Correlations and responses for a system of $n$ coupled linear oscillators with asymmetric interactions

We focus on the asymmetry of the interaction in the optimal velocity (OV) model, which is a model of self-driven particles, and analytically investigate the effects of the asymmetry on the fluctuation-response relation, which is one of the remarkable relationships in statistical physics. By linearizing a modified OV model, i.e., the backward-looking optimal velocity model, which can easily control the magnitude of asymmetry in the interaction, we derive $n$ coupled linear oscillators with asymmetric interactions. We analytically solve the equations of the $n$ coupled linear oscillators and calculate the response and correlation functions. We find that the fluctuation response relation does not hold in the $n$ coupled linear oscillators with asymmetric interactions. Moreover, as the magnitude of the asymmetry increases, the difference between the response and correlation functions increases .

Figure 1

Time evolution of response function ${\mathrm{\Phi }}_{11}\left(t\right)$ and correlation function $C_{11}\left(t\right)$ for $k_{\text{L}}=1.0, n=9$, and $k_{\text{R}}=0.25$ and 0.05. Solid and dashed curves represent the response function ${\varphi }_{11}$ and correlation function $C_{11}$, respectively. Gray regions represent the difference between the response and correlation. (a) $k_{\text{R}}=0.25$. (b) $k_{\text{R}}=0.05$.

Figure 2

Fourier transform of the response and correlation functions. Solid and dashed curves are amplitudes of the response and correlation functions, respectively. We set $k_{\text{L}}=1$ and $k_{\text{R}}=0.25$. (a) $n=9$. (b) $n=18$.