Statistical thermodynamics of a two-dimensional relativistic gas

Abstract

In this article 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 (Γ) as well as the moving frame (Γ ′ ). Our results confirm that Jüttner distribution is the correct generalization of Maxwell-Boltzmann distribution. We obtain the same "temperature" parameter β 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).

Can electron distribution functions be derived through the Sunyaev-Zel’dovich effect?

“…One of the methods to study the relativistic equilibrium DF of particles is to perform fully relativistic molecular dynamics simulations (see Cubero et al 2007, for the 1D case; Montakhab et al 2009, for the 2D case; and Peano et al 2009, for the 3D case). However, to justify the use of the relativistic equilibrium DF in astrophysics, an observational confirmation is required.…”

Time parametrization and stationary distributions in a relativistic gas

“…The model we have used here is the previously proposed two dimensional relativistic hard-disk gas [15]. In this model, the disk-like particles move in straight lines at constant speed and change their momenta instantaneously when they touch at distance σ.…”

“…We further clarify the resulting consequence of these different choices. Using our recently proposed two dimensional hard disk model [15], we perform simultaneous measurements at fixed observer's time 't' and particle's proper time 'τ ' with respect to a rest as well as a moving observer. The obtained results are then compared with stationary distributions consistent with different symmetry considerations.…”

“…Figure 2 shows simulation results for a system described in Fig.1 with u = 0.1. Here, t-averaged PDF is obtained by measuring velocities simultaneously with respect to the moving observer as described in [14,15]. Considering the Lorentz invariance of proper-time it is not difficult to find the τ -averaged PDF in the moving frame.…”

“…To resolve the uncertainty, semi-relativistic [13] and fully relativistic [14,15] molecular dynamics simulations as well as Monte Carlo studies [16] have been performed by different groups in recent years that unequivocally favored Jüttner distribution. However, some recent investigations on relativistic Brownian motions [17,18] have revealed that stationary distributions can differ depending on the underlying time-parameterizations, a problem which never arises in Newtonian physics due to the existence of a universal time for all inertial observers.…”