Ohm's law, Joule heat, and Planckian dissipation
Electric current generation and its dissipation are important physical processes. It ranges from the one follows the Ohm's law to superconductivity. Recently, it has been shown that the gradient of the chemical potential force arises from the time-component of the Berry connection from many-electron wave functions, and we consider its importance for the electric current conduction in this work. We first show that it rectifies the odd explanation in Joule heating by electric current in a metallic wire: Poynting's theorem explains that the energy for the Joule heating enters from the outside of the wire as radiation. We show that this energy is supplied by the chemical potential gradient generated by the battery connection. Next, we consider the discharging of a capacitor problem where the capacitor plays a role of a battery; and the tunneling supercurrent through the Josephson junction problem, where the original derivation did not include the capacitor contribution. Lastly, we argue that the gauge fluctuation of the time-component of the Berry connection included in the chemical potential gradient force might explain the Planckian dissipation observed in high transition temperature cuprate superconductors. The present work suggests the rethinking of the gauge invariance in Maxwell's equations.