Correlations of correlations: Secondary autocorrelations in finite harmonic systems

The momentum or velocity autocorrelation function $C\left(t\right)$ for a tagged oscillator in a finite harmonic system decays like that of an infinite system for short times, but exhibits erratic behavior at longer time scales. We introduce the autocorrelation function of the long-time noisy tail of $C\left(t\right)$ (“a correlation of the correlation”), which characterizes the distribution of recurrence times. Remarkably, for harmonic systems with same-mass particles this secondary correlation may coincide with the primary correlation $C\left(t\right)$ (when both functions are normalized) either exactly, or over a significant initial time interval. When the tagged particle is heavier than the rest, the equality does not hold, correlations show nonrandom long-time scale pattern, and higher-order correlations converge to the lowest normal mode.

Figure 1

Normalized momentum correlation function $C_i\left(t\right)$ for the central particle ($i=i_0=51$) of the harmonic chain with fixed ends with Hamiltonian (1) with $N=101$, given by Eq. (3). The time unit is $1/2\omega$. The time of the crossover from the regular dissipation to “stochastic” regimes is $t_c\approx 200$.

Figure 2

Primary momentum correlation function $C_i\left(t\right)$ given by Eq. (14) (solid line) and secondary correlation function $D_i\left(t\right)$ given by Eq. (17) (dashed line) for the harmonic chain with Hamiltonian (1) with $N=101$ for particles $i=20$ (top), $i=30$ (middle), and $i=40$ (bottom). The difference between $C_i\left(t\right)$ and $D_i\left(t\right)$ becomes noticeable for $t>t_0$, where $t_0$ depends on $i$ nonmonotonically: $t_0\approx 80,120,90$, from top to the bottom.

Figure 3

Secondary correlation functions $D_{30}\left(t\right)$ (top) and $D_{40}\left(t\right)$ (bottom) according to the exact expression (17) (solid lines) and the approximation (23) (dashed lines). The insets show apparently random behavior of $D_{30}\left(t\right)$ and $D_{40}\left(t\right)$ at longer times.

Figure 4

Top plot (a): the functions ${\mathrm{\Delta }}_i\left(t\right)=C_i\left(t\right)-J_0\left(2\omega t\right)$ (solid line) and ${\delta }_i\left(t\right)=C_i\left(t\right)-D_i\left(t\right)$ (dashed line) for particle $i=40$. The characteristic times $t_c$ and $t_0$ are defined as times at which ${\mathrm{\Delta }}_i\left(t\right)$ and ${\delta }_i\left(t\right)$, respectively, have their first local extremum. Bottom plot (b): the characteristic times $t_c$ ($\times$) and $t_0$ ($+$) for particles with indices $i\le 51$ for the left side of the chain described by Hamiltonian (1) with $N=101$. For particles $i_0=51$ and $i_1=34$ the time $t_0$ diverges (illustrated by an arrow pointing upward), indicating the exact equality $C_i\left(t\right)=D_i\left(t\right)$.

Figure 5

Primary (solid line) and secondary (dashed line) momentum correlation functions, $C\left(t\right)$ and $D\left(t\right)$, for the heavy impurity problem described by Hamiltonian (36) with mass ratio $\mu =m/M=0.1$ and $N=50$. The top, middle, and bottom figures show the evolution of the correlations on short ($t<t_c$), intermediate ($t\sim t_c$), and long ($t\gg t_c$) time scales. The inset shows correlations on the short time scale for an impurity that is twice as heavy; $\mu =0.05$. Time is in units of $1/2\omega$.

Figure 6

Primary correlation $C_1\left(t\right)$ (solid line), and two higher correlations $C_3\left(t\right)$ (dashed line) and $C_5\left(t\right)$ (dotted line), defined by Eq. (34), at long time $t\gg t_c$ for the heavy impurity problem with $\mu =0.1$ and $N=50$. Higher correlations converge to the lowest normal mode $cos{\mathrm{\Omega }}_{1}t$ (dot dashed line).