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.
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.
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.
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.
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.