Loading of Relativistic Maxwellian-type Distributions Revisited

Abstract

A simple numerical method for loading of a relativistic Maxwellian-type distribution is proposed based on inverse transform sampling. The relativistic Maxwellian energy distribution is introduced as an alternative to the Maxwell-Jüttner distribution. The cumulative distribution of the shifted-Maxwellian energy distribution is approximated by an invertible function. Random variates of energy is transformed from uniformly distributed random variables. Then, the energy variates are converted to momentum vector variates. Numerical tests are presented to show that the present method successfully reproduce the relativistic Maxwellian energy distribution.

Figures (5)

• Figure 1: Comparison between the cumulative distribution $F_{\cal E}$ in Eq. (\ref{['eq:rela-d-maxwellian-e-cumu']}) and its approximation $F_{\rm app}$ in Eq. (\ref{['eq:rela-d-maxwellian-e-app']}). (a) The profiles of $F_{\cal E}$ and $F_{\rm app}$ for $0 < x \le 8$. The thick line shows $F_{\cal E}$ and the dashed line shows $F_{\rm app}$. (b) The relative error $\eta$ between $F_{\cal E}$ and $F_{\rm app}$.
• Figure 2: Comparison between the Maxwellian energy distribution in Eq. (\ref{['eq:rela-d-maxwellian-e']}) and a histogram of numerically generated random variates. The dashed lines show the profile of Eq. (\ref{['eq:rela-d-maxwellian-e']}) as a function of the normalized energy ${\cal E}=mc^2(\gamma_B-1)/(\gamma_DT)$ in (a) a linear scale and (b) a logarithmic scale. The bars show a histogram of random variates generated by using Eq. (\ref{['eq:rela-d-maxwellian-e-rand']}) with $N_{\rm s}=10^8$ samples and $N_{\rm bin}=10^4$ bins.
• Figure 3: Comparison between the momentum/velocity distribution based on the Maxwellian energy distribution and a histogram of numerically generated random variates. The dashed lines in the left panels show the reduced momentum distribution in the $u_x$ direction with $v_D/c=0.9$ and $mc^2/T=6.25$ in (a) a linear scale and (b) a logarithmic scale. The dashed lines in the right panels show the corresponding reduced velocity distribution in the $v_x$ direction in (c) a linear scale and (d) a logarithmic scale. The bars show a histogram of random variates generated by using Eqs. (\ref{['eq:rela-d-maxwellian-rand']}) and (\ref{['eq:summary']}) with $N_{\rm s}=10^8$ samples and $N_{\rm bin}=10^4$ bins.
• Figure 4: Comparison between the Maxwellian energy distribution in Eq. (\ref{['eq:rela-d-maxwellian-e']}) and the Maxwell-Jüttner distribution in Eq. (\ref{['eq:juttner-d-e']}) with different $mc^2/T$ in (a) a linear scale and (b) a logarithmic scale. The solid line shows the profile of Eq. (\ref{['eq:rela-d-maxwellian-e']}) with $\gamma_D=1$. The circles show the profile of Eq. (\ref{['eq:juttner-d-e']}) with $mc^2/T=100$. The squares show the profile of Eq. (\ref{['eq:juttner-d-e']}) with $mc^2/T=1$. The triangles show the profile of Eq. (\ref{['eq:juttner-d-e']}) with $mc^2/T=0.01$.
• Figure 5: Comparison between tbetween the relativistic Maxwellian energy distribution in Eq. (\ref{['eq:rela-d-maxwellian']}) and the Maxwell-Jüttner distribution in Eq. (\ref{['eq:juttner-d']}) with $v_D/c=0.5$ and $mc^2/T=1$. Left panels show the reduced momentum distribution in the $u_x$ space in (a) a linear scale and (b) a logarithmic scale. Right panels show the reduced velocity distribution in the $v_x$ space in (c) a linear scale and (d) a logarithmic scale. The solid line shows the profile of the reduced Maxwellian energy distribution. The squares show the profile of the reduced Maxwell-Jüttner distribution.