# Thermodynamics of the Fermi gas in a nanotube

## TL;DR

This paper derives the thermodynamic properties of an ideal Fermi gas confined in a cylindrical nanotube, revealing how these properties depend on temperature, chemical potential, and tube radius, especially in low-dimensional regimes.

## Contribution

It provides a comprehensive analytical framework for the thermodynamics of a Fermi gas in a nanotube, considering arbitrary temperatures and treating the tube radius as a thermodynamic variable.

## Key findings

- Entropy and heat capacities are linear at low temperatures in quasi-one-dimensional cases.
- Sharp maxima in entropy and heat capacities occur at the onset of filling new discrete energy levels.
- Thermodynamic dependencies on tube radius differ qualitatively between fixed linear and fixed total density.

## Abstract

For the ideal Fermi gas that fills the space inside a cylindrical tube, there are calculated the thermodynamic characteristics in general form for arbitrary temperatures, namely: the thermodynamic potential, energy, entropy, equations of state, heat capacities and compressibilities. All these quantities are expressed through the introduced standard functions and their derivatives. The radius of the tube is considered as an additional thermodynamic variable. It is shown that at low temperatures in the quasi-one-dimensional case the temperature dependencies of the entropy and heat capacities remain linear. The dependencies of the entropy and heat capacities on the chemical potential have sharp maximums at the points where the filling of a new discrete level begins. The character of dependencies of thermodynamic quantities on the tube radius proves to be qualitatively different in the cases of fixed linear and fixed total density. At the fixed linear density these dependencies are monotonous and at the fixed total density they have an oscillating character. Key words: Fermi particle, nanotube, thermodynamic functions, low-dimensional systems, equation of state, heat capacity, compressibility