What does the time average of an observable along a trajectory mean in the context of ergodicity?

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.

Ergodicity can be demonstrated by comparing the spatial distribution of a single long trajectory with that of a large ensemble of initial conditions. As the time scales increase, the two distributions visually converge to become identical.

Conditions: Applies to the Lorenz equations within a specific parameter range where ergodicity can be proven analytically.; The phase space is projected onto the x and z axes.; An ensemble of approximately 500 million points is used.

According to ergodicity arguments, the butterfly effect cannot change the macroscopic climate statistics because the statistical distribution of the system remains unchanged regardless of the method of averaging. While individual trajectories may diverge due to sensitive dependence on initial conditions, the overall probability distribution remains stable.

Conditions: Assumes ergodicity of the climate system.; Focuses on statistical invariance rather than specific trajectories.

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.

As the time span for a single trajectory and the evolution time for an ensemble increase, their resulting spatial distributions in the phase space converge to being identical. When these distributions become identical, the time average of any function is equal to its ensemble average.

Conditions: The system exhibits ergodicity.; The spatial distributions of the single trajectory and the ensemble are identical.