How is the ensemble average described operationally in the talk?
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.
Reasoning, step by step
- Start an ensemble of trajectories according to a given density.
- Evolve all members of the ensemble under the same dynamics for a long time.
- Average the observable across the set of ensemble members.
- Conclude that this ensemble average coincides with the almost-everywhere-initial-condition-independent time average.
Example
The speaker explains that if one starts an ensemble, as long as the ensemble has a density, evolves it for a long time, and averages over the set of ensemble members, one gets the same average.
Common misconceptions
- Confusing the ensemble average with the time average of a single trajectory.
- Assuming the ensemble average requires a specific measure-theoretic definition that is written out on the slide.
Watch the explanation
Connected concepts
Explore next
Related questions
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.
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.
Answers are generated from source material and independently checked. Consult the original video or creator if something is unclear.