Description
Coarse-graining a macroscopic body of $N\gg1$ identical subsystems makes collective observables self-average, yet for any nonlinear observable $f(\hat A)$ the strict inequality $\langle f(\hat A)\rangle\neq f(\langle\hat A\rangle)$ leaves a residual that is not washed out by coarse-graining and is fixed by the variance $\sigma^2$ of the underlying state.
In a gravitational potential the same residue yields an emergent enclosed mass, dark-matter-like halo, that flattens galactic rotation curves with no new fields and no modification of gravity.
Here, we propose a set of tabletop experiments that test the mechanism itself, establishing that the nonlinearity shifts a macroscopic observable with the predicted variance dependence. For a Bose--Einstein condensate of total mass $M$ in an anharmonic trap
$\Phi(x)=\tfrac{M}{2}\!\left(\omega_0^2 x^2+\tfrac12\beta x^4\right)$, the centre-of-mass equation of motion acquires a force term $\tfrac12\sigma^2\Phi'''$ from the Quantum Central Limit and Ehrenfest theorems, non-zero exactly because of the anharmonicity, with $\sigma^2$ the subsystem variance. This shifts the dipole frequency to $\omega_d^2=\omega_0^2+3\beta\sigma^2$ against the harmonic-limit null $\omega_d=\omega_0$ made exact by Kohn's theorem. The effect survives interatomic interactions: the shift is set by the single-subsystem $\sigma^2$, while short-range correlations keep the collective coordinate self-averaging ($\mathrm{Var}(\hat X)=\mathcal{O}(1/N)$), with the anharmonic coefficient interpolating between the collisionless and rigid-superfluid limits. The condensate observable is thus a concrete, falsifiable test of the coarse-graining mechanism, with a rigorous null and a predicted $1$--$12\%$ frequency shift for standard $^{87}$Rb parameters ($\omega_0/2\pi=50$ Hz, $\sigma=2$--$9\,\mu$m). We give the experimental design with independent variance controls --- atom number, Feshbach-tuned scattering length, and temperature --- together with the dominant systematics. A numerical simulation confirms the linear scaling in $\sigma^2$ and the Kohn null to better than $10^{-2}$ Hz. We further extend the proposal to trapped-ion crystals, levitated nanospheres, and thermal clouds --- for which temperature is the natural knob --- since the mechanism needs only finite variance, quantum or thermal.
| I am the presenting author | Yes |
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