Sigmaeqm

Sigmaeqm is a theoretical construct in probability theory and statistical mechanics used to describe a class of equilibrium states defined with respect to a specified sigma-algebra on the system’s state space. The name signals a synthesis of sigma-algebra (sigma) with equilibrium (eqm) to emphasize coarse-graining or partial observability in modeling.

Definition: Let (X, F, μ) be a probability space with a measure-preserving transformation T: X → X. Let G ⊆ F be a sub-sigma-algebra representing macroscopic observables. A measure μ is called a sigmaeqm relative to G if (i) μ is T-invariant, and (ii) the macroscopic statistics of μ are determined entirely by G; equivalently, μ is determined by its G-marginal distribution. In practice, this means that any two T-invariant measures with the same G-distribution are identified as the same sigmaeqm.

Properties: Sigmaeqm generalize standard invariant measures by incorporating coarse-graining constraints. The class can be non-unique; they can form a convex set and may depend on the choice of G. They are useful for studying reduced dynamics and information loss under partial observation.

Construction and relations: Given a standard invariant measure ν, its projection onto G yields a G-marginal. Sigmaeqm can be obtained by selecting invariant measures consistent with a given G-marginal. They relate to coarse-grained entropy and to ergodic decomposition under observable constraints.

Applications: modeling of macroscopic states in statistical physics, coarse-grained simulations, and information-theoretic analyses of dynamical systems. See also invariant measure, ergodic theory, coarse-graining, sigma-algebra.