Generalising thermodynamic efficiency of interactions: inferential, information-geometric and computational perspectives

TL;DR

The paper generalizes thermodynamic efficiency to multi-parameter, directional protocols and presents two observer-centric formulations: an inferential form tied to macroscopic fluctuations and an information-geometric form based on the Fisher information. It connects efficiency to statistical inference and differential geometry, including natural gradient dynamics and parameter-space compression, and demonstrates the ideas with a 2D Ising model where efficiency peaks near critical regimes. The work contrasts with prior system-centric views and suggests practical uses for guiding self-organization and interpreting how observable fluctuations reveal hidden control parameters. The approach provides a unified framework linking predictability gains, energetic costs, and information geometry in complex, self-organizing systems.

Abstract

Self-organizing systems consume energy to generate internal order. The concept of thermodynamic efficiency, drawing from statistical physics and information theory, has previously been proposed to characterise a change in control parameter by relating the resulting predictability gain to the required amount of work. However, previous studies have taken a system-centric perspective and considered only single control parameters. Here, we generalise thermodynamic efficiency to multiple control parameters and extend the definition of thermodynamic efficiency to protocols in arbitrary directions, by introducing directional efficiency. Taking an observer-centric perspective, we derive two novel formulations. The first, an inferential form, relates efficiency to fluctuations of macroscopic observables, interpreting thermodynamic efficiency in terms of how well the system parameters can be inferred from observable macroscopic behaviour. The second, an information-geometric form, expresses efficiency in terms of the Fisher information matrix, interpreting it with respect to how difficult it is to navigate the statistical manifold defined by the control protocol. This observer-centric perspective is contrasted with the existing system-centric view, where efficiency is considered an intrinsic property of the system.

Figures (9)

• Figure 1: Statistical manifold $\mathcal{M}$. The $\lambda_i$, $\lambda_j$ are coordinate curves, and $\underline{e}_i$, $\underline{e}_j$ are the corresponding tangent vectors. A point on the manifold is a probability density $p(x|\underline{\lambda})$ of a random variable $x$, parameterised by $\underline{\lambda}$.
• Figure 2: To escape from the maximum-entropy (uniform) distribution $p(x|\underline{0})$, the steepest direction on the statistical manifold is given by the natural gradient of $-S(\underline{\lambda})$ (in red). The numerator of directional efficiency represents the extent to which the protocol's velocity vector (in blue) aligns with this direction.
• Figure 3: The denominator of directional efficiency represents the weighted sum of the energies of the paths from the zero-response point to the current point over all dimensions. For each dimension $\lambda_j$ ($j=1,2,\dots,n$), the energy $E_j$ of the path (in green) from the zero-response point ($\lambda_j^*$) to its current position ($\lambda_j$) is computed. The energies of these paths are weighted by the velocity of protocol at $\underline{\lambda}$ (in blue).
• Figure 4: Thermodynamic efficiency computed for the inferential form $\eta^{\bowtie}(J)$, information-geometric form $\eta^{\triangle}(J)$, and computational form $\eta^{\dagger}(J)$, using the canonical 2D Ising model with zero external field.
• Figure 5: Probability landscape and thermodynamic efficiency $\eta^{\bowtie}$ (inferential form). Left: Probability of positive spin at equilibrium for different combinations of external field strength $h$ and coupling strength $J$. $P(\sigma_i=1)$ closer to 0.5 (blue colour) indicates a disordered state of the system, where spins are equally likely to be up or down; $P(\sigma_i=1)$ closer to 1 (red colour) indicates an ordered state, where spins are predominantly up. Middle and Right: Thermodynamic efficiency $\eta^{\bowtie}$ with respect to $J$ and $h$ (inferential form) for the same parameter combinations. Notice that the edge where the transition from disorder to order is steep corresponds to high thermodynamic efficiency.