Definition

1. Adjective:
   • (Mathematics, Statistics, Physics) Describing a stochastic or dynamic system whose statistical properties can be deduced from a single, sufficiently long, random sample of the system. In essence, over time, the system's average behavior is the same as its average behavior across all possible states.
   • (Technical) Pertaining to or involving the equality of time averages and ensemble averages.

Usage

• The term is primarily used in advanced scientific and mathematical contexts, such as statistical mechanics, information theory, and dynamical systems theory.
• It describes a fundamental property of certain random processes or systems.

Examples

• Adjective:
  • The physicist assumed the system was ergodic, allowing her to use time-series data to estimate its long-term statistical properties.
  • In an ergodic process, the average outcome you observe over a long period for one particle will be the same as the average outcome across all particles at a single moment.

Advanced Usage

• Ergodic Hypothesis: A key postulate in statistical mechanics which states that over a long period, a system will spend an equal amount of time in each of its accessible microstates. This hypothesis allows for the connection between microscopic mechanics and macroscopic thermodynamics.
  • The validity of the ergodic hypothesis is central to the foundations of statistical physics.
• Ergodic Theory: A branch of mathematics that studies the long-term average behavior of dynamical systems.
  • His research is in ergodic theory, focusing on the properties of measure-preserving transformations.

Variants and Related Words

• Ergodicity (n): The property of being ergodic.
  • The ergodicity of the market process is a common assumption in some financial models.
• Non-ergodic (adj): Not possessing the ergodic property; where time averages and ensemble averages differ.
  • Systems with broken symmetry are often non-ergodic.

Synonyms

• (Technical) Stationary (in specific contexts where stationarity and ergodicity coincide for mean and covariance).
• (Conceptual) Representative (in the sense that a single sample path represents the whole ensemble).

Notes on Different Meanings

• The reference from the English-Vietnamese dictionary defines "ergodic" in the context of ergonomics (from Greek = work). This is a distinct and less common usage, often considered a confusion with the word "ergonomic." The primary and overwhelmingly dominant meaning in academic English is the mathematical one defined above, relating to stochastic systems and time averages.

Adjective

1. positive recurrent aperiodic state of stochastic systems; tending in probability to a limiting form that is independent of the initial conditions