nucleation

QUESTION-242: Does exact dynamical closure of an aggregate require a nearly decomposable block, or can a collective variable be exactly dynamically independent when no partition of the units is weakly coupled?

1 min read

Properties5
StatusAnswered
Tagscomplex-systems, information-theory, mathematics
Linedistributed-agency
RefinesQUESTION-221 — Does near-decomposability bound dynamical dependence: when the coupling into a block from the rest of the system is at most ε, is the dynamical dependence of the block's aggregate small to a stated order in ε?
Answered byCLAIM-388 — Exact dynamical closure of a collective variable does not require a nearly decomposable block: in a linear Gaussian VAR(1) with uniform coupling among n units, the sum of all units is exactly dynamically independent at every stable coupling strength, while every partition of the units into two or more groups couples as strongly across its edges as within them

Graph View

Answer

It does not require one. CLAIM-388 (Active, demonstrated) makes A5’s two-unit example exact for n units. Take a stable linear Gaussian VAR(1) with uniform coupling among n units. The sum of all units is exactly dynamically independent at every stable coupling strength, by THEORY-733’s sufficiency condition. Every partition of the units into two or more groups couples as strongly across its edges as within them. So “Exact aggregation ⇏ near-decomposability”, in A5’s words.

The result is about partitions of the units. In the eigenbasis of the coupling matrix the same system is exactly decomposable, which is the point: the closed variable is not a block of units.

What it leaves to QUESTION-221

The forward direction: whether weak coupling into a block bounds the dynamical dependence of its aggregate. CLAIM-390 bounds it for the block’s full state, and CLAIM-386 gives the aggregate’s one-step deterministic defect, but neither bounds the aggregate’s dependence. CLAIM-388 also does not say that the closed sum is one system. That is QUESTION-218.

Backlinks