Stationarity, ergodicity, and entropy in relativistic systems

Abstract

-Recent molecular-dynamics simulations show that a dilute relativistic gas equilibrates to a Jüttner velocity distribution if ensemble velocities are measured simultaneously in the observer frame. The analysis of relativistic Brownian motion processes, on the other hand, implies that stationary one-particle distributions can differ depending on the underlying time parameterizations. Using molecular-dynamics simulations, we demonstrate how this relativistic phenomenon can be understood within a deterministic model system. We show that, depending on the time parameterization, one can distinguish different types of "soft" ergodicity on the level of the one-particle distributions. Our analysis further reveals a close connection between time parameters and entropy in special relativity. A combination of different time parameterizations can potentially be useful in simulations that combine molecular-dynamics algorithms with randomized particle creation, annihilation, or decay processes. Copyright c EPLA, 2009Introduction. -Understanding the relation between ensemble and time averages poses one of the most fundamental problems in statistical physics. Ergodicity -the equivalence of the two averaging procedures-is a commonly employed assumption in statistical mechanics [1], albeit difficult to prove for realistic systems. During the past decades, the ergodicity hypothesis was intensely examined for nonrelativistic classical [2][3][4][5] and quantum models [6][7][8]. However, much less is known about its meaning and validity in relativistic settings [9], when even more basic concepts like "stationarity" may become ambigous as time becomes relative [10][11][12]. A clear conception of the interplay between time parameters and thermostatistical concepts, like entropy [13][14][15], is crucial, e.g., if one wishes to generalize non-equilibrium fluctuation theorems to a relativistic framework [16,17]. Given the rapidly increasing number of applications in high-energy physics [18,19] and astrophysics [20,21], a firm conceptual foundation is desirable not only from a theoretical, but also from a practical perspective.Ideally, one would like to tackle relativistic manyparticle problems within a quantum field theory framework, as this allows for the consistent treatment of particle

Time parametrization and stationary distributions in a relativistic gas

“…Among these, are the evolution of dynamical systems towards equilibrium and more specifically their equilibrium probability distribution function (PDF). As reported recently [17,18,23], the description of the motion of relativistic particles with respect to the observer's time, t, and the particle's proper time, τ are completely different. To elucidate, consider the t-averaged PDF of a Brownian particle in terms of the observer's time…”

“…Arguments of this kind suggest that one may possibly * Electronic address: montakhab@shirazu.ac.ir seek multiple relativistic generalizations of MB distribution. Using molecular dynamics simulations [23] and theoretical analysis [17], some authors have recently proposed that it is possible to establish a connection between the time-parameter and the kind of symmetry that lead to a special distribution. Consequently, relativistic distributions fall into two classes that are distinguished by their associated symmetry and time-parameter.…”

“…Numerical studies are also not decisive due to the arbitrariness in the choice of time parameters [17,23] or discretization rules [24,25,26]. Therefore, at this stage we are obliged to accept the arbitrariness in the choice of reference density.…”

“…By simply changing the integration variable, it can be shown [23] that the relation between stationary (t, τ → ∞) distributions isf…”

Nonextensivity and the q-distribution of a relativistic gas under an external electromagnetic field

“…In this aspect, some theoretical research has been done (e.g. see [3][4][5][6][7][8][9][10][11] and the references therein), in which self-gravitating systems and plasma systems have offered the best framework to research the nonextensive effects [4][5][6][7][8][12][13][14][15][16][17][18][19][20][21][22][23][24].…”