Continuum and lattice heat currents for oscillator chains

We show that two commonly used definitions for the heat current give different results—through the Kubo formula—for the heat conductivity of oscillator chains. The difference exists for finite chains, and is expected to be important more generally for small structures. For a chain of $N$ particles that are tethered at the ends, the ratio of the heat conductivities calculated with the two currents differs from unity by $O\left(1/N\right)$. For a chain held at constant pressure, the difference from unity decays more slowly, and is consistent with $O\left(1/N^{\eta }\right)$ with $1>\eta >0.5$.

Figure 1

(Color online) Results for the tethered chain. A log-log plot of the ratio of the continuum thermal conductivity with the heat current defined in Eq. (13) to the lattice thermal conductivity with the heat current defined in Eq. (12), minus one. If the continuum and lattice thermal conductivities agree in the thermodynamic limit, the results should tend to zero for $N\to \infty$. The best fit has a slope of $-0.9$, close to $-1$ instead of $-4/3$, which is explained near the end of Sec. 1.

Figure 2

(Color online) Results for the tethered chain. A log-log plot of same as for Fig. 1, but using the continuum heat current given by Eq. (26).

Figure 3

(Color online) Results for the constant pressure chain. A log-log plot of the ratio of the continuum thermal conductivity with the heat current defined in Eq. (13) to the lattice thermal conductivity with the heat current defined in Eq. (12), minus one. The best fit has a slope of about $-0.8$.

Figure 4

(Color online) Results for the constant pressure chain. A log-log plot of same as Fig. 3 but using the continuum heat current given in Eq. (26). The error bars are sufficiently large that it is not possible to estimate the size dependence reliably.