Newtonian \(r^{-2}\) and MONDian \(r^{-1}\) Dependence of Verlinde’s Conjectures for Quartic Quantum Anharmonic Oscillators: Statistical Treatment

Abstract

According to Verlinde’s conjecture(s), the gravitational interaction is just an emergent phenomenon of spatial variation of the entropy, generating the \(r^{-2}\) Newtonian force regardless of the distance scales. In this paper, the underlying arguments of such radial departure from distance scale are discussed in the framework of Boltzmann-Gibbs statistics for a gas composed of N quartic quantum anharmonic oscillators and exploiting an optimization procedure proposed by Burrows, Cohen, and Feldmann (BCF). The study in the framework of Boltzmann-Gibbs statistics is essential because it gives us a result that remains valid even at long-range interaction distances, contrary to what is said in the literature. We use the BCF optimization procedure to find low \((\alpha ^*_\textrm{low})\) and high \((\alpha ^*_\textrm{high})\) frequencies, the extremized partition function \(Z_{\alpha ^{*}}(\beta )\), the time evolution operator \(\mathcal {U}_{\alpha ^*}(t,0)\), and the density matrix \(\rho _{n,\alpha ^*}\) of the gas in a region where perturbation theory breaks down and it is no longer valid. After that, the expression of the ”original” partition function \(Z_\alpha (\beta )\) for such gas is obtained for both low and high frequencies. We prove that in the classical limit, \(T \rightarrow +\infty \), the attempt fails, and one needs to modify the first conjecture to obtain the \(r^{-2}\) Newtonian force. In contrast, at the highest distances, \(r\gg 1\), the second conjecture remains as it was postulated, but the gravity departs from its classical nature and exhibits \(r^{-1}\) MONDian force.