What question did this study set out to answer?

This work aims to derive the thermal origin of the logarithmic term in the Schrödinger equation and its impact on free energy.

July 9, 2026Open Access

Thermal Origin of the Repulsive Logarithmic Schrödinger Free-Energy Term: Exact Occupancy Dependence, Neutron Bounds, and a Cold-Atom Test of the Counting Class

Key Points

This work aims to derive the thermal origin of the logarithmic term in the Schrödinger equation and its impact on free energy.
Derivation of exact occupancy-dependent entropic potential based on Maxwell-Boltzmann/Gibbs counting.
Monte Carlo simulations conducted to verify high-occupancy behavior and validate the existence of logarithmic terms.
Single-particle experiments to probe lower bounds of free-energy terms using neutron-beam phase-space density measurements.
Derivation shows precise agreement of logarithmic coefficients at high occupancy, differing systematically at low occupancy.
Neutron bounds confirm a marginal existence of the logarithmic term, with amplitude below the established threshold, suggesting more sensitive methods are needed.
Monte Carlo simulations validate the equation of state and demonstrate discriminative properties between different counting classes.

Abstract

This paper derives the thermal origin of the repulsive (acoustic-branch) logarithmic term of the logarithmic Schrödinger equation (LogSE) at theequilibrium free-energy level; the focusing matter branch, and the promotion ofthe free-energy term to a conservative phase term in the dynamical equation, areexplicitly reserved for the dynamical part of the program. Maxwell–Boltzmann/Gibbs counting of n quanta over M micro-nodes yields an exact per-cell entropicpotential V (n) = kB·T·ψ₀ (n+1) − ln M; the discrete insertion potential is ΔF (n) =kB·T·ln (n+1) /G. Both prescriptions give the logarithm with coefficient kB·T athigh occupancy — exact at the equilibrium free-energy level, with all cell-sizeambiguity absorbed by the reference scale — and both reduce to a weak contactlikeresponse at low occupancy, differing only by O (1) coefficients: the lowoccupancyswitch-off is prescription-independent. We define the local logarithmiccoefficient bₗoc (n) = kB·T·n·ψ₁ (n+1) (trigamma), with bₗoc → kB·T for n ≫ 1and bₗoc ≈ (π²/6) kB·T·n for n ≪ 1. Single-particle interferometry bounds onfundamental LogSE terms (Shull et al. 1980; Gähler, Klein in the Bose class thelogarithmic response saturates at high occupancy. A BEC measurement of the ln ρform and its density dependence therefore tests both the thermal coefficient andthe underlying counting class. Monte Carlo simulations with single-quantumtransfer confirm the logarithmic high-occupancy equation of state (slope −1/T;residual 2–13% deviations are systematic K/T coherence corrections, statisticalerrors 0. 5–1%), directly measure the discrete insertion free energy via occupancyhistogramratios (the Gibbs n! fingerprint R· (n+1) = const vs. the Bose fingerprintR = const), and demonstrate the counting-class equation-of-state discriminator insilico (MB slope −1/T at all occupancies; Bose slope − (1+n̄) /T).

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