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By analogy with the fundamental quantum units of electrical conductance$G_0^e=\frac{2 e^2}{h}$ and thermal conductance $K_0^t=\frac{2 K_B^2 T}{h}$ wedefine a fundamental quantum unit of conductance, $G_0^m$, and diffusion of amassive gas of atomic particles, respectively given by $$ G_0^m=\frac{m^2}{h} \, \ D_0=\frac{h}{m}$$ with $h$ the Planck constant, $K_B$ the Boltzmannconstant, $T$ the absolute temperature, $e$ the unit charge and $m$ the mass ofthe atomic gas particle that move balistically in a one dimensional medium oflength $L$. The effect of scattering can be accounted for by introducing anappropriate transmission probability in analogy with the quantum electricalconductance model introduced by Landauer in 1957. For an electron gas$G_0^m=1.25 \times 10^{-27} \ Kg^2/(J s)$ and $D_0 = 7.3 \times 10^{-3} \m^2/s$, and we found a quantum expression for the generalized Einstein relationthat writes $$G_0^e = \frac{2e^2m}{h^2} D_0 $$

The fundamental unit of quantum conductance and quantum diffusion for a gas of massive particles