Quantum Dynamics from a Helmholtz Functional on Probability Densities

Existing information-theoretic derivations of quantum mechanics from Fisher informationlack a thermodynamic embedding: they contain no temperature, no entropy, and no free-energy balance, and therefore cannot address the quantum–classical boundary within thesame variational structure. We show that this limitation is removed by a Helmholtz-typefree-energy functional on probability distributions in which Fisher information provides theenergy and relative Shannon entropy, weighted by an effective temperature ϑ, provides theentropic cost. Chentsov’s theorem and the Shore–Johnson axioms constrain this functionalup to two coefficients, one of which is fixed by consistency with the quantum kinetic energy.At ϑ= 0 the functional reproduces the Fisher-information variational principle of Reginattoand Hall–Reginatto; extremisation of the associated action recovers the Schr¨odinger equationvia the Madelung transformation. Quantum mechanics is thereby identified as the zero-temperature fixed point of the Helmholtz structure. The finite-temperature sector ϑ > 0,which has no counterpart in previous approaches, yields three results: a gauge-invariantbalance scale Lc = ℏ/2√mϑ at which both the free-energy descent and the gradient-flowspreading law change character — the spreading exponent crossing over from the quantum-pressure value 1/4 to the diffusive value 1/2 — together with, under harmonic confinement,a genuine variationally stable minimum interpolating between the exact quantum ground-state and classical equipartition widths; a dissipation bound for localisation transitions,derived from a Wasserstein gradient flow with Lyapunov structure, from which Landauer’sprinciple follows as a corollary in the entropy-dominated regime; and a thermodynamicstability property that singles out the Born weights as the unique stable basin fractionsunder the dissipative dynamics, presented as a consistency check.