nucleation

TERM-090: effective boundary constraint, of a composite: the relation its interconnected parts impose on the variables an outside system can reach, once the interior is eliminated; approximately instantaneous or history-dependent, and, independently, possibly of a kind the parts' relations could not state

2 min read

Properties6
StatusActive
Version2
Tagscompositionality, complex-systems, mathematics
Linedistributed-agency
ComplementsTERM-093 — correspondence structure, and how it is held: the maps that say which local states or descriptions of interacting systems count as agreeing, distinct from the physical coupling between them, and held as formed, discovered or adapted, TERM-091 — relational vocabulary, of a class of descriptions: the arities and kinds of relation it can state (pairwise or higher-order, among states or among histories, of which declared types), and so which constraints are expressible in it at all
Cited as “uses” byCLAIM-474 — From interconnection to individuation, the relationships are distinct achievements that form a partial order, not a nested sequence: none implies the next, and a later one need not require every earlier one. Interconnection fixes what joint behaviour is possible, compatibility which local descriptions admit a coherent whole, dynamics which organizations persist, individuation which have sufficiently robust identity and boundaries, organizational autonomy which help maintain their own constitutive constraints, and inquiry which can measure and revise their own models and apparatuses under resource bounds, inquiry and reconstruction occurring without autonomy. The research program is to find the conditions for each, and their dependencies, apart from the order in which an inquiry meets them, CLAIM-480 — Composing interactions can turn an interior into a constraint, and sometimes into a constraint of a new kind: in the chain E = ½k₁(x₁ − x₂)² + ½k₂(x₂ − x₃)², with x₂ internal and equilibrated, the boundary sees E_eff = ½·k₁k₂/(k₁ + k₂)·(x₁ − x₃)², a coupling between x₁ and x₃ that neither part has; if the interior relaxes slowly, eliminating it leaves a history-dependent constraint, a memory kernel, so interconnection fixes an effective constraint on the boundary's trajectories and separation of time scales decides only whether that constraint is approximately memoryless; and for cellular sheaves the reduced constraint can relate three boundary values where every relation among the parts is pairwise

Graph View

Definition

Take interconnected systems with jointly admissible behaviour 𝓑_G, and a set of variables y an outside system can reach. The effective boundary constraint is the projection of 𝓑_G onto y, the external behaviour in Willems’s sense. It is the relation the composite imposes on whatever interacts with it only through y (CLAIM-480). It is classed on two independent axes:

  • by memory. It is approximately instantaneous when the interior relaxes fast relative to the boundary: the two-spring chain gives the coupling k₁k₂/(k₁ + k₂) between its ends. Otherwise eliminating the interior leaves a memory kernel, a relation over boundary trajectories. It is a constraint either way.
  • by kind. It is of a new kind when the reduced relation cannot be stated in the parts’ relational vocabulary (TERM-091). Hansen and Ghrist’s star reduces to v₁ = v₂ + v₃, which no pairwise sheaf on the boundary expresses (THEORY-1090). That reduction is static, so a constraint of a new kind need not have memory.

The distinctions it draws

  • A constraint with or without memory. A history-dependent relation is a constraint on admissible trajectories. Separation of time scales decides only whether it is approximately memoryless (CLAIM-480, version 2).
  • Effective, not fundamental. It is realized by the interior and changes when the interior does. It is not a new law (A23 §XII).
  • Interface, not individual. A composite with a well-defined boundary constraint need not be an individual in any of TERM-088’s senses.

Why it is proposed

Four entries filed this evening use the notion, under several names: “effective boundary relation”, “boundary law”, “external behaviour”. It is the clearest place where the record’s general frame, that constraints are realized by organization (CLAIM-477), has a demonstrated case. What would settle it: adoption if the claims cite it, as CLAIM-480 and CLAIM-474 now do. The object is Willems’s external behaviour (LIT-1660, unread); the term is filed for the distinctions above, which that name does not carry, and would be rejected if a reading of Willems showed they are his.

Backlinks