Generalized Stefan–Boltzmann Law

Abstract

We reconsider the thermodynamic derivation by L. Boltzmann of the Stefan law and we generalize it for various different physical systems whose chemical potential vanishes. Being only based on classical arguments, therefore independent of the quantum statistics, this derivation applies as well to the saturated Bose gas in various geometries as to “compensated” Fermi gas near a neutrality point, such as a gas of Weyl Fermions. It unifies in the same framework the thermodynamics of many different bosonic or fermionic non-interacting gases which were until now described in completely different contexts.

Appendices

Appendix A: Planck’s natural unit of action

In his 1899 paper [6], more than one year before his discovery of the black body law with quantization of energy exchange, M. Planck found a natural unit of action from his analysis of the Stefan–Boltzmann law and Wien’s law:

$$\begin{aligned} u(\nu ,T)= { 8 \pi b \over c^3} \nu ^3 e^{- a \nu /T} \end{aligned}$$

(36)

where a and b are parameters to be fixed from the experiment. Comparing with experimental estimate of the total energy

$$\begin{aligned} u(T)= \int _0^\infty u(\nu ,T) d \nu = { 48 \pi b \over a^4 c^3}\, T^4 \end{aligned}$$

(37)

and Wien’s displacement law, Planck could extract the parameters $b/a^4$ and a, and obtain the values $b= 6.885 \, 10^{-34}$ J s and $a= 0.4818 \, 10^{-10}$ K s. This was the first estimate of a unit of action, which was not yet related to any idea of quantization, and which will be denoted later as h in the 1900 discovery of the black body law. Moreover, in our modern notations, $a= h/k_B$ so that the obtained value of a leads to an estimate of $k_B = 1.43 \, 10^{-23}$J/K (that Planck will introduce one year later in his seminal paper).

Appendix B: Thermodynamic relations

Here we show the equivalence between these to relations

$$\begin{aligned} U = \left( {\partial \beta \varOmega \over \partial \beta }\right) _{\! \alpha } \quad \Longleftrightarrow \quad \varOmega =U - TS - \mu N \end{aligned}$$

(38)

One first explicitly write the derivative as

$$\begin{aligned} U = \beta \left( {\partial \varOmega \over \partial \beta }\right) _{\! \alpha } + \varOmega = - T \left( {\partial \varOmega \over \partial T}\right) _{\! \alpha }+ \varOmega \end{aligned}$$

(39)

The next step is to rewrite $\left( {\partial \varOmega \over \partial T}\right) _{\! \alpha }$. We first write the grand potential $\varOmega$ as ($k_B=1$ in this section)

$$\begin{aligned} \varOmega \left( \alpha = {\mu \over T},T \right) \ . \end{aligned}$$

(40)

Consider the derivatives

$$\begin{aligned} \left( {\partial \varOmega \over \partial \mu }\right) _{\! T}&= {1 \over T} \left( {\partial \varOmega \over \partial \alpha }\right) _{\! T} \end{aligned}$$

(41)

$$\begin{aligned} \left( {\partial \varOmega \over \partial T}\right) _{\! \mu }&= -{\mu \over T^2} \left( {\partial \varOmega \over \partial \alpha }\right) _{\! T}+ \left( {\partial \varOmega \over \partial T}\right) _{\! \alpha } \end{aligned}$$

(42)

Inserting the first equation into the second one in order to eliminate $\alpha$, we obtain

$$\begin{aligned} T \left( {\partial \varOmega \over \partial T}\right) _{\! \alpha }= \mu \left( {\partial \varOmega \over \partial \mu }\right) _{\! T} + T \left( {\partial \varOmega \over \partial T}\right) _{\! \mu } = - \mu N - TS \end{aligned}$$

(43)

Inserting this last relation in Eq. (39), we find the expected result.

Appendix C: Exact thermodynamic results

We give here for completeness the exact expressions for the density of states and thermodynamic quantities of gases of particles in various dimensions and geometries.

1.1 C.1 Integrated density of states

Here we give complete expressions fot the integrated density of states for particles with various dispersion relations.

• Free massive particles, $\epsilon = {p^2 \over 2 m}$ :

$$\begin{aligned} N_{\!<}(\epsilon )= C_d \left( {L \over 2 \pi }\right) ^d \left( {2 m \epsilon \over \hbar ^2}\right) ^{d/2} \ . \end{aligned}$$

(44)

where $C_d= \pi ^{d/2}/\Gamma (1+d/2)$ is the unit volume in d dimensions.

• Free massless particles, $\epsilon = |p| c$ :

$$\begin{aligned} N_{\!<}(\epsilon )= C_d \left( {L \over 2 \pi }\right) ^d \left( { \epsilon \over \hbar c}\right) ^{d} \ . \end{aligned}$$

(45)

• Particles in a isotropic harmonic potential, $\epsilon = (n_x+n_y+n_z) \hbar \omega$ :

$$\begin{aligned} N_{\!<}(\epsilon )= {1 \over d!} \left( { \epsilon \over \hbar \omega }\right) ^{d} \ . \end{aligned}$$

(46)

• 2D semi-Dirac fermions $\epsilon =\sqrt{\left( {p_x^2 \over 2 m}\right) ^2 + p_{y}^{2} c^2}$ : [14]

$$\begin{aligned} N_{\!<}(\epsilon )=\left( {L \over \hbar }\right) ^2 C {\sqrt{m} \over c } \epsilon ^{3/2} \, \end{aligned}$$

(47)

with $C= {\Gamma (1/4)^2 \over 6 \pi ^{5/2}}$.

• 2D particles in a boxed-harmonic potential, $\epsilon = {p_x^2 \over 2 m} + n_y \hbar \omega$ :

$$\begin{aligned} N_{\!<}(\epsilon )= {2 L \over 3 \pi } {\sqrt{m} \over \hbar ^2 \omega } \epsilon ^{3/2} \ . \end{aligned}$$

(48)

• Particles with dispersion relation $\epsilon = \sum _j A_j n_j^{{\nu }_j}$, $n_j >0$ :

$$\begin{aligned} N_{\!<}(\epsilon )= {\epsilon ^\kappa \over \Gamma \left( 1 + \kappa \right) } \prod _j { \Gamma \left( 1+ {1 / {\nu }_j}\right) \over A_j^{1/{\nu }_j}} \end{aligned}$$

(49)

with $\kappa = d_c + d_m/2$. The previous equations (44-46, 48) may be obtained from this formula with

$$\begin{aligned} {\nu }_j&=1, A_j=\hbar \pi c_j/L_j or A_j=\hbar \omega _j\\ {\nu }_j&=2, A_j=(\hbar ^2 \pi ^2 /(2 m_j L_j^2), \end{aligned}$$

and the integrated density of states may be summarized in the form:

$$\begin{aligned} N_{\!<}(\epsilon )= {\epsilon ^\kappa \over \Gamma \left( 1+ \kappa \right) } \prod _{j=1}^{d_c} \left( {L_j\over \pi \hbar c_j}\right) \prod _{j=1}^{d_m} \left( \sqrt{2 \pi m \over h^2} L_j \right) \end{aligned}$$

(50)

for a massless spectrum in $d_c$ directions and in the form:

$$\begin{aligned} N_{\!<}(\epsilon )= {\epsilon ^\kappa \over \Gamma \left( 1+ \kappa \right) } \prod _{j=1}^{d_c} \left( {1 \over \hbar \omega _j}\right) \prod _{j=1}^{d_m} \left( \sqrt{2 \pi m \over h^2} L_j \right) \ . \end{aligned}$$

(51)

for an harmonic potential in $d_c$ directions.

$\bullet$ Particles with dispersion relation $\epsilon = \sqrt{\sum _j (A_j n_j)^{2 {\nu }_j} }$, $n_j >0$:

$$\begin{aligned} N_{\!<}(\epsilon )= {\epsilon ^\kappa \over \Gamma \left( 1 + \kappa /2 \right) } \prod _j { \Gamma \left( 1+ {1 / (2 {\nu }_j)}\right) \over A_j^{1/{\nu }_j}} \end{aligned}$$

(52)

With the above expressions of $A_j$, the integrated density of states can be summarized as:

$$\begin{aligned} N_{\!<}(\epsilon )&= {\epsilon ^\kappa \over \Gamma \left( 1+ \kappa /2 \right) } \Gamma (3/2)^{d_c} \Gamma (5/4)^{d_m} \nonumber \\&\qquad \times \prod _{j=1}^{d_c} \left( {L_j\over \pi \hbar c_j}\right) \prod _{j=1}^{d_m} \left( \sqrt{\frac{2 m}{\pi ^2 \hbar ^2}} L_j \right) \end{aligned}$$

(53)

which is the same as for the dispersion $\epsilon = \sum _j A_j n_j^{{\nu }_j}$, within different numerical factors.

1.2 C.2 Bosons and Fermions with zero $\mu$

Here we collect the exact thermodynamic quantities for the saturated Bose gas and the compensated Fermi gas ($\mu =0$). We start from a power law for the integrated density of states , $N_{\!<}(\epsilon )= B \epsilon ^\kappa$. The total energy is

$$\begin{aligned} U(T) = B \kappa \int _0^\infty {\epsilon ^\kappa \over e^{{\nu }\epsilon - \alpha } \pm 1} d\epsilon . \end{aligned}$$

(54)

For the saturated Bose gas, the number of particles and the total energy read: [16]

$$\begin{aligned} N(T)&= B \Gamma (1+\kappa ) \zeta (\kappa ) \, T^{\kappa } \nonumber \\ U(T)&= \kappa {\zeta (\kappa +1) \over \zeta (\kappa )} N(T) k_B T \ . \end{aligned}$$

(55)

For the compensated Fermi gas, we have

(56)