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

Abstract

In nonautonomous dynamical systems, like in climate dynamics, an ensemble of trajectories initiated in the remote past defines a unique probability distribution, the natural measure of a snapshot attractor, for any instant of time, but this distribution typically changes in time. In cases with an aperiodic driving, temporal averages taken along a single trajectory would differ from the corresponding ensemble averages even in the infinite-time limit: ergodicity does not hold. It is worth considering this difference, which we call the nonergodic mismatch, by taking time windows of finite length for temporal averaging. We point out that the probability distribution of the nonergodic mismatch is qualitatively different in ergodic and nonergodic cases: its average is zero and typically nonzero, respectively. A main conclusion is that the difference of the average from zero, which we call the bias, is a useful measure of nonergodicity, for any window length. In contrast, the standard deviation of the nonergodic mismatch, which characterizes the spread between different realizations, exhibits a power-law decrease with increasing window length in both ergodic and nonergodic cases, and this implies that temporal and ensemble averages differ in dynamical systems with finite window lengths. It is the average modulus of the nonergodic mismatch, which we call the ergodicity deficit, that represents the expected deviation from fulfilling the equality of temporal and ensemble averages. As an important finding, we demonstrate that the ergodicity deficit cannot be reduced arbitrarily in nonergodic systems. We illustrate via a conceptual climate model that the nonergodic framework may be useful in Earth system dynamics, within which we propose the measure of nonergodicity, i.e., the bias, as an order-parameter-like quantifier of climate change.

The theory of parallel climate realizations as a new framework for teleconnection analysis

“…This means that even if the climate is changing, the ensemble average is able to give a relevant statistics in any instant of time. On the contrary, temporal averages in any nonperiodically forced system always differ from the ensemble average715, dynamics related to climate changes are thus unavoidably nonergodic42.…”

“…For a more detailed analysis of the difference between temporal and ensemble averages, i.e., of the breakdown of ergodicity during climate changes, see ref. 42.…”

Nonergodicity in the Anisotropic Dicke Model

“…For these systems, dynamically emergent conserved quantities are responsible for a variety of distinctive properties of many-body localized phases. Non-ergodic phases could also exist in driven and dissipative quantum systems [11,12]. The existence of nonergodic phases breaking traditional statistical physics by not satisfying the eigenstate thermalization hypothesis [13] can possibly be linked with the existence of a yet unknown quantum version of the classical Kolmogorov-Arnold-Moser theorem [14,15] (qualitatively stating that classical integrable systems remain quasi-integrable under weak perturbations).…”

The Theory of Parallel Climate Realizations

“…A systematic investigation of such questions was carried out by Drótos et al [94]. In order to characterize the difference between the two kinds of averages, the term nonergodic mismatch was coined.…”