Loading a relativistic kappa distribution in particle simulations Seiji Zenitani1and Shinya Nakano2 1Research Center for Urban Safety and Security Kobe University 1-1 Rokkodai-cho

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Loading a relativistic kappa distribution in particle simulations

Seiji Zenitani1and Shin’ya Nakano2

1)Research Center for Urban Safety and Security, Kobe University, 1-1 Rokkodai-cho,

Nada-ku, Kobe 657-8501, Japan.a)

2)The Institute of Statistical Mathematics, 10-3 Midori-cho, Tachikawa,

Tokyo 190-8562, Japan

(Dated: Submitted to Physics of Plasmas)

A procedure for loading particle velocities from a relativistic kappa distribution in

particle-in-cell (PIC) and Monte Carlo simulations is presented. It is based on the

rejection method and the beta prime distribution. The rejection part extends earlier

method for the Maxwell–J¨uttner distribution, and then the acceptance rate reaches

&95%. Utilizing the generalized beta prime distributions, we successfully reproduce

the relativistic kappa distribution, including the power-law tail. The derivation of

the procedure, mathematical preparations, comparison with other procedures, and

numerical tests are presented.

a)Electronic mail: zenitani@port.kobe-u.ac.jp

arXiv:2210.15118v1 [physics.plasm-ph] 27 Oct 2022

I. INTRODUCTION

The kappa distribution is one of the most fundamental velocity distributions in kinetic

studies in space and solar-wind plasmas.11 Since it was introduced in 1960s,16,23 the kappa

distribution has been widely used to study plasmas with suprathermal populations, because

it seamlessly contains both the thermal Maxwellian component and the nonthermal power-

law component in the high-energy part. It is also noteworthy that the kappa distribution is

connected with Tsallis statistical mechanics, as it maximizes a non-extensive entropy.11,12,22

For the theory and applications of the kappa distribution, the readers may refer to a re-

cent monograph11 and references therein. Kinetic plasma processes in a kappa-distributed

space plasma have been investigated by using particle-in-cell (PIC) simulations.1,2,9,13,17 To

simulate a kappa-distributed plasma, one often has to initialize particle velocities according

to the kappa distribution, however, numerical procedures to load kappa distributions may

not be well-documented. Among them, Abdul & Mace 1recognized that the kappa distribu-

tion is equivalent to the multivariate t-distribution, which can be generated from a normal

distribution and a chi-squared distribution.10

To deal with energetic electrons of &0.5 MeV and energetic ions of &1 GeV, special

relativity needs to be considered. Velocity distributions need to be modiﬁed accordingly. A

relativistic Maxwell distribution, often referred to as a Maxwell–J¨uttner distribution,8has

been used in Monte Carlo simulations as well as in PIC simulations in high-energy astro-

physics. It is not straightforward to generate a Maxwell–J¨uttner distribution, because the

relativistic Lorentz factor γ= [1 −(v/c)2]1/2makes the problem diﬃcult. To load Maxwell–

J¨uttner distributions, several rejection-based algorithms have been proposed.4,18–21,26 For

example, Sobol 20 has proposed a simple rejection method, based on the gamma distribu-

tion. Canﬁeld et al. 4have utilized four gamma distributions. Their method achieves good

acceptance rate of &70% for nearly arbitrary initial conditions. Another option is the in-

verse transform method, which refers to a numerical table of the cumulative distribution

function. In such a case, one needs to carefully adjust the size of the table, because the code

often becomes ineﬃcient.

A relativistic kappa distribution will be useful in modeling energetic processes in laser,

space, and astrophysical plasmas. Its mathematical form including the normalization con-

stant was provided by Xiao24 and by Han-Thanh et al.7. Tsallis-type statistics in relativis-

tic collisionless plasmas has gained attention very recently.27 Meanwhile, to the best of our

knowledge, no one has presented a numerical procedure to load a relativistic kappa distribu-

tion in particle simulations. One can similarly consider the inverse transform method, but a

much larger table will be required, because the kappa distribution has a power-law tail that

extends nearly inﬁnitely. We desire a reliable algorithm that is free from tables.

In this article, we propose a numerical algorithm to load a relativistic kappa distribution

in PIC and Monte Carlo simulations. The rest of this manuscript is organized as follows.

As starting points, Sections II and III discuss algorithms to generate Maxwell and kappa

distributions. Section IV presents our extention of Canﬁeld et al.4’s algorithm to generate

a Maxwell–J¨uttner distribution. Section V introduces a new procedure to generate a rela-

tivistic kappa distribution, based on beta prime distributions. Section VI presents numerical

tests of the proposed methods for relativistic distributions. The eﬃciency of the rejection

method are evaluated. Section VII contains discussions and summary.

II. MAXWELL DISTRIBUTION

A Maxwell distribution is a multivariate normal distribution.

fM(v)d3v=NM1

πv2

M3

2exp −v2

Md3v(1)

Here, NMis the number density for a Maxwellian plasma, vMis the most probable velocity,

and the other symbols have their standard meanings. In this case, plasma temperature is

given by TM= (1/2)mv2

M. Throughout the paper, we focus on isotropic distributions.

The three components of the Maxwellian can be obtained by

vx=σn1, vy=σn2, vz=σn3,(2)

where σ2= (1/2)v2

Mis the variance and n1, n2,and n3are the normal random variates. The

normal variates can be generated by the Box–Muller method3or other methods.5,10,25

We examine the Maxwellian from another angle for discussions in later sections. We move

to spherical coordinates via d3v= 4πv2dv. We use a normalized parameter

x≡v2

,(3)

and then we obtain

fM(x)dx =2NM

√πx1/2e−xdx =NMGa x;3

2,1dx (4)

where Γ(x) is the gamma function and

Ga(x;k, λ) = xk−1e−x/λ

Γ(k)λk(5)

is the gamma distribution with a shape parameter kand a scale parameter λ. There are

several procedures to generate the gamma distributions.5,10,14,25 We present some of them in

Appendix A. From a gamma-distributed random variate XGa(α,β), we recover the velocity

v=vMpXGa(3/2,1) =σpXGa(3/2,2).(6)

We can also rewrite this with TM,

v=r2TM

mXGa(3/2,1) =pXGa(3/2,2TM/m).(7)

Finally, we obtain the vvector, by randomly scattering vonto the spherical surface, using

two uniform random variates X1, X2∼U(0,1).











vx=v(2X1−1)

vy= 2vpX1(1 −X1) cos(2πX2)

vz= 2vpX1(1 −X1) sin(2πX2)

(8)

III. KAPPA DISTRIBUTION

A kappa distribution is deﬁned by

fκ(v)d3v=Nκ

(πκθ2)3/2

Γ(κ+ 1)

Γ(κ−1/2)1 + v2

κθ2−(κ+1)d3v(9)

where θis the most probable speed and κis the kappa parameter. The κparameter controls

a power-law index in the high-energy part. It ranges κ > 3/2, so that the eﬀective plasma

temperature T= [κ/(2κ−3)]mθ2remains ﬁnite. Other properties of the kappa distribution

are discussed in Livadiotis 11 and references therein. Assuming isotropy, we obtain

fκ(v)dv =Nκ

π1/2(κθ2)3/2

Γ(κ+ 1)

Γ(κ−1/2)1 + v2

κθ2−(κ+1)v2dv (10)

=NκB0v;3

2,ν

2,2,(κθ2)1/2dv. (11)

Here, ν= 2κ−1 and B0is the generalized beta prime distribution with a shape parameter

pand a scale parameter q,

B0(x;α, β, p, q) = p

qB(α, β)x

qαp−11 + x

qp−(α+β)

=pΓ(α+β)

qαpΓ(α)Γ(β)1 + x

qp−(α+β)

xαp−1.(12)

B(α, β) is the beta function. If we redeﬁne x0≡(x/q)p, with help from (dx0/x0) = p(dx/x),

then x0follows the (standard) beta prime distribution,

B0(x0;α, β) = B0(x0;α, β, 1,1) = Γ(α+β)

Γ(α)Γ(β)(1 + x0)−(α+β)(x0)α−1.(13)

A random variate according to the beta prime distribution, XB0(α,β), is generated by

XB0(α,β)=XGa(α,δ)

XGa(β,δ)

(14)

where XGa(α,δ)is a random variate that follows the gamma distribution Ga(α, δ).6Here

Eq. (14) is independent of choice of δ. Since the relation (Eq. (14)) is not well known, we

provide a brief proof in Appendix B. Then, considering the scaling factors, we obtain

XB0(α,β,p,q)=qXB0(α,β)1/p =qXGa(α,δ)

XGa(β,δ)1/p

(15)

Setting δ= 2, the kappa-distributed velocity vis given by

v=κθ2XGa(3/2,2)

XGa(ν/2,2) 1/2

(16)

Then we generate two gamma distributions. We note that the shape parameter ν/2 in the

denominator can be an integer, half-integer, or ﬂoating-point number, as long as it satisﬁes

ν/2 = κ−1/2>1. Gamma generators for arbitrary k > 1 can be found in Appendix

A and references therein5,10,14,25. After this, we spherically scatter vto the vx, vy, and vz

components by Eq. (8). The entire steps to generate the kappa distribution are presented

in Algorithm 1-1 in Table I.

In Section II, we have already seen that the gamma distribution with k= 3/2 and the

spherical scattering provide the Maxwell distribution (Eqs. (2), (6), and (8)). From Eqs. (16)

and (8), we similarly obtain the three components of the kappa distribution:

vx=√κθ2n1

pχ2

, vy=√κθ2n2

pχ2

, vz=√κθ2n3

pχ2

,(17)

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LoadingarelativistickappadistributioninparticlesimulationsSeijiZenitani1andShin'yaNakano21)ResearchCenterforUrbanSafetyandSecurity,KobeUniversity,1-1Rokkodai-cho,Nada-ku,Kobe657-8501,Japan.a)2)TheInstituteofStatisticalMathematics,10-3Midori-cho,Tachikawa,Tokyo190-8562,Japan(Dated:SubmittedtoPhysicso...

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Loading a relativistic kappa distribution in particle simulations Seiji Zenitani1and Shinya Nakano2 1Research Center for Urban Safety and Security Kobe University 1-1 Rokkodai-cho

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