# Quantum and Relativistic corrections to Maxwell-Boltzmann ideal gas model from a Quantum Phase Space approach

## TL;DR

This paper introduces quantum and relativistic phase space corrections to the Maxwell-Boltzmann ideal gas model, improving the understanding of quantum size, shape effects, and relativistic influences on thermodynamics.

## Contribution

It presents novel quantum and relativistic phase space corrections to the ideal gas model, extending beyond particle statistics to include size, shape, and relativistic effects.

## Key findings

- Quantum corrections describe deviations at low temperatures and confined spaces.
- Relativistic corrections are relevant when thermal energy approaches rest energy.
- Classical thermodynamics are recovered as asymptotic limits.

## Abstract

The quantum corrections related to the ideal gas model that are often considered are those which are related to the particles nature: bosons or fermions. These corrections lead respectively to the Bose-Einstein and Fermi-Dirac statistics. However, in this work, other kinds of corrections which are related to the quantum nature of phase space are considered. These corrections are introduced as improvement in the expression of the partition function of an ideal gas. Then corrected thermodynamics properties of the gas are deduced. Both the non-relativistic quantum and relativistic quantum cases are considered. It is shown that the corrections in the non-relativistic quantum case may be particularly useful to describe the deviation from classical behavior of a Maxwell-Boltzmann gas at low temperature and in confined space. These corrections can be considered as including the description of quantum size and shape effects. For the relativistic quantum case, the corrections could be relevant for confined space and when the thermal energy of each particle is comparable to their rest energy. The corrections appear mainly as modifications in the thermodynamic equation of state and in the expressions of the partition function and thermodynamic functions like entropy, internal energy, and free energy. Classical expressions are obtained as asymptotic limits.

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Source: https://tomesphere.com/paper/2302.13973