The Inaccessible Game

TL;DR

The paper formalizes information dynamics through four axioms, including a novel information isolation postulate, to derive an information-relaxation principle that maximizes entropy production under a fixed total marginal entropy. This yields a GENERIC-like structure with a symmetric dissipative part and an antisymmetric conservative part, arising automatically from the information-geometry of the Fisher information conductance. Through analytical arguments and Ising-like simulations, Curie–Weiss thermodynamics, and a mass–spring example, the work links marginal-entropy conservation to energy-entropy equivalence in the thermodynamic limit and demonstrates how Landauer-like dissipation can emerge from purely information-theoretic constraints. The framework provides a principled bridge between information theory and non-equilibrium thermodynamics, suggesting a constructive path to derive conservative dynamics and offering insights into fundamental limits of computation and information erasure.

Abstract

In this paper we introduce the inaccessible game, an information-theoretic dynamical system constructed from four axioms. The first three axioms are known and define \emph{information loss} in the system. The fourth is a novel \emph{information isolation} axiom that assumes our system is isolated from observation, making it observer-independent and exchangeable. Under this isolation axiom, total marginal entropy is conserved: $\sum_i h_i = C$. We consider maximum entropy production in the game and show that the dynamics exhibit a GENERIC-like structure combining reversible and irreversible components.

Figures (4)

• Figure 1: Temperature scaling of our GENERIC-like decomposition. Left: Component norms show both $\|S\|$ is highest at low coldness and $\|A\|$ tends to increase with higher coldness, but also exhibits a maximum at an intermediate value. Right: The ratio $\tfrac{\|A\|}{\|S\|}$ peaks at intermediate temperature, showing that frustrated systems can exhibit relatively strong conservative dynamics. At low coldness (high temperature) the system is dominated by dissipative dynamics and the ratio drops. At high coldness (low temperature) the ratio is elevated with a peak but the peak is given at a critical inverse temperature given by $\beta=0.21$.
• Figure 2: The parameter trajectories show how $\theta_1$ and $\theta_2$ converge to different points in the constrained and unconstrained cases. Due to symmetries $\theta_3$ follows the same path as $\theta_1$. The interaction parameters converge to the same point where all interaction is removed. For both systems this provides the maximum joint entropy. The constrained system achieves this maximum joint entropy without changing the sum of marginal entropies. Interaction parameter $\theta_{23}$ follows the same trajectory as $\theta_{12}$.
• Figure 3: Order parameter $|m|$ transitions from 0 to finite values at $\beta_c$, marking the crossover between regimes.
• Figure 4: The change in the gradient of the multi-information with respect to the order parameter as the number of interacting spins, $n$, increases. Across different temperatures as $n$ grows the gradient becomes constant. This contrasts with both $\tfrac{\text{d}\sum_i h_i}{\text{d}m}$ and $\tfrac{\text{d}H}{\text{d}m}$ which both scale proportionally to $n$. In physics we would call these extensive quantities in contrast to the multi-information gradient which is intensive. The numerical noise in the $4 \beta_c$ curve at higher values of $n$ comes from catastrophic cancellation as the marginal and joint entropy gradients scale extensively and the multi-information (which is calculated as the difference between the two) scales intensively.