Thermal equilibrium in Einstein's elevator

Bernardo Sánchez-Rey, Guillermo Chacón-Acosta, Leonardo Dagdug, and David Cubero

Phys. Rev. E 87, 052121 – Published 17 May 2013

Abstract

We report fully relativistic molecular-dynamics simulations that verify the appearance of thermal equilibrium of a classical gas inside a uniformly accelerated container. The numerical experiments confirm that the local momentum distribution in this system is very well approximated by the Jüttner function—originally derived for a flat spacetime—via the Tolman-Ehrenfest effect. Moreover, it is shown that when the acceleration or the container size is large enough, the global momentum distribution can be described by the so-called modified Jüttner function, which was initially proposed as an alternative to the Jüttner function.

Received 6 February 2013

DOI:https://doi.org/10.1103/PhysRevE.87.052121

Authors & Affiliations

Bernardo Sánchez-Rey¹, Guillermo Chacón-Acosta², Leonardo Dagdug³, and David Cubero^1,*

¹Departamento de Física Aplicada I, EPS, Universidad de Sevilla, Calle Virgen de África 7, 41011 Sevilla, Spain
²Departamento de Matemáticas Aplicadas y Sistemas, Universidad Autónoma Metropolitana-Cuajimalpa, México D. F. 01120, México
³Departamento de Física, Universidad Autónoma Metropolitana-Iztapalapa, México D. F. 09340, México

^*Corresponding author: dcubero@us.es

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Vol. 87, Iss. 5 — May 2013

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Images

Figure 1
Numerically measured equilibrium distributions of momenta in Einstein's elevator. The results are based on simulations of an accelerated rigid container of length $L$ with a mixture gas inside consisting of $N_{1} = 415$ light particles with mass $m_{1}$ (diamonds) and $N_{2} = 585$ particles with mass $m_{2} = 2 m_{1}$ (crosses) and a transparency probability of $q_{t} = 1 / 2$ . Reduced units are defined such that $L = c = m_{1} = 1$ . In these units, the acceleration of the bottom of the container (located at $z = 0$ ) in its proper inertial frame is $g = 0.5$ . The solid lines correspond to the prediction (20) with same parameter $β = 0.914$ , showing a very good agreement.Reuse & Permissions
Figure 2
Tolman-Ehrenfest effect in Einstein's elevator. Local temperature $T_{0}$ of the light (diamonds) and heavy (crosses) particles as measured locally using the statistical thermometer proposed in Ref. [11]. The solid line is the Tolman-Ehrenfest equation (22) with $k_{B} T = 1.094$ . The rest of the parameters are as described in the caption to Fig. 1.Reuse & Permissions
Figure 3
Particle density in Einstein's elevator for light particles (diamonds) and heavy particles (crosses). The solid lines correspond to (24) for both species. The rest of the parameters are as described in the caption to Fig. 1.Reuse & Permissions
Figure 4
Sensibility of particle density on the point particles’ semipenetrability. The top panel shows the same as in Fig. 5 but for a system of impenetrable particles, i.e., $q_{t} = 0$ . The bottom panel show the relative error $| ρ - ρ_{th} | / ρ_{th}$ , where $ρ$ is the numerically measured particle density and $ρ_{th}$ the analytical prediction given by (24), for the light particles for several transparency probabilities: $q_{t} = 0$ (filled diamonds), $q_{t} = 1 / 4$ (circles), $q_{t} = 1 / 2$ (triangles), and $q_{t} = 3 / 4$ (squares), all simulations starting with the same initial conditions. The rest of the parameters are as described in the caption to Fig. 1.Reuse & Permissions
Figure 5
Momentum distributions $f (z, p)$ at the borders of the container. The simulation box was divided into 50 bins along the $z$ direction: (a) lower bin and (b) upper bin. The diamonds (crosses) correspond to light (heavy) particles and the solid lines to the generalized Jüttner function (18). The rest of the parameters are as described in the caption to Fig. 1.Reuse & Permissions
Figure 6
Equilibrium distributions of momenta in Einstein's elevator for an acceleration $g = 5$ (rest of the parameters as in Fig. 1). The diamonds (crosses) correspond to the light (heavy) particles. The marginal distributions are very well approximated by the modified Jüttner function (25) with $β = 0.71$ (solid lines). This inverse temperature value was obtained by using the same procedure as in Fig. 2. The dotted lines show the special-relativity Jüttner functions (21) with the same $β$ , plotted as a reference. As the distributions are symmetric with respect to the origin, only the positive momentum axis is shown.Reuse & Permissions

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