Fokker-Planck equation (FFPE) in the momentum space. Such study is most
important in using the relativistic laser for any kind of applications. Levy’ s flights
or jumps in the momentum space become important to allow anomalous and
nonlocal transports of electrons in the energy space. Comparing with
computational and experimental data, it is shown that the FFPE is a simple but
strong tool to investigate the collisionless heating of electrons in relativistic laser.
Model experiment of cosmic ray in universe is also discussed based on the FFPE by
strong electromagnetic waves generated by relativistic collisionless shock waves.
1.6 PIC Simulations
There have been many challenges to find the theory to understand the laser-plasma
coupling in the case of non-relativistic laser with relatively long pulse. Theoretical
model of classical absorption in non-relativistic laser-plasma interaction is relatively
easy to derive. However, it is a rather new topic to develop the theory for
understanding and predicting the laser-plasma coupling in the case of relativistic
laser irradiation. This is partially because relativistic nonlinearity makes the problem
complicated.
The laser-plasma interaction and resultant hydrodynamics have been studied by
different simulations. The former has been studied mainly by Particle-in-Cell (PIC)
simulations, while the latter has been studied with hydrodynamic one. The general
method is that some model equations from theory and/or PIC simulations are
installed in hydrodynamic code to see a long time evolution of plasma. Roughly
speaking, hydrodynamics simulation can run about thousand times linger time
compared to the PIC code. This is due to the fact that electron motion is thousand
times faster than the ion motion.
Since in this book, the laser-electron interaction is discussed frequently, and
many PIC simulation results are compared to theories and experiments. It is useful
to briefly understand the principle of PIC simulation. In PIC scheme, (1.3.8) is
regarded as the equation of the average momentum of N G electrons (N G >> 1). Such
a particle is called a giant particle with the mass and charge:
M G ¼ N G m
Q G ¼ N G e
ð1:6:1Þ
Then, regard p and r of (1.3.8) are average momentum and position of N G electrons.
In the codes, the partial differential Maxwell Eqs. (1.3.1) and (1.3.2) are solved with
finite difference method at each grid point in space for a time step Δt determined by
numerical stability condition. However, the equations of motion (1.3.8) for all
particles are solved in Lagrangian frame (particle frame). Time integration of
(1.3.8) is carried out for each Δt by finite difference method or other methods. The
field quantities and particle quantities are defined at different points, and
interpolation method is used to couple both equations.
1.6 PIC Simulations
23
important in using the relativistic laser for any kind of applications. Levy’ s flights
or jumps in the momentum space become important to allow anomalous and
nonlocal transports of electrons in the energy space. Comparing with
computational and experimental data, it is shown that the FFPE is a simple but
strong tool to investigate the collisionless heating of electrons in relativistic laser.
Model experiment of cosmic ray in universe is also discussed based on the FFPE by
strong electromagnetic waves generated by relativistic collisionless shock waves.
1.6 PIC Simulations
There have been many challenges to find the theory to understand the laser-plasma
coupling in the case of non-relativistic laser with relatively long pulse. Theoretical
model of classical absorption in non-relativistic laser-plasma interaction is relatively
easy to derive. However, it is a rather new topic to develop the theory for
understanding and predicting the laser-plasma coupling in the case of relativistic
laser irradiation. This is partially because relativistic nonlinearity makes the problem
complicated.
The laser-plasma interaction and resultant hydrodynamics have been studied by
different simulations. The former has been studied mainly by Particle-in-Cell (PIC)
simulations, while the latter has been studied with hydrodynamic one. The general
method is that some model equations from theory and/or PIC simulations are
installed in hydrodynamic code to see a long time evolution of plasma. Roughly
speaking, hydrodynamics simulation can run about thousand times linger time
compared to the PIC code. This is due to the fact that electron motion is thousand
times faster than the ion motion.
Since in this book, the laser-electron interaction is discussed frequently, and
many PIC simulation results are compared to theories and experiments. It is useful
to briefly understand the principle of PIC simulation. In PIC scheme, (1.3.8) is
regarded as the equation of the average momentum of N G electrons (N G >> 1). Such
a particle is called a giant particle with the mass and charge:
M G ¼ N G m
Q G ¼ N G e
ð1:6:1Þ
Then, regard p and r of (1.3.8) are average momentum and position of N G electrons.
In the codes, the partial differential Maxwell Eqs. (1.3.1) and (1.3.2) are solved with
finite difference method at each grid point in space for a time step Δt determined by
numerical stability condition. However, the equations of motion (1.3.8) for all
particles are solved in Lagrangian frame (particle frame). Time integration of
(1.3.8) is carried out for each Δt by finite difference method or other methods. The
field quantities and particle quantities are defined at different points, and
interpolation method is used to couple both equations.
1.6 PIC Simulations
23