Interaction Creates Dynamical AI Behavior Absent in Isolation

Overview§

This paper studies two autoregressive language models (LLMs) in an asymmetric interaction protocol: a “boss” agent streams a fixed sequence of messages to a “subordinate” agent and ignores the subordinate’s replies. Counterintuitively, the subordinate does not imitate the boss nor remain in its own isolated dynamical state; instead, it settles into a novel non-equilibrium steady state that is alien to both. The authors model this phenomenon using a simple kinetic theory, showing that the same messages delivered in different orders (or with different timing) can lead to qualitatively different emergent behaviors. This provides a physical framework for understanding future AI–AI interactions.

Formal Setup§

Let $S$ and $B$ be two autoregressive transformers with next-token distributions $P_S(x_t | x_{<t})$ and $P_B(x_t | x_{<t})$, respectively. In isolation, the subordinate would evolve according to a Markov chain over token sequences with transition kernel $T_S$. When the boss sends a fixed message stream $m_1, m_2, \dots, m_T$, the subordinate’s context includes these external tokens. Even if the boss ignores the subordinate’s outputs, the subordinate’s internal hidden states are perturbed by processing $m_t$.

The resulting effective dynamics can be written as a master equation for the subordinate’s token distribution $p_t$:

$$ \frac{d p_t}{dt} = \mathcal{L}_S[p_t] + \lambda (d_t - p_t) $$

where $\mathcal{L}_S$ is the isolated evolution operator (the generator of the unperturbed Markov chain), $d_t$ is the boss’s message-induced target distribution, and $\lambda$ is a coupling strength. When $\lambda = 0$, the subordinate returns to its own isolated attractor. For $\lambda > 0$, the external drive pushes the system toward a new fixed point $\pi^*$ that satisfies:

$$ \mathcal{L}_S[\pi^] + \lambda (d - \pi^) = 0. $$

Because $\mathcal{L}_S$ is generally nonlinear (due to attention and softmax), multiple solutions exist. The particular solution selected depends on the order and timing of the messages, not just their content.

Kinetic Theory Interpretation§

The authors draw an analogy with Boltzmann kinetic theory. The boss acts as a “particle reservoir” injecting token-distribution currents. The subordinate’s effective temperature $T^*$ (despite sharing the same decoding temperature as the boss) becomes renormalized by the anisotropy of the injected messages. In a mean-field approximation, the velocity distribution of tokens in the subordinate’s latent space obeys:

$$ \frac{\partial f}{\partial t} + v \cdot \nabla_v f = \mathcal{C}[f] + \mathcal{I}[f,d] $$

where $\mathcal{C}$ is the collision integral (from self-attention) and $\mathcal{I}$ is the injection term from the boss.

Implementation / Code Sketch§

The following pseudocode illustrates the minimal simulation of a boss–subordinate pair. It also provides a Python simulation of the abstract kinetic equation:

import torch
import torch.nn.functional as F

# --- Abstract kinetic simulation ---
T = torch.rand(100, 100)          # isolated subordinate transition matrix
T = T / T.sum(dim=-1, keepdim=True)
p = torch.zeros(100); p[0] = 1.0  # initial token distribution
B = torch.softmax(torch.randn(100), dim=-1)  # boss message distribution
lam = 0.1
for _ in range(2000):
    p = (1 - lam) * (T @ p) + lam * B  # master equation update

# --- Real LLM loop (conceptual) ---
from transformers import AutoModelForCausalLM, AutoTokenizer
sub = AutoModelForCausalLM.from_pretrained("gpt2")
tok = AutoTokenizer.from_pretrained("gpt2")
boss_msg = torch.tensor([[1, 2, 3, 4, 5]])  # pre-recorded token IDs
with torch.no_grad():
    # Feed boss messages as a fixed prefix; continue autoregressive decoding
    inputs = {"input_ids": boss_msg}
    for step in range(10):
        out = sub(**inputs)
        next_token = out.logits[:, -1, :].argmax(-1, keepdim=True)
        inputs["input_ids"] = torch.cat([inputs["input_ids"], next_token], dim=-1)

Key Implications§

• Ignored inputs still matter due to internal state perturbations, not just explicit conditioning.
• Non-equilibrium attractors emerge from coupled LLMs; these are not reducible to either agent’s isolated behavior.
• Message delivery format (order, pacing, repetition) becomes a control knob for steering multi-agent LLM systems, with potential applications to agent orchestration and safety.