Loading various velocity distributions inparticle-in-cell (PIC) simulation

Seiji ZENITANIIWF, Austrian Academy of Sciences, AUSTRIA
RCUSS, Kobe University, JAPANShin’ya Nakano
Institute of Statistical Mathematics, JAPANParticle-in-cell (PIC) simulation• It is very important to randomly preparevelocity distributions of particles in a cellCounts

0Velocity Vx

0. Maxwell distribution• Normal distribution<latexit sha1_base64="p8GmFKZscKg2cwGehqXHWMV4g/A=">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</latexit>• Box=Muller (1958) method• Two random variates: U1, U2 ⊂ (0,1)

n1n2pp2 ln U1 cos 2⇡U2
2 ln U1 sin 2⇡U2

How to initialize various velocity distributions(Kappa, loss-cone, relativistic...)in particle-in-cell (PIC) simulation?

0. Maxwell distribution• Normal distribution<latexit sha1_base64="p8GmFKZscKg2cwGehqXHWMV4g/A=">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</latexit>• Box=Muller (1958) method• Two random variates: U1, U2 ⊂ [0,1]n1n2pp2 ln U1 cos 2⇡U2
2 ln U1 sin 2⇡U2

• Spherical form (d3v → 4πv2dv)
Gamma distribution

Gamma function

1. Kappa distribution• Thermal core + power-law tail• Popular in space physics(Vasyliunas 1968, Olbert 1968)• κ : power-law index• κ → ∞ MaxwellianKappa distribution• Spherical form

• Generalized Beta-prime distribution

Gamma random number• Beta-prime random number

Gamma random number(Kappa distribution)(Normal distribution)qXGa( 1/2,2)<latexit sha1_base64="6mpr7KgzEzH2w7tzjX9uaDpjsQE=">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</latexit>Kappa distribution - Recipe

See also Abdul & Mace 2015

2. Loss-cone distribution (Pitch-angle type)<latexit sha1_base64="jSGhFNpGfFmYgETp02GMcKB069s=">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</latexit>,/⇣ v ⌘2j?

v

c.f. Kennel 1966
(from Katoh-san)Utilizing Beta distributionv='<latexit sha1_base64="ZsS8jInyJjSSWKVdJYS9gWqrmHM=">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</latexit>

Beta function

Beta distribution• One can transform isotropic distributions to loss-cone
distributions via Beta random variate.
• Beta variate can be generated from two gamma variates

Loss-cone distribution - Recipe

Loss-conetransform(Beta variate)3. Relativistic Maxwell distribution(Maxwell=Jüttner distribution)

Modified Bessel functionof the second kindLorentz factor= [1(v/c)2 ] 1/2• Lorentz factor makes
the distribution complex<latexit sha1_base64="IKAIAZzusjD1uREWupf9G3wBFD0=">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</latexit>

px = mvx

Modified Canfield method[Canfield+ 1987, Zenitani & Nakano 2022]• Spherical form, as a function of the energy

Probability

Gamma distribution

Rejection function (0.95 < R < 1.0)

Modified Canfield method - Recipe

Probabilistic switch

Gamma distribution
Rejection• Gamma-distributed variates
are necessary• Rejection : acceptanceefficiency is 95.7-100%.

Velocity distributions
Maxwell
distribution

Relativistic
Maxwell distribution

Box=Muller 1958

Sobol 1976, SZ & Nakano 2022

Kappa
distribution

Abdul & Mace 2015

Loss-cone
distributions
SZ & Nakano 2023

Velocity distributionsMaxwell
distribution

Relativistic
Maxwell distribution

Box=Muller 1958

Sobol 1976, SZ & Nakano 2022

Kappa
distribution

Abdul & Mace 2015

Relativistic
kappa distribution
SZ & Nakano 2022

Loss-cone
distributions
SZ & Nakano 2023

Kappa loss-cone(KLC) distributions

Velocity distributions
Maxwell
distribution

Relativistic
Maxwell distribution

Box=Muller 1958

Sobol 1976, SZ & Nakano 2022

Kappa
distribution

Relativistic
kappa distribution

Abdul & Mace 2015

SZ & Nakano 2022

Loss-cone
distributions
SZ & Nakano 2023

Relativistic losscone distribution

Kappa loss-cone(KLC) distributions

Relativistic kappa loss-cone(KLC) distribution

Various distributions
Relativistic
MaxwellMaxwellKappa

Relativistickappa• Subtracted Maxwellian• Dory• Summers KLC

Loss-cone (LC)distributions• Flattop

Kappa loss-cone(KLC) distributions

Relativistic KLC• Numerical recipes require• Uniform random number• Normal random number• Gamma random number• Flattop losscone

Subtracted Maxwellian[Ashour-Abdalla & Kennel, 1978]

Loss-cone filling factor Δ

Shape factor β*Flattop distribution[Zenitani 2024 RNAAS]

Various distributions
RelativisticMaxwellMaxwell
Kappa

Relativistickappa• Subtracted Maxwellian• Dory• Summers KLC

Loss-cone (LC)distributions• Flattop

Kappa loss-cone(KLC) distributions

Relativistic KLC• Numerical recipes require• Uniform random number• Normal random number• Gamma random number• Flattop losscone

Gamma-distributed random number
α<1

1<α

Best (1983) method isbelieved to be the bestGamma-distributed random number
α<1

1<α

Our proposals

Best (1983) method isbelieved to be the bestNew gamma generator[Zenitani 2024, under review]** Protected **Summary• 1. Kappa distribution• Normal distribution, divided by Gamma distribution

These three
can be combined+Five more• 2. Loss-cone distribution• Beta distribution to scatter• 3. Relativistic distribution• Superpositions of four gamma distributions• 4. Gamma distribution• New generator for shape parameter less than unity• References• Zenitani & Nakano, Phys. Plasmas, 2022
• Zenitani & Nakano, JGR Space Physics, 2023
• Zenitani, Economics Bulletin, under review

Thank you for your attention!