Physics of Information Geometry - Part I: Principle of Least Action on the Probability Simplex
AuthorsC. Emre Koksal, Deniz Sargun
Resources
This work treats probability distributions like physical systems, finding efficient paths between them using information geometry and a least-action principle.
Key results
Dimensionless nonequilibrium free-energy separation in the three-state example, measured in nats.
Per-step KL constraint δ used for the detailed least-action trajectory.
Number of transitions generated with δ=0.001.
Total KL kinetic expenditure for the δ=0.001 path, measured in nats.
Savings versus the 0.297-nat direct jump at δ=0.001.
What the paper found
This paper formulates distributional evolution on the probability simplex as a discrete least-action problem. Relative entropy to a Gibbs reference, D_KL(p∥q0), represents dimensionless nonequilibrium free energy, while D_KL(q_t∥q_{t−1}) is the informational kinetic cost of each transition. Using the Pythagorean inequality for KL divergence, the authors show that every information-projection step converts free-energy increase into kinetic expenditure, with cumulative kinetic cost bounded by the endpoint free-energy separation. A backward greedy construction selects the lowest-free-energy predecessor inside a KL ball, producing a forward path with a finite-step guarantee TδIP ≤ ceil[D_KL(p_pref∥q0)/δ]. Unlike exponential tilting, the chosen KL orientation yields a closed-form projection through the principal Lambert W function. The method is relevant to diffusion-based generative modeling, including Denoising Diffusion Probabilistic Models, as well as Bayesian inference and active inference, but the paper does not claim that its locally greedy path is globally minimum-time. In a three-state numerical example with q0=(1/3,1/3,1/3), p_pref=(0.7,0.2,0.1), and endpoint separation 0.297 nats, δ=0.001 produced an 18-step trajectory whose sequential kinetic cost was 0.017 nats, yielding savings exceeding 94 percent versus a direct jump. State-dependent penalties are incorporated by exponentially tilting the Gibbs reference, preserving the same projection and Lambert-W machinery.
Original abstract
We develop a least-action framework for describing how a probability distribution can evolve from an equilibrium state to a prescribed nonequilibrium state under constrained incremental changes. Taking a Gibbs distribution as the equilibrium reference, the framework gives a direct physical meaning to the geometry of the probability simplex: distance from equilibrium corresponds to nonequilibrium free energy, while changes between successive distributions carry an informational kinetic cost. The Pythagorean structure of relative entropy then provides the central insight of the work. It shows that intermediate distributions chosen via sequential information projections can reduce the kinetic cost of large transitions and establishes an energy-conservation-like relation between the kinetic expenditure along a path and the free energy accumulated in reaching the target distribution. Motivated by this geometry, we construct a greedy least-action path through successive information projections, obtain a closed-form characterization of each projection through the Lambert W function, and establish a finite-step performance guarantee. We further show that state-dependent costs can be incorporated naturally by reshaping the underlying Gibbs reference, providing a thermodynamic interpretation of path penalties as modifications of the effective energy landscape. Together, these results provide a unified view of distributional evolution through least action, information geometry, and nonequilibrium thermodynamics.
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