Thomson's principle

Remarks

Let $i$ be the unit current flow, with voltage $v$, so that $r(x, y)\, i(x, y) = v(x) - v(y)$. For any unit flow $\theta$, the difference $\theta - i$ has net flow $0$ at every vertex. Summing by parts gives $\sum_{\set{x, y} \in E} r(x, y)\, i(x, y)\big(\theta - i\big)(x, y) = \sum_x v(x) \sum_y (\theta - i)(x, y) = 0$. Hence $\mathcal{E}(\theta) = \mathcal{E}(i) + \mathcal{E}(\theta - i)$, which is at least $\mathcal{E}(i)$ with equality only for $\theta = i$. Finally $\mathcal{E}(i) = \sum_x v(x) \sum_y i(x, y) = v(a) - v(b) = R_{\mathrm{eff}}(a, b)$. Example: let the network consist of two paths from $a$ to $b$ that share only their endpoints, with total resistances $r_1$ and $r_2$. A unit flow carries a constant amount $t$ along the first path and $1 - t$ along the second, so its energy is $r_1 t^2 + r_2 (1 - t)^2$. Minimizing over $t$ gives $r_1 r_2/(r_1 + r_2)$, the parallel law. The dual statement, Dirichlet's principle, says that $1/R_{\mathrm{eff}}(a, b)$ is the minimum of $\sum_{\set{x, y} \in E} c(x, y)\big(f(x) - f(y)\big)^2$ over all $f$ with $f(a) = 1$ and $f(b) = 0$.

Going deeper

For a student

The principle turns effective resistance into a minimum, so any unit flow gives an upper bound on $R_{\mathrm{eff}}(a, b)$. Dually, any test function in Dirichlet's principle gives a lower bound. The Rayleigh monotonicity law follows at once, since raising resistances raises the energy of every flow. On infinite graphs, a unit flow of finite energy to infinity exists exactly when the random walk is transient. This gives a proof of the transience of $\Z^3$ by exhibiting such a flow.