Integrability properties for relativistic extended thermodynamics of polyatomic gas

Abstract

In a recent article (Ann Phys 377:414–445, 2017. https://doi.org/10.1016/j.aop.2016.12.012), Pennisi and Ruggeri proposed a relativistic extended thermodynamics theory of polyatomic gas. It was achieved by adopting the closure procedure for the generalized moments of a distribution function that, as in the classical case, depends on an additional continuous variable representing the energy of the internal modes of a molecule; this permits the theory to take into account the energy exchange between the translational and the internal modes of a molecule in binary collisions. In this paper the attention will be focused on some integrals appearing in the field equations of the relativistic theory of polyatomic gas and their integrability will be proven. In the last part of the paper we consider the case of a monatomic gas and we evaluate the ultra-relativistic limit of the differential system of balance laws.

Appendix: Proof of Eqs. (6) and (8)

We start from the Newton’s binomial rule

$$\begin{aligned} (a+b)^n&= \sum _{h=0}^n \begin{pmatrix} n \\ h \end{pmatrix} a^h b^{n-h}= \sum _{h=0}^{\left[ \frac{n}{2} \right] } \begin{pmatrix} n \\ h \end{pmatrix} a^h b^{n-h} + \sum _{h=\left[ \frac{n}{2} \right] +1}^{n} \begin{pmatrix} n \\ h \end{pmatrix} a^h b^{n-h} {\mathop {=}\limits ^{*}} \\&= \sum _{h=0}^{\left[ \frac{n}{2} \right] } \begin{pmatrix} n \\ h \end{pmatrix} a^h b^{n-h} + \sum _{k=0}^{n-\left[ \frac{n}{2} \right] -1} \begin{pmatrix} n \\ n-k \end{pmatrix} a^{n-k} b^k = \sum _{h=0}^{\left[ \frac{n}{2} \right] } \begin{pmatrix} n \\ h \end{pmatrix} a^h b^{n-h} \\&\quad + \sum _{h=0}^{\left[ \frac{n-1}{2} \right] } \begin{pmatrix} n \\ h \end{pmatrix} a^{n-h} b^h = \sum _{h=0}^{\left[ \frac{n-1}{2} \right] } \begin{pmatrix} n \\ h \end{pmatrix} \left( a^h b^{n-h} + a^{n-h} b^h \right) + \begin{pmatrix} n \\ n/2 \end{pmatrix} a^{\frac{n}{2}} b^{\frac{n}{2}}, \end{aligned}$$

where the last term is present only if n is an even number, and in the passage denoted with ${\mathop {=}\limits ^{*}}$ we have changed the index of the second summation according to the rule $n-h=k$.

By using this result with $a=\frac{1}{2} e^s$ and $b=\frac{1}{2} e^{-s}$, we find

$$\begin{aligned} \cosh ^n s&= \sum _{h=0}^{\left[ \frac{n-1}{2} \right] } \begin{pmatrix} n \\ h \end{pmatrix} \frac{1}{2^{n}} \left( e^{(2h-n)s} + e^{-(2h-n)s} \right) + \begin{pmatrix} n \\ n/2 \end{pmatrix} \frac{1}{2^{n}} \\&= \frac{1}{2^{n-1}} \sum _{h=0}^{\left[ \frac{n-1}{2} \right] } \begin{pmatrix} n \\ h \end{pmatrix} \cosh [(2h-n)s] + \begin{pmatrix} n \\ n/2 \end{pmatrix} \frac{1}{2^{n}} \cosh 0. \end{aligned}$$

By multiplying this relation by $e^{-\gamma \cosh s}$ and integrating in ds for $s \in [0, \, + \infty [$, we find Eq. (6); we have only to take into account that, when n is an even number,

$$\begin{aligned} \begin{pmatrix} n \\ n/2 \end{pmatrix} \frac{1}{2^{n}}&= \frac{n!}{[(n/2)!]^2} \frac{1}{2^{n}} = \frac{(n-1)!! \, n!!}{[(n/2)!]^2} \frac{1}{2^{n}} = \frac{(n-1)!! \, 2^{n/2} (n/2)!}{[(n/2)!]^2} \frac{1}{2^{n}}\\&= \frac{(n-1)!!}{(n/2)!} \frac{1}{2^{n/2}} = \frac{(n-1)!!}{n!!}. \end{aligned}$$

Let us prove now Eq. (8).

From $\cosh s + \sinh s = e^s$, $\cosh s - \sinh s = e^{-s}$, it follows

$$\begin{aligned}{} & {} e^{ns} = \sum _{h=0}^{n} \begin{pmatrix} n \\ h \end{pmatrix} \sinh ^h s \cosh ^{n-h} s, \quad e^{-ns} = \sum _{h=0}^{n} \begin{pmatrix} n \\ h \end{pmatrix} (-1)^h \sinh ^h s \cosh ^{n-h} s \, \\{} & {} \text{ from } \text{ which } \text{ it } \text{ follows } \quad \cosh (ns) = \frac{1}{2} \sum _{h=0}^{n} \begin{pmatrix} n \\ h \end{pmatrix} \left[ 1 + (-1)^h \right] \sinh ^h s \cosh ^{n-h} s. \end{aligned}$$

Here, only the terms with h even i.e. $h=2r$ survive and the above relation becomes

$$\begin{aligned}{} & {} \cosh (ns) = \sum _{r=0}^{\left[ \frac{n}{2} \right] } \begin{pmatrix} n \\ 2r \end{pmatrix} \sinh ^{2r} s \cosh ^{n-2r} s = \sum _{r=0}^{\left[ \frac{n}{2} \right] } \begin{pmatrix} n \\ 2r \end{pmatrix} \sum _{p=0}^{r} \begin{pmatrix} r \\ p \end{pmatrix} (-1)^p \cosh ^{n-2p} s. \end{aligned}$$

By multiplying this relation by $e^{-\gamma \cosh s}$ and integrating in ds for $s \in [0, \, + \infty [$, we find

$$\begin{aligned}{} & {} K_n (\gamma ) = \sum _{r=0}^{\left[ \frac{n}{2} \right] } \begin{pmatrix} n \\ 2r \end{pmatrix} J_{2r, n-2r} (\gamma )= \sum _{r=0}^{\left[ \frac{n}{2} \right] } \begin{pmatrix} n \\ 2r \end{pmatrix} \sum _{p=0}^{r} \begin{pmatrix} r \\ p \end{pmatrix} (-1)^p J_{0, n-2p} (\gamma ). \end{aligned}$$

But $\sum _{r=0}^{\left[ \frac{n}{2} \right] } \sum _{p=0}^{r}$ means that the summation is extended to all the indexes (r, p) with $0 \le r \le \left[ \frac{n}{2} \right]$, $0 \le p \le \left[ \frac{n}{2} \right]$, $0 \le r-p$; so we can exchange the order of the two summations and obtain Eq. (8).

It is interesting to note that here the term with $J_{0, n} (\gamma )$ appears only for $p=0$ and its coefficient is

$$\begin{aligned}{} & {} \sum _{r=0}^{\left[ \frac{n}{2} \right] } \begin{pmatrix} n \\ 2r \end{pmatrix} = \left\{ \begin{matrix} 2^{n-1} &{}\quad \text{ if } \quad n \ge 1, \\ 1 &{}\quad \text{ if } \quad n =0. \end{matrix} \right. \end{aligned}$$

In fact, for $n \ge 1$ we have

$$\begin{aligned} (1+1)^n = \sum _{h=0}^{n} \begin{pmatrix} n \\ h \end{pmatrix}, \quad (1-1)^n = \sum _{h=0}^{n} \begin{pmatrix} n \\ h \end{pmatrix} (-1)^h \quad \text{ whose } \text{ sum } \text{ is } \quad 2^n = \sum _{h=0}^{n} \begin{pmatrix} n \\ h \end{pmatrix} \left[ 1 +(-1)^h \right] ; \end{aligned}$$

here, only the terms with h even i.e. $h=2r$ survive and the above relation becomes

$$\begin{aligned}&2^n = \sum _{r=0}^{\left[ \frac{n}{2} \right] } \begin{pmatrix} n \\ 2r \end{pmatrix} 2. \end{aligned}$$