Response and correlation functions of nonlinear systems in equilibrium states

In this paper, we study systematically a serial of correlation functions in some one-dimensional nonlinear lattices. Due to the energy conservation law, they are implicitly interdependent. Various transport coefficients are thus also connected. In the studies of the autocorrelations of local energy density and of local heat current, a general relation between diverging heat conduction and super heat diffusion has been proposed recently. We clarify that such a relation is valid only in systems without temperature pressure. In those with temperature pressure, a constant but nontrivial term appears. This term explains a previously observed fact that heat diffusion in such systems is always ballistic but heat conduction can diverge very slowly. Such a result not only disproves the existence of any general relation between diverging heat conduction and super heat diffusion, but it also breaks the long-term presumption that ballistic heat conduction and diffusion always coexist.

Figure 1

The correlation functions $C_{\dot{h}h}$ (the first row), $C_{hh}$ (the second row), and $C_{jh}$ (the third row) in the FPU-$\beta$ lattice (the left column), the FPU-$\alpha 2\beta$ lattice (the middle column), and the ${\varphi }^4$ lattice (the right column). $N=1024$, $T=0.2$.

Figure 2

Numerical confirmations for the Eqs. (14), (19), and (17) in (a) the FPU-$\beta$ lattice, (b) the FPU-$\alpha 2\beta$ lattice, and (c) the ${\varphi }^4$ lattice. $N=1024$, $T=0.2$. The wave number $k$ along the arrows are $k=\frac{\pi }{8},\frac{2\pi }{8},\frac{3\pi }{8},\frac{4\pi }{8},\frac{5\pi }{8},\frac{6\pi }{8}$, and $\frac{7\pi }{8}$. The five groups of curves are (1) $\omega {\tilde{C}}_{hh}^{\prime }$ (black dashed), (2) $-{\tilde{C}}_{\dot{h}h}^{\prime }$(green short dashed), (3) $2{\tilde{C}}_{jh}^{\prime }$ (blue dotted), (4) $2tan\frac{k}{2}{\tilde{C}}_{jh}^{″}$ (purple short dotted), and (5) $k{\tilde{C}}_{jh}^{″}$ (red solid). All of them, except for the last one $k{\tilde{C}}_{jh}^{″}$, overlap each other exactly.

Figure 3

The correlation functions $C_{\dot{h}j}$ (the first row), $C_{hj}$ (the second row), and $C_{jj}$ (the third row) in the FPU-$\beta$ lattice (the left column), the FPU-$\alpha 2\beta$ lattice (the middle column), and the ${\varphi }^4$ lattice (the right column) $N=1024$, $T=0.2$.

Figure 4

Numerical confirmations for the Eqs. (22), (27), and (25) in (a) the FPU-$\beta$ lattice, (b) the FPU-$\alpha 2\beta$ lattice, and (c) the ${\varphi }^4$ lattice. $N=1024$, $T=0.2$. The wave number $k$ along the arrows are $k=\frac{\pi }{8},\frac{2\pi }{8},\frac{3\pi }{8},\frac{4\pi }{8},\frac{5\pi }{8},\frac{6\pi }{8}$, and $\frac{7\pi }{8}$. The four groups of curves are (1) $\omega {\tilde{C}}_{hj}^{″}$ (black dashed), (2) ${\tilde{C}}_{\dot{h}j}^{″}$ (green short dashed), (3) $sink{\tilde{C}}_{jj}^{\prime }$ (blue dotted), and (4) $k{\tilde{C}}_{jj}^{\prime }$ (red solid). All of them, except for the last one $k{\tilde{C}}_{jj}^{\prime }$, overlap each other exactly.

Figure 5

Numerical confirmations for the Eqs. (29) and (31) in (a) the FPU-$\beta$ lattice, (b) the FPU-$\alpha 2\beta$ lattice, and (c) the ${\varphi }^4$ lattice. $N=1024$, $T=0.2$. The wave number $k$ along the arrows are $k=\frac{\pi }{8},\frac{2\pi }{8},\frac{3\pi }{8},\frac{4\pi }{8},\frac{5\pi }{8},\frac{6\pi }{8}$, and $\frac{7\pi }{8}$. The three groups of curves are (1) ${\omega }^2{\tilde{C}}_{hh}(k,\omega )$ (black dashed), (2) ${(2sin\frac{k}{2})}^2{\tilde{C}}_{jj}(k,\omega )$ (green short dashed), and (3) $k^2{\tilde{C}}_{jj}(k,\omega )$ (red solid). The first two overlap each other very well but the third one differs apparently.