Two representations of the current density in charge-transport problems

Several fundamental problems of transport phenomena are discussed: First, the definition of the transport current density is examined. Two different definitions of the transport current density are distinguished and their explicit expressions are derived. One is the response-current density ${\mathit{j}}_{\mathit{R}}$, which refers to the local change of the current density induced by the presence of transport. The other is the Fermi-surface current density ${\mathit{j}}_{\mathit{F}}$, which represents the fraction of the current density that arises from the electrons around the Fermi surface in the presence of transport. ${\mathit{j}}_{\mathit{R}}$ and ${\mathit{j}}_{\mathit{F}}$ are equal to each other in the zero magnetic field, but different in magnetic fields. The controversy about the local current distribution in the integer quantum Hall effects substantially arises from the confusion between ${\mathit{j}}_{\mathit{R}}$ and ${\mathit{j}}_{\mathit{F}}$, viz., ${\mathit{j}}_{\mathit{F}}$ is finite only at the edge state, while ${\mathit{j}}_{\mathit{R}}$ spreads out into the interior region of a conductor. Second, the scattering-theoretic approach to transport is reexamined. The Büttiker formula is rederived through straightforward calculations from the expression of ${\mathit{j}}_{\mathit{F}}$ to unambiguously establish the validity of the formula in the presence of electric fields. Third, apart from the distinction between ${\mathit{j}}_{\mathit{R}}$ and ${\mathit{j}}_{\mathit{F}}$, there are two contributions to the transport current. One is the component driven by an external electric field, which is referred to as the electrostatic-potential current ${\mathit{j}}_{\mathit{E}}$. The other is the component carried by the extra electrons and holes added to the conductor, referred to as the chemical-potential current ${\mathit{j}}_{\mathit{c}}$. Both the response-current density ${\mathit{j}}_{\mathit{R}}$ and the Fermi-surface current density ${\mathit{j}}_{\mathit{F}}$ are, respectively, the sum of these two contributions, ${\mathit{j}}_{\mathit{R}}$=${\mathit{j}}_{\mathrm{RE}}$+${\mathit{j}}_{\mathit{c}}$ and ${\mathit{j}}_{\mathit{F}}$=${\mathit{j}}_{\mathrm{FE}}$+${\mathit{j}}_{\mathit{c}}$, where the electrostatic-potential current has different expressions ${\mathit{j}}_{\mathrm{RE}}$ and