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  "body": "\n## Definition\n\nLet $H \\colon \\R^{2n} \\to \\R$ be a smooth Hamiltonian and let $\\varphi_t$ be the flow of the [[Hamiltonian system|0EDKGA2FAK]] of $H$ on the [[phase space|CPEC2NW2KF]] $\\R^{2n}$.\nLet $E$ be a regular value of $H$ such that the energy surface $\\Sigma_E = H^{-1}(E)$ is compact.\nThe flow preserves $\\Sigma_E$ and a finite measure $\\mu_E$ on it, the Liouville (microcanonical) measure $\\mu_E = d\\sigma / \\norm{\\nabla H}$, where $\\sigma$ is the surface measure.\nNormalise $\\mu_E$ to a probability measure.\nThe system satisfies the *ergodic hypothesis* on $\\Sigma_E$ if for every $g \\in L^1(\\Sigma_E, \\mu_E)$ and for $\\mu_E$-almost every $x \\in \\Sigma_E$,\n$$\\lim_{T \\to \\infty} \\frac{1}{T} \\int_0^T g(\\varphi_t(x)) \\, dt = \\int_{\\Sigma_E} g \\, d\\mu_E .$$\n\n## Remarks\n\nBy Birkhoff's ergodic theorem, this holds exactly when the flow on $\\Sigma_E$ is [[ergodic|AZRPHS478B]] for $\\mu_E$: every measurable invariant subset of $\\Sigma_E$ has measure $0$ or $1$.\nExample: for $H(q, p) = \\tfrac12 (p^2 + q^2)$ on $\\R^2$ and $E \u003e 0$, the energy surface is a circle, the flow is a rotation at constant speed, and $\\mu_E$ is the normalised arc length.\nEvery orbit runs once around the circle in time $2\\pi$, so time averages equal the circle average and the hypothesis holds.\nNon-example: for two uncoupled oscillators, $H = \\tfrac12(p_1^2 + q_1^2) + \\tfrac12(p_2^2 + q_2^2)$, each energy $h_1 = \\tfrac12(p_1^2 + q_1^2)$ is conserved.\nThe set where $h_1 \u003c E/2$ is then an invariant subset of $\\Sigma_E$ of measure strictly between $0$ and $1$, so the hypothesis fails.\n\n## Explanations\n\n### L1 human\n\nThe definition makes precise the slogan \"time average equals space average\".\nOne fixes the total energy, follows a typical state forever, and compares the long-run average of a measured quantity with its average over all states of that energy.\n",
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