The Relativistic Boltzmann Equation: Theory and Applications

One-dimensional non-relativistic and relativistic Brownian motions: a microscopic collision model

“…[23][24][25]). In two recent papers [26,27] we have discussed in detail how one can construct Langevin equations for relativistic Brownian motions (see Debbasch et al [28,29] and Zygadlo [30] for similar approaches, and also Dunkel and Ha¨nggi [31]).…”

“…[26,27] we have analyzed the three most popular discretization rules for Langevin equations with multiplicative noise, namely, Ito's pre-point discretization rule [32,33], the Fisk-Stratonovich mid-point rule [34][35][36][37], and the Ha¨nggi-Klimontovich (HK) post-point rule [38][39][40][41]. As a main result it was found that only the HK interpretation of the Langevin equation yields a FPE, whose stationary solution coincides with the one-dimensional relativistic Ju¨ttner-Maxwell distribution, as known from Ju¨ttner's early work on the relativistic gas [42,43] and also from the relativistic kinetic theory [23,24]. Thus, in absence of other qualifying criteria, one may conclude that the post-point discretization rule is favorable.…”

From conduction to convection of thermally relativistic fluids between two parallel walls under gravitational force

“…In particular, the temperature jumps on two walls increase in accordance with the decrease in the temperature, because the representative relaxation rate [10] decreases in accordance with the decrease in the temperature, when Knudsen number is fixed. As a result, we cannot use the Rayleigh number as a parameter, which characterizes the transition of the thermal conduction to the thermal convection, under the transition regime between the rarefied and continuum regimes.…”

“…Provided that thermally relativistic fluids are composed of hard spherical partons with mass 7 m and diameter d, we obtain Ra using definitions of η and λ for hard spherical partons, which were calculated by Cercignani and Kremer on the basis of Israel-Stewart theory [10], as Ra = 256 45…”

“…5 where η is the viscosity coefficient, ζ is the bulk viscosity, and λ is the thermal conductivity, which are calculated for hard spherical partons by Cercignani and Kremer [10] on the basis of Israel-Stewart theory.…”