From conductance viewed as transmission to resistance viewed as reflection. An extension of Landauer quantum paradigm to the classical case at finite temperature

In this paper we present an extension of Landauer paradigm, conductance is transmission, to the case of macroscopic classical conductors making use of a description of conductance and resistance based on the application of the fluctuation dissipation (FD) theorem. The main result is summarized in the expressions below for conductance $G$ and resistance $R$ at thermodynamic equilibrium, with the usual meaning of symbols. $G$ is given in terms of the variance of total carrier number fluctuations between two ideal transparent contacts in an open system described by a grand canonical ensemble as $$ G =\frac{e^2 \overline{v_x'^2} τ}{L^2 K_BT} \overline{δN^2} %= \frac{e^2 \sqrt{\overline{v_x'^2}} Γ}{L K_BT} \overline{δ%N^2} %= \frac{e^2 \overline{N} Γ} {Lm\sqrt{\overline{v_x'^2}}} \ \ \ \ $$ By contrast $R$ is given in terms of the variance of carrier drift-velocity fluctuations due to the instantaneous carrier specular reflection at the internal contact interfaces of a closed system described by a canonical ensemble as $$ R= \frac{(m L)^2}{e^2 K_BT τ} \overline{δv_d^2} %= \frac {Lm\sqrt{\overline{v_x'^2}}} {e^2 \overline{N} Γ} $$ The FD approach gives evidence of the duality property of conductance related to transmission and resistance related to reflection. Remarkably, the expressions above are shown to recover the quantum Landauer paradigm in the limit of zero temperature for a one-dimensional conductor.