The Geometric Origin of the Arrow of Time: Deriving the Second Law from the Principle of Least Entropic Stress

We investigate a class of information–geometric field theories in which a scalar “entropy field” S(x, t) evolves to minimize a quadratic entropic stress functional I[S]. We call the principle the Principle of Least Entropic Stress (PLES). Treating S(x, t) as a coarse–grained descriptor of macroscopic information geometry, we show that if its dynamics are given by gradient flow with respect to I[S], then the total entropic stress is a Lyapunov functional and decreases monotonically: dI/dt ≤ 0. We then place I[S] in a statistical mechanics framework by postulating a canonical measure P [S] ∝ exp(−I[S]/Θ) and invoking Einstein’s definition Stherm = kB ln P [1] to establish an inverse relationship Stherm ∝ −I. Combining these ingredients yields a derivation of the Second Law, dStherm/dt ≥ 0, as a geometric consequence of PLES. We support this abstract argument with two explicit toy models. In a deterministic one–dimensional diffusion model for S(x, t), the PLES stress I(t) decreases monotonically while a natural thermodynamic entropy functional Stherm(t), defined from the normalized field profile, increases monotonically, with a strong inverse correlation between the two. In a stochastic lattice implementation with Langevin dynamics, the same functional I[S] that drives gradient flow is shown numerically to define the equilibrium measure P [S] ∝ e−I/Θ with a measured temperature scale Θ that matches the input value to within a few percent and with R2 ≃ 0.999 for the straight–line test. We do not assume any specific microscopic realization of S(x) in the main derivation; the results apply to any field whose macroscopic dynamics are governed by a PLES functional of this form. In separate work, the same stress functional has been developed by the present author as the core of a unified entropic gravity framework, Modular Entropic Gravity (MEG), in which S(x) is a scalar entropy field sourcing spacetime curvature and reproducing the Newtonian Poisson equation in the static limit. The PLES arrow–of–time mechanism presented here islogically independent of that gravitational application, but it uses the same underlying geometric structure.