銭谷誠司 京都大学生存圏研究所

相対論的高速流プラズマの数値実験技法(PICシミュレーション)

Loading Relativistic Maxwell Distributions in Particle SimulationsZenitani, Phys. Plasmas 22, 042116 (2015)

9P56

Dissipation in relativistic pair-plasma reconnection: revisitedZenitani, Plasma Phys. Control. Fusion 60, 014028 (2018)

Three technical barriers in relativistic PIC simulations

• 1. Setup – Loading velocity distribution functions by using random variables

– Not clearly documented ==> Zenitani 2015

• 2. Computation – Electromagnetic field (Haber 1974, Vay+ 2011)

– Particle (Vay 2008)

• 3. Diagnosis & Interpretation – Definition of “fluid” frame and further decomposition

– Not clearly documented ==> Zenitani 2018

• Maxwell=Jüttner distribution

• Shifted Maxwell distribution

-20 0 201x10-71x10-6

0.000010.00010.0010.010.11

Loading relativistic particles

ux

T=mc2

0 50 100 150 200 250 3001x10-71x10-6

0.000010.0001

0.0010.01

0.11

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

Γ=10

� exp�� �(�� � �u�

x)T

�d3u�

f(u)d3u � exp�� �mc2

T

�d3u

0 ux =γvx

N

ux

log N

log N

T=mc2

Recent attempts

• 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

Our strategy

•Swisdak (2013) -- Two-step rejection method

•Melzani+ (2013) -- Cylindrical transformation + numerical table

Modified Sobol algorithm [Sobol 1976, Pozdnyakov+ 1983, Zenitani 2015]

• Sobol method – Stationary Maxwellian

– Reject some particles from 3rd-order Gamma distribution

• SZ’s addition – Adjust particle density

for Volume transform

– Without this, we will see a big error (33%) in the energy flow

≠

0 0

Observer frame (S’)Fluid rest frame (S)

Recycle

0

1

Acceptance factor

0

•OCLC/WorldCat database suggested 5 libraries

•We found it at U. Illinois Urbana-Champaignat the 4th attempt

Quest for the original article

☹☹

☹?

☺

Sobol (1976)’s article

• He did it right 40 years ago...

?

Numerical test

-20 -10 0 10 20 30 40 501e-05

1e-04

1e-03

1e-02

1e-01

1

Γ=1Γ=1.1Γ=10Γ=10*

Without the volume transformation

• Excellent agreement between numerical results vs analytic curve

• Volume transform factor corrects energy flux by ~33%

Ohm’s law in a kinetic plasma

-2 -1 0 1 2-0.05

0

0.05

0.1

0.15

Composition of Electric field

z

– 1 –

Ey (1)

(�ve ⇥B)y (2)

�(1/qne)(r · !P e)y (3)

�(me/q)(ve ·r)vey (4)

�(me/q)(@/@t)vey (5)Zenitani+ 2011 PRL

2D particle-in-cell (PIC) simulation

Bulk inertiaThermal inertia (Local momentum transport)

Hesse+ 1999, 2011

• Which term (& what physics) violates the ideal condition?

Ohm’s law in a relativistic kinetic plasma• Stress-energy tensor

• Eckart (1940) decomposition • See also Mihalas & Mihalas (1999)

• Energy momentum equation for relativistic plasmas

• Relativistic Ohm’s law (with ∂t=0)

Heat flow inertia (new in relativistic regime)

Thermal inertia (Local momentum transport)Bulk inertia

(including relativistic pressure)

projection operator

heat flow

heat flow tensor

rest-frame energy

pressure tensor

γ = 100

v ~ +c

γ = 200

v ~ -c γ = 10

v ~ +c

+X

What is the relativistic bulk velocity?

Energy flow Number flow

Eckart frameN i = 0

Landau frameW i0 = W 0j = 0

We set the bulk velocity to the Eckart velocity

2D Particle-in-Cell simulation

• Relativistic electron-positron plasma

• T/mc2=1, nbg/n0=0.1, vdrift/c=0.3

• 109.5 particles: 104 pairs in a cell (⇔ 102 in typical works)

Density

Vpx

z

z

x

20 40 60 80

-0.05

0

0.05

0.1

0.15

0.2

2D Particle-in-Cell simulation

• Normalized energy dissipation (~ j.E/n ~ ηj2/n)

• Composition of Ohm’s law: heat flow term appears

De/ne

z

x

Ey

B0

zHeat flow = energy flow = mass flow

Momentum transport• Strong particle acceleration gives rise to Qyz

• Scale hight: Q is more confined than P

z

x

z

x

(�v)z

(�v)y

z

B

Summary

• Part 1: we have proposed basic algorithms to load relativistic velocity distributions – We reintroduced Sobol (1976) method.

– We developed a volume-transform method.

•Part 2: we have analyzed kinetic Ohm’s law in relativistic magnetic reconnection – We can evaluate relativistic fluid properties from the

stress-energy tensor by using Eckart decomposition.

– In addition to thermal inertia, new dissipation term (heat flow inertia) appears

– They are related to energetic particles.