Ballistic Thermal Transistor of Dielectric Four-terminal Nanostructures

We report a theoretical model for a thermal transistor in dielectric four-terminal nanostructures based on mesoscopic ballistic phonon transport, in which a steady thermal flow condition of system is obtained to set up the temperature field effect of gate. In the environment, thermal flow shows the transisting behaviors at low temperatures: saturation, asymmetry, and rectification. The phenomena can be explained reasonably by the nonlinear variation of the temperature dependence of propagating phonon modes in terminals. The results suggest the possibility of the novel nano-thermal transistor fabrication

Variation comparison between the FF-distribution and the normal distribution

Let $X_{d_1,d_2}$ be an $F$-random variable with numerator and denominator degrees of freedom $d_1$ and $d_2$, respectively. We investigate the inequality: $P\{|X_{d_1,d_2}-E[X_{d_1,d_2}]|\le \sqrt{{\rm Var}(X_{d_1,d_2})}\}\ge P\{|W-E[W]|\le \sqrt{{\rm Var}(W)}\}$, where $W$ is a standard normal random variable or a $\chi^2(d_1)$ random variable. We prove that this inequality holds for $d_1\in\{1,2,3,4\}$ and $5\le d_2\in\mathbb{N}$