Statistical thermodynamics of a two-dimensional relativistic gas

In this paper we study a fully relativistic model of a two-dimensional hard-disk gas. This model avoids the general problems associated with relativistic particle collisions and is therefore an ideal system to study relativistic effects in statistical thermodynamics. We study this model using molecular-dynamics simulation, concentrating on the velocity distribution functions. We obtain results for $x$ and $y$ components of velocity in the rest frame $\left(\Gamma \right)$ as well as the moving frame $\left({\Gamma }^{\prime }\right)$. Our results confirm that Jüttner distribution is the correct generalization of Maxwell-Boltzmann distribution. We obtain the same “temperature” parameter $\beta$ for both frames consistent with a recent study of a limited one-dimensional model. We also address the controversial topic of temperature transformation. We show that while local thermal equilibrium holds in the moving frame, relying on statistical methods such as distribution functions or equipartition theorem are ultimately inconclusive in deciding on a correct temperature transformation law (if any).

Figure 1

(Color online) Equilibrium velocity distributions in the rest frame $\Gamma$: numerically obtained $x$ component of single-particle velocity distribution $(+)$ from simulation of $N=100$ particles of mass $m=0.1$. (a) Here, $ϵ=2.28m$ and the corresponding temperature parameters are ${\beta }_J=11.4$, ${\beta }_{\text{MJ}}=7.8$. (b) $ϵ=1.04m$, ${\beta }_J=253.1$, and ${\beta }_{\text{MB}}=259.8$. A significant deviation from modified Jüttner function is evident in the relativistic regime. Similar results are obtained for $f\left(v_y\right)$.

Figure 2

(Color online) Equilibrium velocity distributions in the moving frame ${\Gamma }^{\prime }$ with relative velocity $u=0.5$. The system parameters are the same as Fig. 1a in particular ${\beta }_J=11.4$ and ${\beta }_{\text{MJ}}=7.8$. Note the breaking of symmetry in part (a) where the inset shows more details of the peak.

Figure 3

(Color online) Local thermal equilibrium in the box as seen by a moving observer: The profile $\mathcal{C}\left(X,Y\right)$ is measured ${\Gamma }^{\prime }$ simultaneously for the same system as in Fig. 2 with $L_x$ and $L_y$, each divided into ten parts. The expected value of ${\left[\beta \gamma \left(u\right)\right]}^{-1}$ is 0.076 which agrees well with the obtained value of $\mathcal{C}$ throughout the lattice.