# Sampling Speeds from a Relativistic Maxwell Distribution

## Abstract

Monte Carlo calculations in astrophysics or other fields of physics involving hot ionized gases may require sarmpling from a relativistic Maxwellian distribution. Winslow ^{4} described a previously unpublished technique attributed to Von Neumann for such sampling. The technique, based on an approxir.,ation to the relativistic equation, seems adequate 2 for temperatures up to about 1/4 the rest energy (i.e., kT$$\sim\atop{<}$$ $$\frac{mc^2}{4}$$) except for inherent inaccuracies in the high energy tail. For much higher temperatures, the approximation is clearly in error. It is the purpose of this paper to give a method for sampling from a relativistic Maxwellian distribution which is mathematically exact and valid for all temperatures. Most (or all) the mathematics involved is scattered through the literature on probability theory. But since many of the arguments are perhaps new to the prospective Monte Carlo user, we have included, for the sake of completeness, a good deal of mathematical rigor in the report.

- Authors:

- Lawrence Livermore National Lab. (LLNL), Livermore, CA (United States)

- Publication Date:

- Research Org.:
- Lawrence Livermore National Lab. (LLNL), Livermore, CA (United States)

- Sponsoring Org.:
- USDOE National Nuclear Security Administration (NNSA)

- OSTI Identifier:
- 1466131

- Report Number(s):
- LLNL-TR-756170

932665

- DOE Contract Number:
- AC52-07NA27344

- Resource Type:
- Technical Report

- Country of Publication:
- United States

- Language:
- English

- Subject:
- 97 MATHEMATICS AND COMPUTING

### Citation Formats

```
Canfield, Eugene, and Barnett, Charles.
```*Sampling Speeds from a Relativistic Maxwell Distribution*. United States: N. p., 2018.
Web. doi:10.2172/1466131.

```
Canfield, Eugene, & Barnett, Charles.
```*Sampling Speeds from a Relativistic Maxwell Distribution*. United States. doi:10.2172/1466131.

```
Canfield, Eugene, and Barnett, Charles. Fri .
"Sampling Speeds from a Relativistic Maxwell Distribution". United States.
doi:10.2172/1466131. https://www.osti.gov/servlets/purl/1466131.
```

```
@article{osti_1466131,
```

title = {Sampling Speeds from a Relativistic Maxwell Distribution},

author = {Canfield, Eugene and Barnett, Charles},

abstractNote = {Monte Carlo calculations in astrophysics or other fields of physics involving hot ionized gases may require sarmpling from a relativistic Maxwellian distribution. Winslow4 described a previously unpublished technique attributed to Von Neumann for such sampling. The technique, based on an approxir.,ation to the relativistic equation, seems adequate 2 for temperatures up to about 1/4 the rest energy (i.e., kT$\sim\atop{<}$ $\frac{mc^2}{4}$) except for inherent inaccuracies in the high energy tail. For much higher temperatures, the approximation is clearly in error. It is the purpose of this paper to give a method for sampling from a relativistic Maxwellian distribution which is mathematically exact and valid for all temperatures. Most (or all) the mathematics involved is scattered through the literature on probability theory. But since many of the arguments are perhaps new to the prospective Monte Carlo user, we have included, for the sake of completeness, a good deal of mathematical rigor in the report.},

doi = {10.2172/1466131},

journal = {},

number = ,

volume = ,

place = {United States},

year = {Fri Mar 09 00:00:00 EST 2018},

month = {Fri Mar 09 00:00:00 EST 2018}

}