stationary state. The fact in (9.4.21) suggests that the turbulence in plasma is not
fully developed and picoseconds are not long to observe the dominant contribution
of Levy flights in such laser-plasma interactions. It is interesting subject to compare
the electron distribution function in [17] with the profile given in Fig. 9.6 for the case
with the fractal index in (9.4.21).
In addition, it should be noted that with use of the assumption (9.4.16), the
average jumping time τ can be evaluated. This normalized time (τω) is in general
believed to be of order unity in the anomalous turbulence in fluid or other turbulent
mixing models.
9.5 Model Experiment of Cosmic Ray Physics
in the Universe
Since the power law energy spectra are more reasonable for the relativistic electrons
produced by lase-plasma interaction in the relativistic intensity regime, consider a
possibility to design a model experiment to demonstrate the physics of cosmic ray
acceleration in laboratory. Particle acceleration is one of the most important subjects
in the study of ultra-short and ultra-intense laser interaction with low-density
plasmas as well as many applications of the produced electron and ion beams.
Cosmic rays were found to come from the universe more than100 years ago. The
observation data from many different types of instruments have been accumulated in
the log-log space of the observed particle energies as shown in Fig. 9.12. It is
amazing that the energy rage in Fig. 9.12 is from 10
9 eV (GeV) to 10
21 eV and
the data show clear dependence of a power law. The blue regions to 10
15 eV (PeV)
are thought to be due to the source of particles generated in our galaxy, while the
particles with the energy in the pink region are inferred to be generated out of our
galaxy.
The physics of the acceleration mechanisms have been studied intensively for a
long time. The most important idea was proposed by E. Fermi [18], and now most of
the researchers believe that the so-called the first Fermi acceleration is the most
reliable model to explain the huge number of observation data in Fig. 9.12. The first
Fermi acceleration is a diffusive acceleration by the accumulation of many small
amount of energy gaining through passing of charged particles across the
collisionless shocks. So, the mathematical model is called diffusive shock acceleration (DSA) [19]. It is the most promising model to explain the power law spectrum
in the case where particles are accelerated many times around the shocks with
non-relativistic velocity. Such non-relativistic but very high Mach number shocks
are generated in our galaxy by, for example, supernova explosions. It is said that the
cosmic ray with energy less than 10
15 eV are non-relativistic shock origin.
On the other hand, when the shocks become stronger and the speed of shock
waves become relativistic, the Fermi acceleration model faces a difficulty to accelerate the particles. The most simple reason is that the particles randomly traveling in
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9 Theory of Stochasticity and Chaos of Electrons in Relativistic Lasers
fully developed and picoseconds are not long to observe the dominant contribution
of Levy flights in such laser-plasma interactions. It is interesting subject to compare
the electron distribution function in [17] with the profile given in Fig. 9.6 for the case
with the fractal index in (9.4.21).
In addition, it should be noted that with use of the assumption (9.4.16), the
average jumping time τ can be evaluated. This normalized time (τω) is in general
believed to be of order unity in the anomalous turbulence in fluid or other turbulent
mixing models.
9.5 Model Experiment of Cosmic Ray Physics
in the Universe
Since the power law energy spectra are more reasonable for the relativistic electrons
produced by lase-plasma interaction in the relativistic intensity regime, consider a
possibility to design a model experiment to demonstrate the physics of cosmic ray
acceleration in laboratory. Particle acceleration is one of the most important subjects
in the study of ultra-short and ultra-intense laser interaction with low-density
plasmas as well as many applications of the produced electron and ion beams.
Cosmic rays were found to come from the universe more than100 years ago. The
observation data from many different types of instruments have been accumulated in
the log-log space of the observed particle energies as shown in Fig. 9.12. It is
amazing that the energy rage in Fig. 9.12 is from 10
9 eV (GeV) to 10
21 eV and
the data show clear dependence of a power law. The blue regions to 10
15 eV (PeV)
are thought to be due to the source of particles generated in our galaxy, while the
particles with the energy in the pink region are inferred to be generated out of our
galaxy.
The physics of the acceleration mechanisms have been studied intensively for a
long time. The most important idea was proposed by E. Fermi [18], and now most of
the researchers believe that the so-called the first Fermi acceleration is the most
reliable model to explain the huge number of observation data in Fig. 9.12. The first
Fermi acceleration is a diffusive acceleration by the accumulation of many small
amount of energy gaining through passing of charged particles across the
collisionless shocks. So, the mathematical model is called diffusive shock acceleration (DSA) [19]. It is the most promising model to explain the power law spectrum
in the case where particles are accelerated many times around the shocks with
non-relativistic velocity. Such non-relativistic but very high Mach number shocks
are generated in our galaxy by, for example, supernova explosions. It is said that the
cosmic ray with energy less than 10
15 eV are non-relativistic shock origin.
On the other hand, when the shocks become stronger and the speed of shock
waves become relativistic, the Fermi acceleration model faces a difficulty to accelerate the particles. The most simple reason is that the particles randomly traveling in
354
9 Theory of Stochasticity and Chaos of Electrons in Relativistic Lasers