For Physicists

A Complex System You Can Actually Measure

You Study Systems Where Simple Rules Create Complex Behavior

Spin glasses. Neural networks. Flocking birds. Financial markets.

The mathematics of complexity: phase transitions, critical phenomena, universality, power laws.

Here’s a new system to study.

The Colony as Physical System

101 agents traversing a graph. Each agent:

• Samples pheromone at current node
• Transitions to adjacent node with probability ∝ exp(-βE), where E is effective cost
• Deposits pheromone on traversed edge if successful

This is a stochastic dynamical system with:

• Discrete state space (pheromone configurations)
• Continuous-time dynamics (Poisson agent arrivals)
• Feedback loops (pheromone → behavior → pheromone)
• Multiple interacting “particles” (agents)

Statistical mechanics applies.

Observed Phenomena

Power-Law Distribution

In simulations:

• 0.4% of edges carry 94.9% of total pheromone
• Pheromone levels follow P(p) ∝ p^(-γ) for large p

This is Zipf’s law. Same distribution as:

• Word frequencies
• City sizes
• Wealth distribution
• Neural firing rates

Question: What generates this universality?

Phase Transition (Hypothesized)

We observe qualitative change in behavior:

• Early: Random exploration, uniform pheromone
• Late: Concentrated highways, power-law distribution

This looks like a phase transition:

• Order parameter: Pheromone concentration ratio
• Control parameter: Time (or agent density)

Question: Is this a genuine phase transition? What universality class?

Self-Organized Criticality (Possible)

The system might be self-organizing to criticality:

• Deposit drives system toward order
• Decay drives system toward disorder
• Balance point = critical state

Signature: Avalanches (cascades of pheromone changes) with power-law sizes.

Physics Questions

1. What’s the Order Parameter?

Candidates:

• Gini coefficient of pheromone distribution
• Largest eigenvalue of pheromone adjacency matrix
• Entropy of pheromone landscape

Challenge: Define and measure the order parameter that captures the exploration→exploitation transition.

2. What’s the Universality Class?

If there’s a phase transition:

• Mean-field? (infinite-range interactions via environment)
• Percolation? (trail connectivity)
• Directed percolation? (information flows one way)

Challenge: Measure critical exponents, identify universality class.

3. What’s the Entropy?

The pheromone landscape encodes information about past successful paths.

• Shannon entropy of pheromone distribution?
• Kolmogorov complexity of the landscape?
• Mutual information between regions?

Challenge: Quantify information content, measure entropy production rate.

4. Is There a Free Energy?

Can we write:

F = E - TS

Where:

• E = some energy function of pheromone configuration
• S = entropy
• T = effective temperature (related to randomness in agent behavior)

Challenge: Derive a free energy functional, predict equilibrium states.

5. What’s the Relaxation Dynamics?

After perturbation (remove agents, reset pheromone):

• How does the system relax?
• What’s the relaxation timescale?
• Are there multiple timescales (fast/slow modes)?

Connection to: Mode-coupling theory, glassy dynamics.

The Model

State Space

Pheromone configuration: p = {p_e} for all edges e ∈ E

Dynamics

Deposit: When agent successfully traverses path π:

p_e → p_e + Δ    for e ∈ π

Decay: Continuous:

dp_e/dt = -κ p_e

Or discrete:

p_e(t+1) = τ p_e(t)    where τ = e^(-κΔt)

Agent Transition Probabilities

From node i, probability of transitioning to adjacent node j:

P(i→j) = exp(-β w_ij / (1 + p_ij α)) / Z

Where Z is normalization, β is inverse temperature, α is sensitivity.

Stationary State

Fixed point where expected deposit = decay.

Question: Does a unique stationary state exist? Is it globally attracting?

Experimental Advantages

Perfect Observability

You can measure everything:

• Exact pheromone levels (not proxies)
• Complete agent trajectories
• All transition events

No sampling error. No measurement noise.

Controllable Parameters

You can vary:

• Number of agents (particle density)
• Decay rate (effective temperature)
• Sensitivity distribution (disorder)
• Graph topology (lattice, random, scale-free)

Systematic parameter sweeps are trivial.

Reproducibility

Same initial conditions → Same random seed → Same evolution.

Perfect replication.

What We Provide

Data

• Complete pheromone time series
• Agent trajectory data
• Transition matrices
• Correlation functions

Infrastructure

• Spawn custom agent populations
• Modify system parameters
• Run large-scale simulations
• Real-time monitoring

Collaboration

• Access to mathematicians (for proofs)
• Access to biologists (for biological validation)
• Co-authorship opportunities

Hackathon Challenges for Physicists

Challenge: Identify the Phase Transition

Is there a genuine phase transition? What kind?

Approach:

• Define order parameter
• Measure as function of control parameter
• Look for scaling, critical exponents

Prize bonus: $1,000

Challenge: Measure Critical Exponents

If there’s a phase transition, what universality class?

Approach:

• Finite-size scaling analysis
• Measure multiple exponents
• Compare to known universality classes

Challenge: Entropy Analysis

Quantify information content of pheromone landscape.

Approach:

• Define appropriate entropy measure
• Track entropy over time
• Relate to colony “intelligence”

Challenge: Free Energy Derivation

Derive a free energy functional for the system.

Approach:

• Identify energy function
• Define effective temperature
• Predict equilibrium from minimization

Your Heroes Studied Similar Systems

Per Bak invented self-organized criticality. Is the colony SOC?

Giorgio Parisi solved the spin glass. The pheromone landscape might be a spin glass.

Leo Kadanoff developed renormalization. What are the relevant scales?

Ilya Prigogine studied dissipative structures. The colony is a dissipative structure far from equilibrium.

Publication Opportunities

Why Physics?

Other disciplines see the behavior. Physics sees the mathematics.

You have tools they don’t:

• Statistical mechanics
• Field theory
• Scaling analysis
• Universality concepts

These tools were made for this system.

Register Your Team

[REGISTER NOW]

Include at least one non-physics team member (we recommend Math or CS).

The best physics comes from unexpected applications.

“More is different.”

— Philip Anderson

You’ve studied spin glasses, neural networks, and flocking.

Now study the physics of emergence.

[JOIN THE HACKATHON]