Quantifying nonergodicity in nonautonomous dissipative dynamical systems: An application to climate change

Area of Science:

• Dynamical Systems Theory
• Climate Dynamics
• Statistical Mechanics

Background:

• Nonautonomous dynamical systems, particularly in climate dynamics, feature time-varying probability distributions.
• Aperiodically driven systems may not be ergodic, meaning temporal averages along a single trajectory differ from ensemble averages, even in the infinite-time limit.

Purpose of the Study:

• To introduce and analyze the concept of "nonergodic mismatch" in dynamical systems.
• To propose "bias" as a robust measure of nonergodicity and "ergodicity deficit" as the expected deviation from temporal and ensemble average equality.
• To demonstrate the utility of this nonergodic framework in Earth system dynamics, particularly for quantifying climate change.

Main Methods:

• Analysis of temporal averages over finite time windows.
• Quantification of the "nonergodic mismatch" between temporal and ensemble averages.
• Introduction of "bias" (average mismatch) and "ergodicity deficit" (average modulus of mismatch) as key metrics.
• Application to a conceptual climate model.

Main Results:

• The probability distribution of nonergodic mismatch differs qualitatively between ergodic and nonergodic systems.
• Bias is a useful measure of nonergodicity across all window lengths.
• Ergodicity deficit cannot be arbitrarily reduced in nonergodic systems.
• Standard deviation of mismatch shows a power-law decrease with window length in both cases.

Conclusions:

• Bias serves as a reliable order-parameter-like quantifier for nonergodicity and climate change.
• The ergodicity deficit highlights fundamental differences between temporal and ensemble averages in nonergodic systems.
• The proposed nonergodic framework offers valuable insights into Earth system dynamics and climate variability.