Orderparametrilla

Orderparametrilla is a term used in theoretical discussions of phase transitions and complex systems to denote an extended order parameter that captures higher-order correlations beyond conventional order parameters. Unlike a scalar magnetization or a two-point correlation function, orderparametrilla comprises a hierarchy of tensor-valued quantities that describe multi-body coherence in a system. The concept is often introduced in pedagogical or speculative contexts to illustrate how order in a system can manifest through collective behavior.

Formally, the orderparametrilla of order m for a system with local state variables x_i is defined as the m-th order cumulant tensor O^{(m)} with components O_{i1 i2 ... im} = ⟨f_{i1}(x_{i1}) f_{i2}(x_{i2}) ... f_{im}(x_{im})⟩_c, where ⟨...⟩_c denotes connected correlators. In translationally invariant systems these components often reduce to a function of relative positions. The second-order orderparametrilla recovers familiar pair correlations, while third- and higher-order terms encode genuine multi-body order.

Applications include diagnosing phases with hidden or complex order not visible through the standard order parameter, characterizing microstructures in soft matter, and probing multi-body correlations in neural or social networks. Computation typically relies on ensemble sampling (Monte Carlo, molecular dynamics) or experimental data with careful noise and finite-size handling.

Relationship to conventional order parameters is hierarchical: the traditional order parameter emerges as a projection or low-order block of the full orderparametrilla set. The term remains unofficial and is not part of standard nomenclature, but it appears in theoretical expositions exploring higher-order coherence.