Universal Relation between Heat conduction and Diffusion

Details Universal Relation between Heat conduction and Diffusion Universal Relation between Heat conduction and Diffusion

2011-03-15

arxiv.org/abs/1103.2835v2

Physics

Condensed Matter

Statistical Mechanics

Scientific paper

We derive analytically a universal equality relating heat current autocorrelation function $C(t)=\lim_{L\to\oo}\frac{1}{L}\mean{J_L(t)J_L(0)}$ to the variance of the energy distribution $\mean{x^2(t)}$ of the corresponding energy diffusion process, namely, $mathrm{d}^2\mean{x^2(t)}/\mathrm{d}t^2=2C(t)/(k_BT^2c_V)$, where $c_V$ is the heat capacity. From this equality, we can recover the existing theories for normal heat conduction using the Green-Kubo formula. More importantly, We are also able to obtain a connection between anomalous heat conduction and anomalous heat diffusion, namely, $\kappa_L=\frac{c_V}{2}\dif{\mean{x^2(t)}}{t}\Big|_{t\sim L/v_s}$, with $L$ the system size and $v_s$ the sound speed. Our results can be applied to all isotropic systems in any dimension. These facts make it possible to understand anomalous heat conduction in the framework of anomalous diffusion.

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