# Boltzmann equation solver adapted to emergent chemical non-equilibrium

## Abstract

We present a novel method to solve the spatially homogeneous and isotropic relativistic Boltzmann equation. We employ a basis set of orthogonal polynomials dynamically adapted to allow for emergence of chemical non-equilibrium. Two time dependent parameters characterize the set of orthogonal polynomials, the effective temperature T(t) and phase space occupation factor ϒ(t). In this first paper we address (effectively) massless fermions and derive dynamical equations for T(t) and ϒ(t) such that the zeroth order term of the basis alone captures the particle number density and energy density of each particle distribution. We validate our method and illustrate the reduced computational cost and the ability to easily represent final state chemical non-equilibrium by studying a model problem that is motivated by the physics of the neutrino freeze-out processes in the early Universe, where the essential physical characteristics include reheating from another disappearing particle component (e{sup ±}-annihilation)

- Authors:

- Program in Applied Mathematics, The University of Arizona, Tucson, AZ, 85721 (United States)
- (United States)
- Department of Mathematics and Lawrence Berkeley National Laboratory, University of California, Berkeley, CA 94721 (United States)
- Department of Physics, The University of Arizona, Tucson, AZ, 85721 (United States)

- Publication Date:

- OSTI Identifier:
- 22382173

- Resource Type:
- Journal Article

- Resource Relation:
- Journal Name: Journal of Computational Physics; Journal Volume: 281; Other Information: Copyright (c) 2014 Elsevier Science B.V., Amsterdam, The Netherlands, All rights reserved.; Country of input: International Atomic Energy Agency (IAEA)

- Country of Publication:
- United States

- Language:
- English

- Subject:
- 71 CLASSICAL AND QUANTUM MECHANICS, GENERAL PHYSICS; 72 PHYSICS OF ELEMENTARY PARTICLES AND FIELDS; ANNIHILATION; BOLTZMANN EQUATION; CAPTURE; ELECTRON-POSITRON INTERACTIONS; ENERGY DENSITY; EQUILIBRIUM; FREEZING OUT; NEUTRINOS; PHASE SPACE; POLYNOMIALS; RELATIVISTIC RANGE; TIME DEPENDENCE; UNIVERSE

### Citation Formats

```
Birrell, Jeremiah, E-mail: jeremey.birrell@gmail.com, Department of Physics, The University of Arizona, Tucson, AZ, 85721, Wilkening, Jon, and Rafelski, Johann.
```*Boltzmann equation solver adapted to emergent chemical non-equilibrium*. United States: N. p., 2015.
Web. doi:10.1016/J.JCP.2014.10.056.

```
Birrell, Jeremiah, E-mail: jeremey.birrell@gmail.com, Department of Physics, The University of Arizona, Tucson, AZ, 85721, Wilkening, Jon, & Rafelski, Johann.
```*Boltzmann equation solver adapted to emergent chemical non-equilibrium*. United States. doi:10.1016/J.JCP.2014.10.056.

```
Birrell, Jeremiah, E-mail: jeremey.birrell@gmail.com, Department of Physics, The University of Arizona, Tucson, AZ, 85721, Wilkening, Jon, and Rafelski, Johann. Thu .
"Boltzmann equation solver adapted to emergent chemical non-equilibrium". United States.
doi:10.1016/J.JCP.2014.10.056.
```

```
@article{osti_22382173,
```

title = {Boltzmann equation solver adapted to emergent chemical non-equilibrium},

author = {Birrell, Jeremiah, E-mail: jeremey.birrell@gmail.com and Department of Physics, The University of Arizona, Tucson, AZ, 85721 and Wilkening, Jon and Rafelski, Johann},

abstractNote = {We present a novel method to solve the spatially homogeneous and isotropic relativistic Boltzmann equation. We employ a basis set of orthogonal polynomials dynamically adapted to allow for emergence of chemical non-equilibrium. Two time dependent parameters characterize the set of orthogonal polynomials, the effective temperature T(t) and phase space occupation factor ϒ(t). In this first paper we address (effectively) massless fermions and derive dynamical equations for T(t) and ϒ(t) such that the zeroth order term of the basis alone captures the particle number density and energy density of each particle distribution. We validate our method and illustrate the reduced computational cost and the ability to easily represent final state chemical non-equilibrium by studying a model problem that is motivated by the physics of the neutrino freeze-out processes in the early Universe, where the essential physical characteristics include reheating from another disappearing particle component (e{sup ±}-annihilation)},

doi = {10.1016/J.JCP.2014.10.056},

journal = {Journal of Computational Physics},

number = ,

volume = 281,

place = {United States},

year = {Thu Jan 15 00:00:00 EST 2015},

month = {Thu Jan 15 00:00:00 EST 2015}

}