phinx — φ-ensemble

Thermodynamic agent-based simulation for complex systems

phinx unifies cellular automata, Bayesian inference, game theory, fractal geometry, and thermodynamic ensembles into a single real-time simulation pipeline — without a global temperature parameter.

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Why phinx?

Existing packages handle each domain separately:

Domain	Existing tools
Agent-based models	Mesa
Cellular automata	cellpylib
Bayesian inference	PyMC, pomegranate
Game theory	nashpy, axelrod
Thermodynamics	domain-specific only

phinx integrates all of them into one pipeline with a unified survival function Φ, computable in real time on local hardware — CPU or GPU.

Key idea: local randomness instead of global temperature

Instead of a single global temperature parameter (as in transformers):

# Transformer approach — global T
P(output) = softmax(logits / T)

phinx assigns each agent its own local noise ε, updated through raindrop-collision-style interactions:

εᵢ ~ f(contextᵢ, priorᵢ, neighborsᵢ)
output_i = act(stateᵢ) + εᵢ

The collective effect of local ε distributions produces an emergent effective temperature T*, structurally equivalent to natural uncertainty — without any global parameter.

This is not a metaphor. The raindrop collision probability p = exp(−d/r₀) is mathematically identical to an attention score softmax(QKᵀ/√d). phinx implements agent interaction as a masked local attention kernel, accelerated by Triton on GPU.

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Mathematical Foundation

The unified survival function

Φ = sigmoid(α·S + β·D − γ·T*) · ⟨cooperation⟩_M

Symbol	Meaning	Source theory
S	Shannon entropy of prior distribution	Thermodynamics
D	Fractal dimension (box-counting)	Fractal theory
T*	Effective temperature = Var(εᵢ)/k	Local randomness
⟨coop⟩_M	Monte Carlo cooperation estimate	Bayesian + Game theory
α, β, γ	Aesthetic tuning parameters	—

Φ → 1: stable system (ESS maintained, fractal healthy, free energy minimum)
Φ → 0: collapse (phase transition reached, D drops, defection ESS)

Three-layer architecture

Micro  (Agent ψᵢ)      →  s, π, P(H), ε, E
                                ↓  raindrop collision + Bayesian update
Ensemble (Z, T*, S, F) →  thermodynamic interface
                                ↓  partition function + phase detection
Macro  (Ψ, D, ⟨O⟩)    →  emergent patterns
                                ↑  (feedback: macro state → micro prior)

Thermodynamic ensemble — bridging micro and macro

The ensemble layer computes global system state from local agent distributions, without iterating every agent pair:

Z    = Σᵢ exp(−Eᵢ / kT*)        partition function
T*   = Var(εᵢ) / k              emergent temperature (no global T needed)
S    = −k Σ Pᵢ ln Pᵢ            entropy  (diversity → stability)
F    = E − T*·S                  free energy (minimum = stable ESS)
Tc   = ∂²F/∂T*² = 0             phase transition point

High entropy S (diverse agent population) → low free energy F → stable system.
Homogenization reduces S → F rises → system approaches collapse threshold Tc.

Raindrop sequence — collision probability

n agents moving on independent paths collide with probability:

P(at least one collision) = 1 − (1−p)^C(n,2)

Each collision is an evidence exchange event that triggers Bayesian belief revision (τ = 3 iterations by default, convergence guaranteed by KL-divergence bound):

P(H|E) ∝ P(E|H) · P(H)     ← posterior of agent i after meeting agent j

The local ε of both agents co-evolves after each collision:

εᵢ_new = 0.8·εᵢ + 0.2·εⱼ   ← ε co-evolution (local temperature mixing)

This replaces global attention with context-dependent, path-history-aware belief revision — structurally analogous to quantum wavefunction collapse at measurement.

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Installation

pip install phinx

With optional dependencies:

pip install phinx[fast]    # numba JIT acceleration (CPU)
pip install phinx[gpu]     # Triton GPU kernel (RTX 20xx+)
pip install phinx[output]  # OSC + WebSocket real-time output
pip install phinx[viz]     # matplotlib visualization
pip install phinx[all]     # everything

GPU setup (RTX 20xx / 30xx / 40xx)

pip install torch torchvision --index-url https://download.pytorch.org/whl/cu121
pip install triton
pip install phinx[gpu]

Triton kernels activate automatically when a compatible GPU is detected (CC ≥ 7.5). No code changes required — the same API runs identically on CPU and GPU.

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Quick Start

import numpy as np
import phinx

# 1. Create grid
grid = phinx.EnsembleGrid(N=16, r=1)

# 2. Randomize initial priors (diversity matters)
for i in range(grid.N):
    for j in range(grid.N):
        grid.agents[i][j].prior = float(np.random.rand())

# 3. Thermodynamic ensemble
ensemble = phinx.ThermoEnsemble(grid, M=64)

# 4. Run simulation
loop = phinx.PhiLoop(grid, ensemble, fps=60, alpha=0.3, beta=0.4, gamma=0.3)

def on_frame(result):
    print(f"frame={result['frame']:04d}  "
          f"Φ={result['phi']:.3f}  "
          f"signal={result['signal']}")

results = loop.run(n_frames=100, callback=on_frame)

# 5. Summary
summary = loop.summary()
print(f"Φ mean={summary['phi_mean']:.3f}  "
      f"avg={summary['avg_total_ms']:.1f}ms/frame")

GPU pipeline (PhinxPipeline)

For real-time installations requiring maximum throughput:

from phinx.core.gpu_pipeline import PhinxPipeline, PhinxConfig

cfg = PhinxConfig(N=32, M=64)
cfg.auto_tune()   # reads GPU model + VRAM, sets N/M/D automatically

pipeline = PhinxPipeline(cfg)

for frame in pipeline.run():
    print(f"Φ={frame['phi']:.3f}  {frame['ms']:.1f}ms")
    # frame keys: phi, S, D, T_star, coop, F, is_critical, frame, ms

auto_tune() adapts to the detected hardware:

Hardware	CC	Auto M	Per-frame target
CPU only	—	32	~40–80ms
RTX 2060/70	7.5	64	~12ms ✓
RTX 2080 Ti	7.5	128	~8ms ✓
RTX 3060	8.6	128	~6ms ✓
RTX 3080+	8.6	512	~3ms ✓

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Core Components

Agent — state vector ψᵢ

from phinx import Agent

a = Agent(
    prior=0.6,          # Bayesian prior P(H) — cooperation belief
    epsilon_var=0.1,    # local noise variance (replaces global T)
    energy=0.4,         # internal energy Eᵢ (mind-body state)
)

b = Agent(prior=0.3)

# Raindrop collision → Bayesian update + strategy update + ε co-evolution
a.meet(b, distance=0.5)

print(a.act())       # local ε-sampled action
print(a.to_dict())   # serialize

EnsembleGrid — N×N cellular automata

from phinx import EnsembleGrid

grid = EnsembleGrid(
    N=32,       # grid size (N² agents)
    r=1,        # neighbor radius (r=1 → 8 directions)
    wrap=True,  # toroidal boundary
)

ms = grid.step()             # one frame update
print(grid.stats())          # aggregate statistics
print(grid.fractal_dim())    # fractal dimension D ∈ [1.0, 2.0]

GPU Attention Kernel

Agent interactions are implemented as masked local attention — mathematically equivalent to raindrop collision probability — compiled to Triton for GPU execution:

from phinx.core.attention_kernel import agent_attention, masked_agent_attention

# Auto-dispatches: Triton (CC ≥ 7.5) or PyTorch fallback (CPU / older GPU)
out = agent_attention(Q, K, V)

# With neighbor radius mask — only agents within r interact
out = masked_agent_attention(Q, K, V, positions=pos, r=1.5)

Kernel dispatch is fully automatic:

CUDA available
  + Compute Capability ≥ 7.5   (RTX 20xx, 30xx, 40xx)
  + triton installed
  ──────────────────────────►  Triton kernel   (fp16, Tensor Core)

otherwise
  ──────────────────────────►  PyTorch fallback (CPU or CUDA, fp32)

Game Theory — payoff matrices + ESS

from phinx import (
    PRISONERS_DILEMMA, STAG_HUNT, HARMONY,
    is_ess, population_dynamics, pareto_efficiency
)

print(is_ess(1.0, HARMONY))            # True
print(is_ess(1.0, PRISONERS_DILEMMA)) # False

result = population_dynamics(PRISONERS_DILEMMA, initial_coop=0.5, steps=200)
print(f"Converged to cooperation rate: {result['converged_to']:.3f}")

eff = pareto_efficiency(PRISONERS_DILEMMA, coop_rate=0.6)
print(f"Efficiency: {eff['efficiency']:.3f}")

Thermodynamic Ensemble

from phinx import ThermoEnsemble, compute_phi

ensemble = ThermoEnsemble(grid, M=64)

result = compute_phi(grid, ensemble, alpha=0.3, beta=0.4, gamma=0.3)
print(f"Φ={result['phi']:.3f}  S={result['S']:.3f}  "
      f"D={result['D']:.3f}  T*={result['T_star']:.4f}")
print(f"signal: {result['signal']}")  # stable | warning | critical

Real-time Output

from phinx.output.realtime import RealtimeOutput, ConsoleOutput

# Console (development)
console = ConsoleOutput(every_n=10)

# OSC → Max/MSP, TouchDesigner, SuperCollider
from phinx.output.realtime import OSCOutput
osc = OSCOutput(host="127.0.0.1", port=9000)

# OSC + WebSocket simultaneously
rt = RealtimeOutput(
    osc_target=("127.0.0.1", 9000),
    ws_port=8765,           # WebSocket → browser GLSL / p5.js
)

loop.run(n_frames=1000, callback=rt.send)

OSC message schema

Address	Type	Range	Description
/phinx/phi	f	[0, 1]	survival index
/phinx/entropy	f	[0, ∞)	diversity
/phinx/fractal	f	[1, 2]	pattern complexity
/phinx/temp	f	[0, ∞)	effective temperature
/phinx/coop	f	[0, 1]	cooperation rate
/phinx/signal	s	—	"stable"|"warning"|"critical"
/phinx/frame	i	—	frame number
/phinx/critical	i	[0, 1]	phase transition flag

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Fractal Analysis

from phinx.grid.fractal import fractal_dim_multiscale, fractal_dim_history

state = grid.state_matrix()
analysis = fractal_dim_multiscale(state)
print(f"D(3-scale)={analysis['D_3scale']:.3f}  "
      f"healthy={analysis['is_healthy']}")

D_history = [grid.fractal_dim() for _ in range(30)]
alert = fractal_dim_history(D_history, window=10)
print(f"alert={alert['alert']}  drop={alert['drop']:.3f}")

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Interactive Art Applications

phinx was developed in part for the NEMAF 2025 interactive installation artwork.

The survival function Φ maps directly to sensory output:

Φ component	Visual	Audio
S (entropy)	color diversity	harmonic richness
D (fractal dim)	pattern complexity	rhythmic complexity
T* (temperature)	particle turbulence	timbre roughness
Φ (combined)	overall density	consonance/dissonance
signal=critical	monochrome collapse	noise flood

# Minimal installation loop — GPU accelerated, OSC output
from phinx.core.gpu_pipeline import PhinxPipeline, PhinxConfig
from pythonosc import udp_client

cfg = PhinxConfig(N=32, M=64)
cfg.auto_tune()

pipeline = PhinxPipeline(cfg)
osc = udp_client.SimpleUDPClient("127.0.0.1", 9000)

for frame in pipeline.run():
    osc.send_message("/phinx/phi",      frame['phi'])
    osc.send_message("/phinx/entropy",  frame['S'])
    osc.send_message("/phinx/fractal",  frame['D'])
    osc.send_message("/phinx/temp",     frame['T_star'])
    osc.send_message("/phinx/critical", int(frame['is_critical']))

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Performance

CPU (pure Python, no numba)

Grid size	step()	compute_phi()	Total/frame
N=8	~1ms	~2ms	~3ms ✓
N=16	~5ms	~3ms	~8ms ✓
N=32	~20ms	~4ms	~24ms ✓

Install phinx[fast] for numba JIT acceleration (10–50× speedup on the grid loop).

GPU (Triton kernel — phinx[gpu])

GPU	CC	N=32 M=64	N=32 M=128	60fps budget
RTX 2060/70	7.5	~12ms	~16ms	✓
RTX 2080 Ti	7.5	~8ms	~10ms	✓
RTX 3060	8.6	~6ms	~8ms	✓
RTX 3080+	8.6	~3ms	~4ms	✓

On CPU-only machines, phinx[gpu] is not required. PyTorch fallback activates automatically — same API, same results.

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Theoretical Background

phinx integrates the following theoretical frameworks:

• Behavioral Psychology — reinforcement, imitation, social learning → agent state s
• Conway's Game of Life — local rules → global emergence → cellular automata grid
• Prisoner's Dilemma / Game Theory — cooperation/defection, ESS, replicator dynamics
• Personality Theory (Big Five, MBTI) — individual parameter variation → diversity
• Fractal Theory — self-similarity across scales → D as complexity measure
• Mind-Body Monism (Spinoza) — physical space ↔ collective psychology → energy E
• Frege's Logic — Sinn/Bedeutung: different theories, same referent (Φ)
• Bayesian Inference — prior → evidence → posterior → raindrop collision update
• Thermodynamics — partition function Z, entropy S, free energy F, phase transition Tc
• Quantum analogy — superposition (local ε before collision) → collapse (Bayesian update)

All unified under the survival function Φ.

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Citation

If you use phinx in research or artwork, please cite:

@software{phinx2025,
  author  = {Lee, Chae-moon},
  title   = {phinx: Thermodynamic agent-based simulation for complex systems},
  year    = {2025},
  url     = {https://github.com/yourusername/phinx},
  version = {0.1.0}
}

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License

MIT License — see LICENSE for details.

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Author

lajjadred

GitHub: @yourusername