CRR Non-Markovian Swarm
Coherence • Rupture • Rebirth Framework
C(x,t) = ∫₀ᵗ L(x,τ) dτ
Swarm coherence through accumulated positional memory integration
// Coherence Phase - Historical Memory Integration
p.history.push([p.x, p.y]);
if (p.history.length > memoryDepth) p.history.shift();
// Memory-weighted centroid calculation
let hx = p.history.reduce((a,b) => a + b[0], 0) / p.history.length;
let hy = p.history.reduce((a,b) => a + b[1], 0) / p.history.length;
// Non-Markovian force based on entire trajectory
p.vx += (hx - p.x) * coherenceStrength;
p.vy += (hy - p.y) * coherenceStrength;
Phase: Coherence Building
Simulation Controls
300
50
0.20
0.99
System Metrics
Active Particles:
300
Avg Memory Length:
0
System Energy:
0.000
Coherence Index:
0.000
Time Since Rupture:
0s
Total Ruptures:
0
CRR Framework
0.002
100
5
0.05
∫ L(x,τ) dτ:
0.000
δ(t-t₀):
0.000
Non-Markovian vs Traditional Swarm Systems
Traditional Markovian Swarm
- Current state only determines future behavior
- No memory of trajectory or past experiences
- Purely reactive to immediate environment
- Predictable, deterministic dynamics
- Limited emergent complexity
CRR Non-Markovian Swarm
- Entire history influences current behavior
- Accumulated memory creates attractor dynamics
- Exhibits learning and adaptation over time
- Rich emergent patterns from memory integration
- Rupture events enable system reorganization
The CRR Framework Applied
Coherence Phase: Particles accumulate positional history and are drawn back toward their personal trajectory centroids. This creates stable, memory-dependent motion patterns that evolve over time.
Rupture Events: Click interactions trigger localised δ(t-t₀) disruptions that clear memory and inject impulses. This breaks established patterns and enables system reorganisation.
Regeneration Phase: After rupture, particles rebuild their behavioral patterns through the rebirth operator R[χ], integrating new experiences with residual memory traces weighted by the field parameter Ω.
Key Mathematical Innovations
- Memory Integration C(x,t) = ∫ L(x,τ) dτ: Each particle maintains a weighted integral of its positional history, creating non-local temporal correlations
- Mnemonic Force Fields: Particles experience forces based not just on current position, but on their entire accumulated trajectory
- Rupture Dynamics: Discrete memory-clearing events that enable rapid behavioral transitions and learning
- Historical Attractors: Emergent patterns arise from the interplay between memory, randomness, and rupture events
Biological and Computational Parallels
This framework models aspects of:
- Neural Plasticity: How memories influence current behavior while remaining updatable through experience
- Collective Memory: Group behaviors that depend on shared historical experiences
- Adaptive Systems: Learning algorithms that balance stability with adaptability
- Swarm Intelligence: Emergent coordination through memory-mediated interactions
Emergent Properties
Unlike Markovian systems, this swarm exhibits:
- Path Dependence: Different histories lead to different current states even with identical conditions
- Temporal Coherence: Particles develop consistent movement patterns based on accumulated experience
- Memory-Mediated Clustering: Spatial organisation emerges from historical trajectory preferences
- Adaptive Resilience: System can maintain organisation while adapting to disruptions
Research Applications
This approach enables modeling of:
- Animal migration patterns with memory of previous routes
- Social dynamics where past interactions influence current behavior
- Distributed computing systems with historical optimisation
- Economic markets where trader behavior depends on past experiences
- Evolutionary processes where fitness depends on historical adaptation