Chapter 9
Theory of Stochasticity and Chaos
of Electrons in Relativistic Lasers
9.1 Vlasov-Fokker-Planck Equations
In the previous chapter, it is shown that a random perturbation to the relativistic
electron motions results in the heating of the electrons. This is called stochastic
heating and one of the most important processes of the hot electron production due
to relativistic laser-plasma interaction. Such a random perturbation to the system, in
general, is described with additional diffusion term mathematically to the governing
equation of statistical distribution of the electrons in the momentum space, f(t, p x , p y )
in the present case.
As the diffusion model in real space, Brownian motion is well-known, and the
distribution of any particles under Brownian motion in space is said, in general, to be
described with the following diffusion equation:
∂
∂t
f t, x
ð Þ ¼ D
∂
2
∂x 2 f t, x
ð Þ
ð9:1:1Þ
For simplicity, one dimension in space is assumed. In (9.1.1), D is the diffusion
coefficient and is usually given in the form:
D ¼ A
Δ
2
τ
ð9:1:2Þ
where Δ is the mean free path and τ is the collision time. It is noted that A is a
coefficient depending on the situation and usually A ¼ 1/3 or 1/2.
In the present case, the perturbation is expected to contribute to the heating of
electrons, and the hot electrons are generated. Then, Brownian motion is random
motion in the velocity or momentum space. Langevin equation is well-known as a
simple model equation to describe such a heating by external random force.
© Springer Nature Switzerland AG 2020
H. Takabe, The Physics of Laser Plasmas and Applications - Volume 1, Springer
Series in Plasma Science and Technology,
https://doi.org/10.1007/978-3-030-49613-5_9
331
Theory of Stochasticity and Chaos
of Electrons in Relativistic Lasers
9.1 Vlasov-Fokker-Planck Equations
In the previous chapter, it is shown that a random perturbation to the relativistic
electron motions results in the heating of the electrons. This is called stochastic
heating and one of the most important processes of the hot electron production due
to relativistic laser-plasma interaction. Such a random perturbation to the system, in
general, is described with additional diffusion term mathematically to the governing
equation of statistical distribution of the electrons in the momentum space, f(t, p x , p y )
in the present case.
As the diffusion model in real space, Brownian motion is well-known, and the
distribution of any particles under Brownian motion in space is said, in general, to be
described with the following diffusion equation:
∂
∂t
f t, x
ð Þ ¼ D
∂
2
∂x 2 f t, x
ð Þ
ð9:1:1Þ
For simplicity, one dimension in space is assumed. In (9.1.1), D is the diffusion
coefficient and is usually given in the form:
D ¼ A
Δ
2
τ
ð9:1:2Þ
where Δ is the mean free path and τ is the collision time. It is noted that A is a
coefficient depending on the situation and usually A ¼ 1/3 or 1/2.
In the present case, the perturbation is expected to contribute to the heating of
electrons, and the hot electrons are generated. Then, Brownian motion is random
motion in the velocity or momentum space. Langevin equation is well-known as a
simple model equation to describe such a heating by external random force.
© Springer Nature Switzerland AG 2020
H. Takabe, The Physics of Laser Plasmas and Applications - Volume 1, Springer
Series in Plasma Science and Technology,
https://doi.org/10.1007/978-3-030-49613-5_9
331