How can ergodicity be demonstrated using the Lorenz equations?
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.
Reasoning, step by step
- Plot the distribution of a single trajectory of length 1000 and compare it to an ensemble evolved for 5 seconds; they look remarkably different.
- Increase the trajectory length to 10000 and the ensemble evolution time to 10 seconds; the distributions now look more similar.
- Further increase the trajectory length to 100000 and the ensemble evolution time to 50 seconds; the distributions look identical.
- Conclude that the visual convergence confirms the ergodic hypothesis for the Lorenz system.
Example
The video shows side-by-side heatmaps where the left plot tracks a single trajectory over increasing time lengths (1,000 to 100,000) and the right plot shows a massive ensemble evolved over corresponding short times (5 to 50 seconds).
Common misconceptions
- Assuming the distributions are identical at short time scales.
- Thinking that the visual proof replaces the analytical proof of ergodicity.
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