9.1.2 Vlasov-Fokker-Planck Equation
The Vlasov-Fokker-Planck (VFP) equation governing the distribution function f is
given in the form:
d
dt
f t, p x , p y
¼
∂f
∂t
þ
dp x
dt
∂f
∂p x
þ
dp y
dt
∂f
∂p y
¼
df
dt
random
ð9:1:16Þ
where RHS is the contribution by the random force. Therefore, the equation for the
distribution of electrons under the relativistic lasers and the stochastic force is
obtained:
∂f
∂t
À
a
2
0
2γ
sin 2ξ
ð Þ
∂f
∂p x
þ a 0 sin ξ
∂f
∂p y
¼
∂
∂p x
D x
∂
∂p x
f
þ D y
∂
2
∂p 2
y
f
ð9:1:17Þ
In deriving (9.1.17), (8.4.8) and (8.4.9) are inserted to (9.1.16).
The role of the diffusion on RHS in (9.1.17) is in general small compared to the
second and third terms in LHS, while depending on the problem, the diffusion plays
an important role to dramatically change of the distribution function. Small diffusion
in p x direction near p x ¼ 0 causes the decrease of the values of α in (8.1.23), and the
maximum energy of electrons increases dramatically due to the diffusion. In addition, D x in (9.1.15) is proportional to 1/γ
2 which is large near p x ¼ 0 point, and this
diffusion effect is very efficient to heat the electrons stochastically.
9.2 Stochastic Heating by Laser Filamentation
It is frequently seen that the incident laser becomes unstable to the filamentation
instability and the wave phase is not uniform in the y-direction but random in this
direction. For example, a snap shot of laser intensity obtained by 2D PIC simulation
is plotted in Fig. 9.1 [2]. In such a case, a jump of an electron from a filament to a
filament is affected by a rapid change of the phase of the electromagnetic waves. It is
shown in Refs. [3, 4] that such a random motion of electrons in relativistic laser field
can be modeled by random walk due to the random force induced by rapid phase
change.
The basic equations are (8.4.8) and (8.4.9). The laser field is given as a ¼ a 0 sin(ξ),
although we assume that the laser phase ξ is not coherent,but randomly changes in
the y-direction because of the filamentation. The time derivative is replaced by the
derivative by the proper time τ defined in (8.1.12). Then, (8.1.12) becomes
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9 Theory of Stochasticity and Chaos of Electrons in Relativistic Lasers
The Vlasov-Fokker-Planck (VFP) equation governing the distribution function f is
given in the form:
d
dt
f t, p x , p y
¼
∂f
∂t
þ
dp x
dt
∂f
∂p x
þ
dp y
dt
∂f
∂p y
¼
df
dt
random
ð9:1:16Þ
where RHS is the contribution by the random force. Therefore, the equation for the
distribution of electrons under the relativistic lasers and the stochastic force is
obtained:
∂f
∂t
À
a
2
0
2γ
sin 2ξ
ð Þ
∂f
∂p x
þ a 0 sin ξ
∂f
∂p y
¼
∂
∂p x
D x
∂
∂p x
f
þ D y
∂
2
∂p 2
y
f
ð9:1:17Þ
In deriving (9.1.17), (8.4.8) and (8.4.9) are inserted to (9.1.16).
The role of the diffusion on RHS in (9.1.17) is in general small compared to the
second and third terms in LHS, while depending on the problem, the diffusion plays
an important role to dramatically change of the distribution function. Small diffusion
in p x direction near p x ¼ 0 causes the decrease of the values of α in (8.1.23), and the
maximum energy of electrons increases dramatically due to the diffusion. In addition, D x in (9.1.15) is proportional to 1/γ
2 which is large near p x ¼ 0 point, and this
diffusion effect is very efficient to heat the electrons stochastically.
9.2 Stochastic Heating by Laser Filamentation
It is frequently seen that the incident laser becomes unstable to the filamentation
instability and the wave phase is not uniform in the y-direction but random in this
direction. For example, a snap shot of laser intensity obtained by 2D PIC simulation
is plotted in Fig. 9.1 [2]. In such a case, a jump of an electron from a filament to a
filament is affected by a rapid change of the phase of the electromagnetic waves. It is
shown in Refs. [3, 4] that such a random motion of electrons in relativistic laser field
can be modeled by random walk due to the random force induced by rapid phase
change.
The basic equations are (8.4.8) and (8.4.9). The laser field is given as a ¼ a 0 sin(ξ),
although we assume that the laser phase ξ is not coherent,but randomly changes in
the y-direction because of the filamentation. The time derivative is replaced by the
derivative by the proper time τ defined in (8.1.12). Then, (8.1.12) becomes
334
9 Theory of Stochasticity and Chaos of Electrons in Relativistic Lasers