SH53B-2507

Loading Relativistic Maxwell
Distributions in Particle Simulations

Seiji ZENITANI
National Astronomical Observatory of Japan
Acknowledgement: T.-N. Kato (NAOJ)

Zenitani, Phys. Plasmas 22, 042116 (2015)
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Kinetic modeling of relativistic plasma processes
Relativistic shock (90’s)

Gallant et al. 1992 Astrophys. J.

Relativistic reconnection (00’s)

Zenitani & Hoshino 2005 Phys. Rev. Lett.

Relativistic flow shear (10’s)

Alves et al. 2012 Astrophys. J.

2

Three technical barriers
in relativistic PIC simulations• 1. Setup– Loading velocity distribution functions by using random variables– Usually undocumented• 2. Computation– Electromagnetic field (Haber 1974, Vay+ 2011)– Particle (Vay 2008)• 3. Diagnosis & Interpretation– Presented in ASJ 2015a meeting (SZ 2015d in prep.)3

Nonrelativistic Maxwell distributions
ff (v)

exp

v2
2
vth0

V0vx

• Uniform variable• Galilei transformvx
vx + V0
Xn (0 < Xn < 1)
• Box=Muller (1958) transformvx =2 ln X1 sin(2 X2 )vy =2 ln X1 cos(2 X2 )4

Relativistic Maxwell distributions• Jüttner=Synge distribution
10.1

f (u)d u
3T=mc20.010.001

mc2 3d uT
mc2 1 + (u/c)2 3d u
Texp
exp

0.00010.00001
1x10-6
1x10-7-20020

u= vux= [1• Shifted Maxwell distribution
exp
10.1
0.01
0.001
0.00010.00001
1x10-6
1x10-7(v/c)2 ] 1/2(ux )Td3 u

Γ=1 (analytic)
Γ=1.1 (analytic)
Γ=10 (analytic)

Γ=10050

100

150

200

250

3005Recent attempts• Swisdak (2013) -- Two-step rejection method (Shown on p.15)• Melzani+ (2013) -- Cylindrical transformation + numerical table

Our strategy• Stationary Maxwellian– Inverse transform method– Sobol (1976) method• Lorentz boost → relativistic shifted Maxwellian– Rejection method - 50% efficiency for the stationary case
– Flipping method- 100% efficiency
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Sobol method[Sobol 1976, Pozdnyakov+ 1977, 1983]• Spherical form

m=1, c=1...

• Gamma (Earlang) distribution
P (u; 3, T )

u 2uT

exp

u=

• Rejection method
expu

1 + u2
T> X4• Spherical scatteringux = u (2X5T ln X1 X2 X3• Another form
[Pozdnyakov+ 1977]

u2 > 1

2=T ln X1 X2 X3 X41)

uy = 2u X5 (1X5 ) cos(2 X6 )uz = 2u X5 (1X5 ) sin(2 X6 )7Loading efficiency• Efficiency quickly decreases for T << 1– We should revert to alternative methods (SZ 2015)

1
0.8

Eff.

0.6

Eq. (13)

0.4
0.2
00

1

2

3

45T

8

Quest for the original article• OCLC/WorldCat database suggested 5 libraries• We found it at U. Illinois Urbana-Champaignat the 4th attempt☹

☺

?

☹

☹

9Sobol (1976)’s article?
• He did it right 40 years ago...

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Lorentz transformation• Energy-momentum– Straightforwarduyuyuzuz• Space-time– Contraction of a volume element

Volume transformfactor≠• Undocumented workarounds– 1. Do nothing (incorrect result)– 2. Analytic formula (CfCA, Colorado)– 3. Adjust particle density (SZ, 4D2U)– 4. Adjust particle weight (Harvard)

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Volume transform - Flipping method
Fluid rest frame (S)0Observer frame (S’)

0Acceptance factor

Volume transform factor(rescaled by 1/Γ)

Recycle100• One can adjust the density by recycling the grey partin arbitrary Vx(Ux)-symmetric distributions
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Numerical test
1Without the volume transformation

1e-01
1e-02
Γ=1
Γ=1.1
Γ=10
Γ=10*

1e-03
1e-04
1e-05
-20-10

0

10

20

30

4050

• Excellent agreement between numerical results vs analytic curve• Volume transform factor corrects energy flux by ~25%
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Summary• Stationary Maxwellian– Inverse transform method– Sobol (1976) method• Lorentz boost– Rejection method– Flipping method• Let’s get started inrelativistic PIC simulations• More detail:– Zenitani, Phys. Plasmas 22, 042116 (2015)14

Pseudo codes
Sobol=Zenitani (2015)

Swisdak’s two-step method (2013)

Zenitani 2015

Nonrelativistic:
Box=Muller (1958)

Swisdak 201315