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Operationally, the ensemble average is described by starting an ensemble of trajectories, assuming the ensemble has a density, evolving it for a long time, and averaging the observable $J$ across the ensemble.

Conditions: One considers an ensemble of trajectories rather than a single trajectory.; The ensemble is assumed to have a density.; The average of $J$ is taken over the ensemble after long-time evolution.

Operationally, the ensemble average is described by starting an ensemble of trajectories, assuming the ensemble has a density, evolving it for a long time, and averaging the observable $J$ across the ensemble.

Conditions: One considers an ensemble of trajectories rather than a single trajectory.; The ensemble is assumed to have a density.; The average of $J$ is taken over the ensemble after long-time evolution.

The statement says almost everywhere because the equality of the time average and the ensemble average is asserted for all initial conditions except possibly a negligible exceptional set. The clip does not specify the exact nature of this exceptional set.

Conditions: The system is governed by an ODE $du/dt=f(u)$.; $J$ is a function of the state $u$.; The time average is defined by $\lim_{T\to\infty}\frac{1}{T}\int_0^T J(u(t))dt$.; The comparison ensemble has a density.

The statement says almost everywhere because the equality of the time average and the ensemble average is asserted for all initial conditions except possibly a negligible exceptional set. The clip does not specify the exact nature of this exceptional set.

Conditions: The system is governed by an ODE $du/dt=f(u)$.; $J$ is a function of the state $u$.; The time average is defined by $\lim_{T\to\infty}\frac{1}{T}\int_0^T J(u(t))dt$.; The comparison ensemble has a density.

The time average of an observable $J$ along a trajectory $u(t)$ is defined as the limit of the average of $J$ over a finite time window of length $T$ as $T$ approaches infinity. It represents the long-run average behavior of the observable.

Conditions: The system is described by an ODE with trajectory $u(t)$.; $J$ is a function of the state $u$.; The averaging interval starts at $t=0$ and has length $T$.; The limit $T\to\infty$ is taken.

The time average of an observable $J$ along a trajectory $u(t)$ is defined as the limit of the average of $J$ over a finite time window of length $T$ as $T$ approaches infinity. It represents the long-run average behavior of the observable.

Conditions: The system is described by an ODE with trajectory $u(t)$.; $J$ is a function of the state $u$.; The averaging interval starts at $t=0$ and has length $T$.; The limit $T\to\infty$ is taken.