---
slug: ergodicity
name: Ergodicity
one_liner: "The property of a system for which the time average along one long-run trajectory equals the ensemble average over many parallel copies at an instant — so that a single history is representative of the whole distribution, and its failure marks precisely when expected-value reasoning misleads."
type: prime
schema_version: 2
status: density_v2_draft
categories: [structural, mathematical, physical]
origin_domain: statistical_mechanics
alternate_origin_domains: [probability_theory, dynamical_systems, economics, evolutionary_biology]
aliases: [time_average_equals_ensemble_average, ergodic_hypothesis, ergodic_property, time_vs_ensemble_average, path_average_equals_population_average]
related: []
solution_archetypes: []
review_flags:
- "OPERATOR-DERIVED (op-11 analogy completion — cross-relational proportional analogy, NOT a single-operator re-run) — reached by completing `time_average : ensemble_average :: stationarity : ergodicity`: ergodicity is the distinct structural property naming the EQUALITY of the two averaging routes, obtained as the fourth term of the proportion rather than as an operator transform of `stationarity`. Discovered in the op-11 pilot 2026-07-04; the STRONGEST find of that pilot. STRONG hold (verified absent from corpus; established scientific name — Boltzmann's ergodic hypothesis; Birkhoff/von Neumann ergodic theory; recently prominent via Peters' ergodicity economics). STATUS: HOLD pending owner keep/cut."
- "Created 2026-07-04; STUB ONLY (frontmatter + Core Idea + Notes). SI/SF/similarity are PROPOSER ESTIMATES pending triangulation. Body sections (Structural Signature onward) NOT written."
- "Phase-C action: re-similarity-check vs `stationarity` (nearest existing prime — related but distinct), `expected_value`/`expected_utility` (ergodicity names exactly WHEN averaging-over-uncertainty is valid — a deep complement), `stochastic_process`, `stationary_distribution` (an op-06 domain-specific backlog item), and `law_of_large_numbers` (an op-05 hold — LLN is the ergodic-like convergence; ergodicity is the time=ensemble identity); confirm it is a distinct property and not a restatement of stationarity or LLN; draw a related edge to `stationarity`, a complement/composition relation to `expected_value`, and note the LLN relationship at incorporation; write a v1 draft and sync the one_liner across v1/v2 per CLAUDE.md before merge."
substrate_independence:
  composite_substrate_independence: 4
  domain_breadth: 5
  structural_abstraction: 4
  transfer_evidence: 4
  reasoning: 'The bare structure — a system in which the *time average* taken along one long-run trajectory coincides with the *ensemble average* taken across many parallel copies at a single instant, so that one history samples the whole state space in the right proportions — is a genuinely substrate-neutral property. It recurs wherever there is a distinction between "one path over time" and "many paths at once": equilibrium averages in statistical mechanics, Markov chains reaching their stationary distribution in probability and dynamical systems, non-ergodic wealth growth in economics (why the population''s average return tells you nothing about your one trajectory), and lineage-versus-population averages in evolution. These are structurally-unlike siblings sharing only the abstract time-vs-ensemble identity, so domain_breadth scores 5. Structural abstraction is 4 because the core is a pure statement about the equality of two averaging operations over a state space, requiring no physical substance to hold; it stays just short of 5 because the natural framing still leans on the notion of "average" and a measure over states, a light quantitative residue. Transfer evidence is strong — the construct is independently named and load-bearing in physics, probability/dynamical-systems theory, and economics.'
similarity_to_nearest_existing_prime:
  slug: stationarity
  similarity: 0.59
structural_framed:
  label: structural
  aggregate: 0.32
  criteria: {vocab_travels: 0.75, evaluative_weight: 0.15, institutional_origin: 0.1, human_practice_bound: 0.15, import_vs_recognize: 0.35}
  rationale: 'Reads structural. The core claim — that the time average along one trajectory equals the ensemble average across parallel copies, so a single history represents the whole distribution — is recognized rather than imported, and travels cleanly into probability, dynamical systems, economics, and evolution without carrying statistical-mechanics vocabulary. The frame residue is light: "average," "trajectory," and "ensemble" import a mild quantitative/measure-theoretic cast (keeping import_vs_recognize and vocab_travels off the ceiling), but the underlying structure — an identity between two ways of averaging over a state space, and the divergence of the two when it fails — is substrate-neutral. Lower framing than the surrounding framed-stratum candidates; comparable to its neighbour `stationarity`.'
phase_b_passes_completed: []
---
# Ergodicity

## Core Idea

Ergodicity is the property of a system for which the *time average* — the value you get by following one trajectory for a long time — equals the *ensemble average* — the value you get by averaging over many parallel copies of the system at a single instant. The distinctive anatomy is that a single long-run sample path visits the whole state space in the right proportions, so that one history is *representative* of the whole distribution, and, crucially, the two otherwise-different routes to an average *coincide*. Where the base notion of a distribution asks "what is the average outcome across all the ways things could go?", ergodicity asks the sharper, structurally distinct question: "does following one realization long enough recover that same average?" When the answer is yes, one trajectory and the whole population carry the same information; the system is ergodic.

The load-bearing content lives just as much in the *failure*. When a system is non-ergodic, the time average and the ensemble average *diverge*: the average outcome computed over the population tells you nothing reliable about the fate of any single individual following one path. This is the deep reason expected-value reasoning can mislead for one-shot, path-dependent, or absorbing-state situations — the gambler who computes a favorable expected return but cannot bank on it, because they live exactly one trajectory rather than the whole ensemble, and a single ruinous drawdown ends their path before the average can assert itself. Ergodicity is thus not merely a technical regularity condition; it is the precise structural marker of *when averaging over uncertainty is trustworthy*. Its presence licenses the substitution of an ensemble average for a lived history; its absence quietly invalidates that substitution.

Read structurally, the same time-versus-ensemble identity recurs wherever there is a distinction between "one path over time" and "many paths at once," and the siblings are genuinely unlike. In *statistical mechanics* it is the ergodic hypothesis underwriting the replacement of a system's long-time behavior with an equilibrium ensemble average. In *probability and dynamical systems* it is the condition under which a Markov chain's long-run occupation of states matches its stationary distribution, and time-averaged observables converge to their space-averaged expectations. In *economics* it is the non-ergodicity of multiplicative wealth dynamics — why the average return across parallel investors cannot stand in for the growth rate along any one investor's actual path. In *evolution* it is the gap between a lineage average traced over generations and a population average taken across contemporaries. Across these the shared abstract engine is a single identity — one path's long-run average equals many paths' instantaneous average — with the biological, physical, and economic cases as species of that genus.

## Notes

**Provenance.** Produced by operator 11 (analogy completion), as the strongest find of the op-11 pilot on 2026-07-04. Reached not by a single-operator transform of an existing prime but by completing the cross-relational proportional analogy `time_average : ensemble_average :: stationarity : ergodicity` — i.e. ergodicity is the fourth term that stands to `stationarity` as `ensemble_average` stands to `time_average`, naming the distinct structural property that the *two averaging routes coincide*. Established scientific construct: Boltzmann's ergodic hypothesis in statistical mechanics; the ergodic theorems of Birkhoff and von Neumann in ergodic theory / dynamical systems; recently prominent via Ole Peters' "ergodicity economics" and its treatment of non-ergodic wealth dynamics. Claimed siblings: Markov chains reaching their stationary distribution (probability / dynamical systems), non-ergodic multiplicative growth (economics), lineage-versus-population averages (evolution). Verified absent from the corpus. Manifest: STRONG hold.

**For owner evaluation (keep/cut).** The case FOR is strong: ergodicity is a *canonical, named scientific construct* (not a proposer coinage), verified absent, with a clean structural anatomy — the equality of the time average and the ensemble average, and the informative divergence of the two when it fails. It is genuinely foundational: the corpus's `stationarity`, `expected_value`, and `law_of_large_numbers` primes *presuppose* the time-vs-ensemble distinction but never name it, and ergodicity names precisely *when averaging over uncertainty is valid* — a major complement to `expected_value` (an op-06 hold). Its cross-domain reach (physics, probability, economics, evolution, decision-making) is deep, and it is the clearest single find of the op-11 pilot. The RISKS Kurt should weigh: (1) **closeness to `stationarity`** (present) — the discriminating claim is that stationarity is time-*invariance* of the distribution, whereas ergodicity is the *equality of the two averaging routes*; these are distinct properties (a process can be stationary without being ergodic), so this is a genuine addition, not a restatement. (2) **Closeness to `law_of_large_numbers`** (an op-05 hold) — argue that LLN is the *convergence* of a sample mean, while ergodicity is the stronger *time = ensemble identity* along a single trajectory; note the relationship rather than merging. (3) **May read as physics-specific** — argue the economics/evolution/decision generalizations, which are structurally load-bearing and not metaphorical. Sharp keep/cut question: *is "the time average equals the ensemble average" a distinct prime, or is it already covered by `stationarity` plus `law_of_large_numbers`?* — the proposer's answer is that the two-averages-coincide identity is a level distinct from time-invariance (stationarity) and from sample-mean convergence (LLN), and is the exact structural condition under which expected-value reasoning is licensed. Recommend evaluating alongside `stationarity`, `expected_value`, and `law_of_large_numbers`.

**At incorporation.** Re-run similarity against `stationarity` (nearest existing prime — related, distinct), `expected_value` / `expected_utility` (ergodicity names exactly when averaging-over-uncertainty is valid — a deep complement), `stochastic_process`, `stationary_distribution` (an op-06 domain-specific backlog item — a specific ergodic-limit object), and `law_of_large_numbers` (op-05 hold — the ergodic-like convergence) to confirm no collapse. If kept: draw a *related* edge to `stationarity` (distinct-but-adjacent properties of a stochastic process), a complement / composition relation to `expected_value` (ergodicity is the validity condition on ensemble-average reasoning), and add a missing-edge note flagging the `law_of_large_numbers` relationship (LLN = convergence; ergodicity = time-equals-ensemble identity) and the `stationary_distribution` connection explicitly. Write a v1 draft and sync the one_liner across v1 and v2 per CLAUDE.md before merge.
