# Why does current arrange itself in such a way as to minimize power loss in resistors connected in parallel?

In problem 2 in Problem Set 6, it is said:

"Electricity prefers to flow in the way that minimizes energy loss to resistance."

Using Lagrange multipliers I was able to show that assuming the above statement, in parallel, $$I_kR_k=V$$, a constant, for each resistor attached in parallel. Physically, this makes sense since if we assume Ohm's law is obeyed, this result is just an immediate corollary of putting resistors in parallel: the terminals of all the resistors coincide so the potential differences across all the resistors, which are path-independent, are the same.

So my question is, is there a deeper physical reason why current distributes itself so that it minimizes power loss?

## 1 Answer

The minimum entropy production principle (MINEP) has been suggested as a key rule in non-equilibrium thermodynamics in steady states. The idea is that the thermodynamic fluxes (in this case currents) organise themselves to minimize entropy production (in this case indicated by power dissipation as heat) because the thermodynamic "forces" set themselves up to do it (basically by constrained optimization like you did for the currents).

MINEP is not always valid. It assumes a system that is relatively close to equilibrium with linear, time-independent responses. If you increase the voltage enough there will presumably be a current breaking through and frying your circuit while maximizing entropy - far from equilibrium maximum entropy production is common. There is a confusing array of results and principles here; my impression is that this field is not settled yet.