Paper review: A grid-based coulomb collision model for PIC codes (Jones_JCP_1996)

1. Paper review: A grid-based coulomb collision model for PIC codes (Jones_JCP_1996) Kim Youngkuk 2012

2. 1 • Plasma as a fluid (hydrodynamic) − Velocity distribution near Maxwellian • Plasma as a particle (kinetic) − non-collision processes − non-Maxwellian distribution • Hybrid (fluid+particle) − Electrons are treated as a fluid (electron mass is 0) − Ions are considered as particle

3. 2 • Coulomb collision − semi-collisional − changing rapidly and continuously in time or space from a strongly collision state to a collisionless state (or vice versa) • Short range Coulomb collision involves the pairing of scattering particles in each computational cell to ensure that energy and momentum are conserved locally. − Standard approach − Somewhat arbitrary • Collision field method − more random manner − Collision field force is chosen so that energy and momentum are conserved − Collision rates reduce to the appropriate values

4. 3 • Local momentum and energy conservation are an essential parts. − fixed background ion + electron → only consider electron − fixed background ions + electron → can not conserve momentum and energy • Particle pairing − Resolve conservation problem • Forces between charged particles are divided into two parts − Long range force (∆x > electron debye length) − Short range force involving particles in the same cell • Collision field method − can be adopted to particle-in-cell code − involves the short-range collision Ref) Takizuka. J. Comput. Phys. 25, 205(1977)

5. Interspecies Collisions : dynamical friction : temperature equilibration Ref) O.Larroche, Phys. Fluid B. 5, 2816(1993) Ref) R.Berger, Phys. Fluid B. 3, 3(1991) 이 두 ref 에 나오는 collision frequency 에는 exp항이 없다. 그렇다고 해서 ref 에 나오는 collision frequency가 잘못된 것은 아니다.

6. Interspecies Collisions • particle velocity determines the force • The fluid quantities, are obtained by linear interpolation onto the grid • The collision field is evaluated on the grid • Next time step, collision force is included with the Lorentz force • Avoid , particle experiences more than one collision per time step Ref) P. W. Rambo. Phys. Plasmas. 2, 3130(1995) [그래프 b의 dot] 하얀 점은 몬테카를로 시뮬레이션, 검은 점은 fluid solution.

7. Interaspecies Collisions • Intraspeciese collision relax the velocity distribution to an isotropic Maxwellian • Collision field method(x) → Langevin equation A is a random, isotropic vector chosen to provided thermalization is a dynamical friction related to the fluid collision frequency Assumption: random collision rate with is much greater than small angle collision dominant Ref) S. Chandrasekhar, Rev. Mod. Phys. 15, 1(1943)

8. Interaspecies Collisions • The distribution of velocities approaches a Maxwellian for a proper choice of • is distributed according to the function P() • • • This algorithm conserves energy for any size ∆t