Non-parametric application of Tsallis statistics to systems consisting of M hydrogen molecules

TL;DR

This study applies nonextensive Tsallis statistics to analyze thermodynamic properties of small hydrogen molecule systems, revealing deviations from classical physics for systems with fewer than about 1000 molecules.

Contribution

It introduces a non-parametric approach using Tsallis statistics with a specific relation between system size and entropic index, incorporating detailed molecular interactions and external field effects.

Findings

Deviations from classical thermodynamics observed for systems with fewer than ~1000 molecules.

Entropy, energy, and specific heat calculated for hydrogen molecules using Hubbard Hamiltonian.

External forces and magnetic fields significantly influence thermodynamic parameters.

Abstract

We have determined the entropy, the total energy, and the specific heat of the systems consisting of $M\geq 3$ hydrogen molecules. The calculations were conducted in the framework of the nonextensive Tsallis statistics. The relation between $M$ and the entropic index $q$ is given by $q = 1 + 1/M$, which results from the fact that the temperature of the nanosystems fluctuates around the temperature of the reservoir (Wilk and W{\l}odarczyk, Phys. Rev. Lett. {\bf 84}, 2770 (2000)). The electron energy states of the hydrogen molecule have been determined with the help of the Hubbard Hamiltonian, which models all two-body interactions. The Hubbard Hamiltonian integrals have been calculated by using the variational method, whereas the Wannier function has been associated with $1s$ Slater-type orbitals. We have included the contributions to the energy of the hydrogen molecule coming from the oscillatory (either harmonic or anharmonic), rotational and translational degrees of freedom. In addition, we have investigated the impact of the external force ($F$) or the magnetic field ($h$) on the thermodynamic parameters of the systems. In each case the noticeable deviation from the results of the classical statistical physics can be observed for the systems consisting of $M<M_{c}\sim 10^{3}$ molecules.