What is the relationship between the time average distribution and the ensemble average distribution as time increases?
Conditions
- The system exhibits ergodicity.
- The spatial distributions of the single trajectory and the ensemble are identical.
Reasoning, step by step
- Observe the initial comparison where the distributions look remarkably different.
- Note the intermediate comparison where the distributions begin to look much more similar.
- See the final comparison where the two heatmaps are visually identical.
- Conclude that because the underlying spatial distributions are the same, calculating the average of any function over the single long trajectory will yield the exact same result as calculating the average over the large ensemble at a specific instant.
Example
The speaker explains the implication of this visual identity: because the underlying spatial distributions are the same, calculating the average of any function J over the single long trajectory (time average) will yield the exact same result as calculating the average over the large ensemble at a specific instant (ensemble average).
Common misconceptions
- Assuming the distributions remain different regardless of the time scale.
- Thinking that the equality of averages only applies to specific functions rather than any function.
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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 .; is a function of the state .; The time average is defined by .; 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.
The time average of an observable along a trajectory is defined as the limit of the average of over a finite time window of length as approaches infinity. It represents the long-run average behavior of the observable.
Conditions: The system is described by an ODE with trajectory .; is a function of the state .; The averaging interval starts at and has length .; The limit is taken.
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 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 is taken over the ensemble after long-time evolution.
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