An efficient way to examine thermodynamics of relativistic ideal Fermi and Bose gases

doi:10.1016/j.physa.2020.124638

Physica A: Statistical Mechanics and its Applications

Volume 555, 1 October 2020, 124638

https://doi.org/10.1016/j.physa.2020.124638 Get rights and content

Highlights

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A complete analytical method of the evaluation of the pressure term of relativistic ideal Fermi and Bose Gases have been proposed.

Abstract

The determination of thermodynamic properties such as entropy, energy, pressure, free energy and etc. is very important for all fields of thermodynamic physics. Also giving reliable analytical formulas for evaluating thermodynamic properties of relativistic ideal gases is essential. One of the way of solving thermodynamic properties of ideal Fermi and Bose Gases is determining the expression of pressure integral. In this study we have presented a new computational method for the evaluation of the pressure term of the relativistic ideal Fermi and Bose gases. The purposed method shows that obtained analytically series expansions works for any values of parameters at high temperatures.

Introduction

Thermodynamics can be defined as the science of energy and entropy. Thermodynamics has a wide range of applications from automobiles to planes and spacecrafts, from power plants to air conditioning systems and computers and other fields of engineering [1], [2]. Because of wide range of applications, the developments in thermodynamic physics have attracted to the scientists. It is necessary to say that if a thermodynamic property of particle at a given state is known, it is possible to determine other properties such as internal energy, enthalpy, entropy and etc. The state of a thermodynamic system is defined by a number of thermodynamic variables such as temperature, volume or pressure [3], [4], [5], [6].

The ideal gas can be expressed as a gas of non-interacting atoms in the limit of low concentration. It is clear from quantum mechanics all particles are either fermions or bosons. The fermions obey the Pauli exclusion principle with half-integer spins. Bosons are the particles which follows Bose–Einstein statics and can occupy the same quantum state with an integer or zero spin. The Pauli principle does not apply to the bosons so there is not any limitation on the occupancy of any orbital. The term of “ideal” comes from a collection of non-interacting fermions or bosons for the case of Fermi gas or Bose gas, respectively [3].

The determining thermodynamic properties of relativistic ideal Fermi and Bose gases is a fundamental step of quantum thermodynamics. In literature there are significant studies on the expressing thermodynamic properties of an ideal Fermi and Bose gases including integral forms or series of some special functions [7], [8], [9], [10], [11], [12]. Unfortunately, most of these integrals cannot be solved exactly because of some limitations. The authors have used different numerical methods to overcome these deficiencies. The numerical methods avoid low temperature limits but cannot applied to the high temperature expansions. Also numerical methods does not give the exact solutions. It is well known that the accuracy and computational efficiency are the most important part of this kind of work. Also the numerical integration methods give correct results for a restricted range of parameters. Apart from these, there are computational problems, some of which are expected to expose the limitations of numerical methods. It is not generally possible to give an algorithmic description for the choice of the best numerical method for a given problem [13]. It is clear that analytically evaluations have no limitations in their uses. Many authors have tried to obtain the analytical evaluation procedures for the calculation of thermodynamic properties of ideal Fermi and Bose gases [14], [15]. But these methods are generally complicated because of using complex special functions.

In this study, we aimed to provide an alternative analytical method for the evaluation of the pressure term of ideal Fermi and Bose gases at high temperatures. It is clear that the other thermodynamical properties can be easily derived from the term of pressure. The novelty of this study is to give an efficient approximation discarding temperature limitations at high values. Estimating the behavior of the ideal Fermi and Bose gases at high temperatures with respect to the relativistic effects will be very useful in plasma and astronomy studies and also lead future studies.

Section snippets

Basic formulas and definitions

The pressure integral of an ideal Bose or Fermi gas can be defined as following form in a.u [12]: $P (T, μ, m) = - α T \int \frac{d^{3} k}{{(2 π)}^{3}} ln [1 - α e^{β (μ - E)}]$ here $E = \sqrt{k^{2} + m^{2}}$ is the relativistic energy, $m$ is the mass of particles, $μ$ is the chemical potential, $T$ is the temperature, $β = 1 ∕ k_{B} T$ and $α$ has special values in following form: $α = \{\begin{matrix} - 1 & f e r m i o n s \\ + 1 & b o s o n s \end{matrix}$ By the use of $λ = \frac{m}{T}$ and $ν = \frac{μ}{T}$ dimensionless variables Eq. (1) can be expressed as: $P (λ, ν) = \frac{1}{6 π^{2}} \int_{λ}^{\infty} \frac{{(x^{2} - λ^{2})}^{3 ∕ 2}}{(e^{x - ν} - α)} d x .$ In literature there are some studies to evaluate Eq. (3) with

Numerical results and discussion

In order to prove the accuracy of this work, the analytical results of the pressure term of ideal Bose and Fermi gases have been compared with the Mathematica numerical integration methods. For this purpose all calculations have been performed with the help of Mathematica 10.0 programming language. In Table 1, the calculation results obtained from Mathematica integration method and the given analytical method in this study have been listed in case of $λ > ν$ . We have concluded that as the

Conclusion

In this paper by the use of binomial expansion theorem, we present a new alternative way to calculate the pressure term of ideal Bose and Fermi gases for the case of $λ > ν$ . We can say that the present scheme give us improving the estimation of some other thermodynamical properties of ideal Bose and Fermi gases. Because the determinations of the particle number, entropy and the energy densities depend on the derivatives of the pressure term. For this purpose, we have decided to focus on new

CRediT authorship contribution statement

Ebru Çopuroğlu: Conceptualization, Methodology, Writing - review & editing.

Declaration of Competing Interest

The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.

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An efficient way to examine thermodynamics of relativistic ideal Fermi and Bose gases

Highlights

Abstract

Introduction

Section snippets

Basic formulas and definitions

Numerical results and discussion

Conclusion

CRediT authorship contribution statement

Declaration of Competing Interest

Physica A

New Astron.

Thermodynamics of Natural Systems

Introduction to the Thermodynamics of Materials

Introduction to Solid State Physics, Vol. 8

Thermodynamics and Statistical Mechanics

Introduction to Statistical Physics

The relativistic bose gas

J. Phys. A: Math. Gen.