, and (e,f) $\beta =1$.

Figure 11

(Color online) Static Lyapunov structure factors of a Lennard-Jones chain of $N=200$ particles with binary disordered masses with ratio 4:1, density $\rho =1$, and increasing temperatures. Inverse temperatures: (a) $\beta =100$, (b) $\beta =10$, (c) $\beta =1$ and (d) $\beta =0.1$ (cf. as well Fig. 12 for a detailed view).

Figure 12

(Color online) (a) Static Lyapunov structure factor and (b) $\lambda \text{−}k$ dispersion relation of a Lennard-Jones chain of $N=200$ particles, with binary disordered masses with ratio 4:1 and density $\rho =1$. Note that (a) is an enlargement of the panel (d) of Fig. 11 to show the linear dispersion.

Figure 13

(Color online) Debye-Waller approximation for the dynamic structure factor of a harmonic chain of alternating masses with ratio 2:1, temperature $\beta =100$, and open boundary conditions [cf. Eq. (B5)]. The approximation yields finite values only for frequencies covered by the dispersion relation and the eigenmode in the spectral gap is clearly visible [cf. Fig. 2c for the exact result].

Figure 14

(Color online) Static approximation for the dynamic structure factor of a harmonic chain of alternating masses with ratio 2:1, temperature $\beta =100$, and open boundary conditions [cf. Eq. (B7)]. The approximation yields finite values only for frequencies covered by the dispersion relation and the eigenmode in the spectral gap is clearly visible [cf. Fig. 2c for the exact result].