ergodicity: one construct, many systems

In statistical mechanics, ergodicity is the property that a single system left to evolve long enough explores its accessible states so thoroughly that the long-run average of a quantity along one trajectory in time equals the average taken over the whole ensemble of possible states at one instant. Boltzmann's ergodic hypothesis first asserted this equivalence, and Birkhoff's 1931 Proof of the ergodic theorem gave it a rigorous foundation, establishing the conditions under which the time average of a measure-preserving system converges and coincides with the space average. The referent is an equivalence between averaging over time and averaging over states for a physical system, tested through invariant measures and mixing. It is the parent statement of the same property that economics measures on growth rates and that psychology measures across people.[1]

Unit

The equality of a time average along one trajectory and the ensemble average over states

Recovery means

Not applicable

Measured by

Established by showing the dynamics preserve a measure and that time averages converge almost everywhere to the ensemble average

In psychology, ergodicity is the condition a process must satisfy for statistics gathered across a group of people to also describe the dynamics of any single person over time; when it fails, group averages simply do not generalise to the individual. Molenaar's 2004 A manifesto on psychology as idiographic science showed that most psychological processes are non-ergodic, because the structure of variation between people differs from the structure of variation within one person across occasions, and argued this makes person-specific, time-series study indispensable. The referent is whether between-person and within-person structure coincide for a psychological attribute, examined with individual longitudinal data. It is the same time-versus-ensemble equivalence that physics defines for dynamical systems, applied here to human variation.[2]

Unit

Whether between-person structure equals within-person structure for a psychological process

Recovery means

Not applicable

Measured by

Diagnosed by comparing a variable's covariance structure across people with its covariance structure within a person over many measurement occasions

In economics, ergodicity is the question of whether the time-average growth rate experienced by a single individual equals the ensemble-average growth rate computed across many parallel individuals, a distinction that decides whether expected-value reasoning describes what one person actually lives through. Peters's 2019 The ergodicity problem in economics argued that many gambles and investment processes are non-ergodic, so that expected wealth across a population keeps rising while the typical individual's wealth stagnates or falls, dissolving long-standing puzzles about risk and utility. The referent is the gap between what happens to one economic agent over time and the population average at a moment. It is the same equivalence physics defines for a gas exploring its states, here asked of an individual's wealth trajectory.[3]

Unit

The gap between an individual's time-average growth rate and the population's ensemble-average growth rate

Recovery means

Not applicable

Measured by

Assessed by comparing the growth rate of a single trajectory over time with the expectation across many realisations of the same process