# A Langevin approach to multi-scale modeling

## Abstract

In plasmas, distribution functions often demonstrate long anisotropic tails or otherwise significant deviations from local Maxwellians. The tails, especially if they are pulled out from the bulk, pose a serious challenge for numerical simulations as resolving both the bulk and the tail on the same mesh is often challenging. A multi-scale approach, providing evolution equations for the bulk and the tail individually, could offer a resolution in the sense that both populations could be treated on separate meshes or different reduction techniques applied to the bulk and the tail population. In this paper, we propose a multi-scale method which allows us to split a distribution function into a bulk and a tail so that both populations remain genuine, non-negative distribution functions and may carry density, momentum, and energy. The proposed method is based on the observation that the motion of an individual test particle in a plasma obeys a stochastic differential equation, also referred to as a Langevin equation. Finally, this allows us to define transition probabilities between the bulk and the tail and to provide evolution equations for both populations separately.

- Authors:

- Princeton Plasma Physics Lab. (PPPL), Princeton, NJ (United States)

- Publication Date:

- Research Org.:
- Princeton Plasma Physics Lab. (PPPL), Princeton, NJ (United States)

- Sponsoring Org.:
- USDOE Office of Science (SC), Fusion Energy Sciences (FES) (SC-24)

- OSTI Identifier:
- 1437600

- Alternate Identifier(s):
- OSTI ID: 1433047

- Grant/Contract Number:
- AC02-09CH11466; SC0016268

- Resource Type:
- Accepted Manuscript

- Journal Name:
- Physics of Plasmas

- Additional Journal Information:
- Journal Volume: 25; Journal Issue: 4; Journal ID: ISSN 1070-664X

- Publisher:
- American Institute of Physics (AIP)

- Country of Publication:
- United States

- Language:
- English

- Subject:
- 70 PLASMA PHYSICS AND FUSION TECHNOLOGY; plasma collisions; stochastic processes; tokamaks; plasma physics; Markov processes; multiscale methods

### Citation Formats

```
Hirvijoki, Eero. A Langevin approach to multi-scale modeling. United States: N. p., 2018.
Web. doi:10.1063/1.5025716.
```

```
Hirvijoki, Eero. A Langevin approach to multi-scale modeling. United States. doi:10.1063/1.5025716.
```

```
Hirvijoki, Eero. Fri .
"A Langevin approach to multi-scale modeling". United States. doi:10.1063/1.5025716. https://www.osti.gov/servlets/purl/1437600.
```

```
@article{osti_1437600,
```

title = {A Langevin approach to multi-scale modeling},

author = {Hirvijoki, Eero},

abstractNote = {In plasmas, distribution functions often demonstrate long anisotropic tails or otherwise significant deviations from local Maxwellians. The tails, especially if they are pulled out from the bulk, pose a serious challenge for numerical simulations as resolving both the bulk and the tail on the same mesh is often challenging. A multi-scale approach, providing evolution equations for the bulk and the tail individually, could offer a resolution in the sense that both populations could be treated on separate meshes or different reduction techniques applied to the bulk and the tail population. In this paper, we propose a multi-scale method which allows us to split a distribution function into a bulk and a tail so that both populations remain genuine, non-negative distribution functions and may carry density, momentum, and energy. The proposed method is based on the observation that the motion of an individual test particle in a plasma obeys a stochastic differential equation, also referred to as a Langevin equation. Finally, this allows us to define transition probabilities between the bulk and the tail and to provide evolution equations for both populations separately.},

doi = {10.1063/1.5025716},

journal = {Physics of Plasmas},

number = 4,

volume = 25,

place = {United States},

year = {2018},

month = {4}

}

*Citation information provided by*

Web of Science

Web of Science

#### Figures / Tables:

*M*= 10 and different

*N*. Notice the steepening of Φ with increasing

*N*which reflects that the particles become less collisional at higher energies.

Works referenced in this record:

##
A backward Monte-Carlo method for time-dependent runaway electron simulations

journal, September 2017

- Zhang, Guannan; del-Castillo-Negrete, Diego
- Physics of Plasmas, Vol. 24, Issue 9

##
Random Generation of Stochastic Area Integrals

journal, August 1994

- Gaines, J. G.; Lyons, T. J.
- SIAM Journal on Applied Mathematics, Vol. 54, Issue 4

##
Itô versus Stratonovich

journal, January 1981

- van Kampen, N. G.
- Journal of Statistical Physics, Vol. 24, Issue 1

##
A backward Monte Carlo approach to exotic option pricing

journal, April 2017

- Bormetti, G.; Callegaro, G.; Livieri, G.
- European Journal of Applied Mathematics, Vol. 29, Issue 1

##
Current in wave-driven plasmas

journal, January 1986

- Karney, Charles F. F.; Fisch, Nathaniel J.
- Physics of Fluids, Vol. 29, Issue 1

##
On the kinetic theory of rarefied gases

journal, December 1949

- Grad, Harold
- Communications on Pure and Applied Mathematics, Vol. 2, Issue 4

##
Higher-order time integration of Coulomb collisions in a plasma using Langevin equations

journal, June 2013

- Dimits, A. M.; Cohen, B. I.; Caflisch, R. E.
- Journal of Computational Physics, Vol. 242

##
Collisional delta- *f* scheme with evolving background for transport time scale simulations

journal, December 1999

- Brunner, S.; Valeo, E.; Krommes, J. A.
- Physics of Plasmas, Vol. 6, Issue 12

##
Adjoint Fokker-Planck equation and runaway electron dynamics

journal, January 2016

- Liu, Chang; Brennan, Dylan P.; Bhattacharjee, Amitava
- Physics of Plasmas, Vol. 23, Issue 1

##
On the simulation of iterated Itô integrals

journal, January 2001

- Rydén, Tobias; Wiktorsson, Magnus
- Stochastic Processes and their Applications, Vol. 91, Issue 1

##
Fokker–Planck kinetic modeling of suprathermal α -particles in a fusion plasma

journal, December 2014

- Peigney, B. E.; Larroche, O.; Tikhonchuk, V.
- Journal of Computational Physics, Vol. 278

*Figures/Tables have been extracted from DOE-funded journal article accepted manuscripts.*