Thermal Equilibrium and Statistical Thermometers in Special Relativity

There is an intense debate in the recent literature about the correct generalization of Maxwell’s velocity distribution in special relativity. The most frequently discussed candidate distributions include the Jüttner function as well as modifications thereof. Here we report results from fully relativistic one-dimensional molecular dynamics simulations that resolve the ambiguity. The numerical evidence unequivocally favors the Jüttner distribution. Moreover, our simulations illustrate that the concept of “thermal equilibrium” extends naturally to special relativity only if a many-particle system is spatially confined. They make evident that “temperature” can be statistically defined and measured in an observer frame independent way.

Figure 1

Equilibrium PDFs in the lab frame $\Sigma$: Numerically obtained one-particle velocity PDFs (⋄) based on a simulation with $N_1=5000$ light particles of mass $m_1$ and $N_2=5000$ heavy particles with mass $m_2=2m_1$. The mean energy per particle in $\Sigma$ is $ϵ=E_{\mathrm{tot}}/(N_1+N_2)=2.5m_1{c}^2$. The solid curves correspond to Jüttner functions (2) with the same parameter ${\beta }_J=0.702(m_1{c}^2{)}^{-1}$ but different particle masses, respectively. The dashed lines show the corresponding modified distribution (3) with ${\beta }_{MJ}=0.402(m_1{c}^2{)}^{-1}$. As the distributions are symmetric with respect to the origin, only the positive velocity axis is shown. The simulation data are consistent with the standard Jüttner distribution (2) and thus provide evidence against the modified distribution (3).

Figure 2

Equilibrium PDFs in a moving frame ${\Sigma }^{\prime }$: Velocity PDFs as measured by an observer who moves with velocity $u=0.25c$ relative to the lab frame $\Sigma$. Parameter values are the same as in Fig. 1. The solid lines correspond to Jüttner functions $f_J^{\prime }$ from Eq. (8) with the same parameter ${\beta }_J=0.702(m_1{c}^2{)}^{-1}$ as in Fig. 1 and different masses $m_1$ and $m_2$, respectively.