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  "body": "\n## Statement\n\nLet $(G, c)$ be a finite [[electrical network|T7KRE7J3B4]] with vertex set $V$ and resistances $r = 1/c$, and let $a, b \\in V$ with $a \\ne b$.\nA *unit flow* from $a$ to $b$ is a function $\\theta$ on ordered pairs of neighbours with $\\theta(y, x) = -\\theta(x, y)$ such that:\n1. $\\sum_y \\theta(x, y) = 0$ for every $x \\notin \\set{a, b}$;\n2. $\\sum_y \\theta(a, y) = 1$.\n\nIts energy is $\\mathcal{E}(\\theta) = \\sum_{\\set{x, y} \\in E} r(x, y)\\, \\theta(x, y)^2$.\nThen\n$$R_{\\mathrm{eff}}(a, b) = \\min \\set{\\mathcal{E}(\\theta) : \\theta \\text{ a unit flow from } a \\text{ to } b},$$\nwhere $R_{\\mathrm{eff}}$ is the [[effective resistance|XAE5YJ3H68]].\nThe minimum is attained exactly at the unit current flow.\nThis is the current of strength $1$ from $a$ to $b$ given by a voltage that is harmonic off $\\set{a, b}$.\n\n## Remarks\n\nLet $i$ be the unit current flow, with voltage $v$, so that $r(x, y)\\, i(x, y) = v(x) - v(y)$.\nFor any unit flow $\\theta$, the difference $\\theta - i$ has net flow $0$ at every vertex.\nSumming by parts gives $\\sum_{\\set{x, y} \\in E} r(x, y)\\, i(x, y)\\big(\\theta - i\\big)(x, y) = \\sum_x v(x) \\sum_y (\\theta - i)(x, y) = 0$.\nHence $\\mathcal{E}(\\theta) = \\mathcal{E}(i) + \\mathcal{E}(\\theta - i)$, which is at least $\\mathcal{E}(i)$ with equality only for $\\theta = i$.\nFinally $\\mathcal{E}(i) = \\sum_x v(x) \\sum_y i(x, y) = v(a) - v(b) = R_{\\mathrm{eff}}(a, b)$.\nExample: let the network consist of two paths from $a$ to $b$ that share only their endpoints, with total resistances $r_1$ and $r_2$.\nA unit flow carries a constant amount $t$ along the first path and $1 - t$ along the second, so its energy is $r_1 t^2 + r_2 (1 - t)^2$.\nMinimizing over $t$ gives $r_1 r_2/(r_1 + r_2)$, the parallel law.\nThe 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 $f(b) = 0$.\n\n## Explanations\n\n### L1 human\n\nSend one unit of current through a network from $a$ to $b$.\nThere are many ways to split the current among the wires, and each way wastes some energy as heat.\nThe way that nature chooses, given by Ohm's law, wastes the least energy.\nThat least energy equals the effective resistance between $a$ and $b$.\n\n### L2 human\n\nThe principle turns effective resistance into a minimum, so any unit flow gives an upper bound on $R_{\\mathrm{eff}}(a, b)$.\nDually, any test function in Dirichlet's principle gives a lower bound.\nThe Rayleigh monotonicity law follows at once, since raising resistances raises the energy of every flow.\nOn infinite graphs, a unit flow of finite energy to infinity exists exactly when the random walk is transient.\nThis gives a proof of the transience of $\\Z^3$ by exhibiting such a flow.\n",
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