This is an autocatalytic network, a system of reactions where products accelerate their own education. This is the fundamental mechanism for the transition from chemistry to protobiology.
1 - I - exceeding the percolation threshold (structural coherence). What's that supposed to mean?
Imagine the many molecules and possible reactions between them. This is Count:
- Nodes - Molecules
- ribs - possible reactions
In percolation theory, there is a critical threshold p_c:
- below the threshold - small disparate clusters
- above the threshold - a giant connected component appears
Formally (for a random graph): \langle k \rangle > 1, where:
- \langle k \rangle — average node degree
When average connectivity exceeds 1, the network becomes globally connected. Why is it important?
An autocatalytic network is only possible if:
- The products of one reaction can become reagents of another.
- Chains are closed in cycles
Without enough connection:
- There are reactions
- No network
- There is no self-support
2 F - external energy flow
Any autocatalytic system is thermodynamically open.
No external flow:\Delta G \to 0
→ reactions come to equilibrium
→ Concentrations
→ autocatalytic cycle goes out
Flow source:
- Geochemical gradients
- UV radiation
- Hydrothermal springs
- Redox gradients
Formally: \frac{dX}{dt} = f(X) + J_{in} - J_{out}, where J_{in} \neq 0.
If the stream = 0 → The network is in equilibrium.
3 S - Negative feedback
Autocatalysis is dangerous.
If no stabilization: X(t) \sim e^{kt}
→ Explosion of Concentrations
→ Exhaustion of resources
→ collapse
Mechanisms are needed:
- Competition for Substrate
- Inhibitor
- Volume limitation
- Disintegration of products
The simplest stabilization: \frac{dX}{dt} = kX(1 - \frac{X}{K})
(logistical dynamics) This is negative feedback.
Now we put it all together
An autocatalytic network occurs when:
1. Structure \langle k \rangle > 1
→ closed reaction cycle appears
2. Flow J_{in} > 0
→ system is held out of balance
3. Stabilisation
\exists \text - negative feedback
→ Exponential explosion prevented
What happens when conditions are violated?
Violated |
Result |
|
I Low |
Disjointed reactions |
|
F is absent |
balance, "chemical death" |
|
S is absent |
Explosion and collapse |
Life arises only at the intersection of these three areas.
A deep conclusion. The autocatalytic network is:
Structurally connected, energy-powered, dynamically stabilized reaction system.
And this is the same principle that:
- the Metabolism of Cells
- the Neural Network
- the Collective Intelligence
Percolation theory is the foundation. Without it, you will not understand where the “system” arises, and where the scattering of elements remains.
1 - What is percolation
Percolation is a theory of the emergence of global connectivity in a random system.
There are many knots. Relationships between them appear with probability p.
Question:
At what point does a giant connected component appear?
2 - The simplest model is the random Erdjos-Renyi graph
Let:
- N Nodes
- Probability of P
Average node degree: \langle k \rangle = pN
Critical threshold:
\langle k \rangle = 1 Or p_c = \frac{1}{N}
3️⃣ Phase transition. If:
🔹 \langle k \rangle < 1
The network consists of small clusters.
No global structure.
🔹 \langle k \rangle = 1 Critical point.
The size of the component begins to grow sharply \langle k \rangle > 1
A giant size component appears: S \sim O(N)
This is a phase transition of the second kind.
4 - Continuous percolation (grids)
On the grid (for example 2D): p_c \approx 0.5927
If the probability of the node employment exceeds ~59%, a End-to-End Cluster.
5 - Mathematical meaning
The solution to the equation: S = 1 - e^{-\langle k \rangle S}
For \langle k \rangle < 1 The only solution is S = 0
For \langle k \rangle > 1 A non-trivial S > 0
This is the birth of the global structure.
6 - Why is it fundamental
Percolation describes the moment when:
- → becomes a reaction network
- Neurons → become brains
- People → become a society
- Agents → become collective intelligence
To the threshold is a set of elements. After the threshold, the system.
7 - Communication with autocatalytic networks
An autocatalytic cycle is possible only if:\langle k \rangle > 1
Otherwise, a closed path simply does not form.
Essentially: Percolation = is a condition for the appearance of cycles. A Cycles = Self-maintenance condition
8 - Critical area. The most interesting behavior is near the threshold.
- large-scale-invariant clusters
- The power laws of distribution
- High sensitivity to fluctuations
This is the area where:
- Life
- Consciousness
- Collective Intelligence
9 - Universal Law
The global system appears when:I > I_c, where I \sim \langle k \rangle
This is a structural condition of self-organization.
1️⃣ What is percolation
Percolation is the theory of the emergence of global connectivity in a random system. Classical formulation: There are many nodes.The connections between them appear with probability p. Question: At what p does a giant connected component appear?
2 - The simplest model is the random Erdjos-Renyi graph
- N Nodes
- Probability of P
Average node degree: \langle k \rangle = pN
Critical threshold: \langle k \rangle = 1 or p_c = \frac{1}{N}
3 - Phase transition. If: ? \langle k \rangle < 1
The network consists of small clusters. No global structure.
🔹 \langle k \rangle = 1
Critical point. The size of the component begins to grow sharply.
🔹 \langle k \rangle > 1 A giant size component appears: S \sim O(N)
This is a phase transition of the second kind.
4 - Continuous percolation (grids)
On the grid (for example 2D): p_c \approx 0.5927
If the probability of the node employment exceeds ~59%, a End-to-End Cluster.
5 - Mathematical meaning
The solution to the equation: S = 1 - e^{-\langle k \rangle S}
For \langle k \rangle < 1 The only solution is S = 0
For \langle k \rangle > 1A non-trivial S > 0
This is the birth of the global structure.
6 - Why is it fundamental
Percolation describes the moment when:
- → becomes a reaction network
- Neurons → become brains
- People → become a society
- Agents → become collective intelligence
To the threshold is a set of elements. After the threshold, the system.
7 - Communication with autocatalytic networks
An autocatalytic cycle is possible only if: \langle k \rangle > 1
Otherwise, a closed path simply does not form. Essentially:
Percolation = the condition of occurrence of cycles A cycles = self-maintenance condition
8 - Critical area. The most interesting behavior is near the threshold.There:
- large-scale-invariant clusters
- The power laws of distribution
- High sensitivity to fluctuations
This is the area where:
- Life
- Consciousness
- Collective Intelligence
9 - Universal law The global system appears when:
I > I_c, where I \sim \langle k \rangle