∂
∂t
f ¼ D
∂
α
∂ p
j j
α f
ð9:3:16Þ
It is obvious that for the case of α ¼ 2, (9.3.16) is the same as the normal diffusion
Eq. (9.3.10).
For example, consider the case of α ¼ 1. Inserting (9.3.14) to (9.3.12) and time
integration for Δt, it is found that the probability density becomes Lorentzian
distribution:
ψ L Δp, Δt
ð
Þ¼
1
2π
Z 1
0
e
Àk DΔtÀiΔp
ð
Þ
þ e
Àk DΔtþiΔp
ð
Þ
n
o
dk
¼
2=DΔt
1 þ Δp=DΔt
ð
Þ
2
ð9:3:17Þ
Note that (9.3.17) has an asymptotic form proportional to 1/(Δp)
2 for large Δp,
namely, in Levy jump region. It is clear that the power law decay is much slower
than the exponential decay at a large argument; consequently, the contribution by
Levy jumps is modeled in the probability density.
It is useful to know the property of Eq. (9.3.16) for α ¼ 1. In the case of α ¼ 1,
(9.3.16) is the equation of convection to the positive direction in p > 0,and the
negative direction for p < 0. Some physical quantities are transported on the flow
with the velocity D, so (9.3.16) is also the equation to describe such convective
transport.
9.4 Time Evolution of Distribution and Fractal Index α
Let us derive the time evolution of the distribution function by external stochastic
force governed by the FFPE in (9.3.16) for a simple condition. In the case of α ¼ 2,
the solution of the diffusion equation is well-known to be given in the form for the
initial distribution of the delta function f(p,0) ¼ δ(p):
f p, t
ð Þ ¼
1
ffiffiffiffiffiffiffiffiffiffiffiffi
πD Ã t
p
exp À
p
2
D Ã t
,
D
Ã
¼ 4D
ð9:4:1Þ
For arbitral α, the solution is given as the inverse-Fourier integral form for FFPE
in the form with use of (9.3.15):
9.4 Time Evolution of Distribution and Fractal Index α
345
∂t
f ¼ D
∂
α
∂ p
j j
α f
ð9:3:16Þ
It is obvious that for the case of α ¼ 2, (9.3.16) is the same as the normal diffusion
Eq. (9.3.10).
For example, consider the case of α ¼ 1. Inserting (9.3.14) to (9.3.12) and time
integration for Δt, it is found that the probability density becomes Lorentzian
distribution:
ψ L Δp, Δt
ð
Þ¼
1
2π
Z 1
0
e
Àk DΔtÀiΔp
ð
Þ
þ e
Àk DΔtþiΔp
ð
Þ
n
o
dk
¼
2=DΔt
1 þ Δp=DΔt
ð
Þ
2
ð9:3:17Þ
Note that (9.3.17) has an asymptotic form proportional to 1/(Δp)
2 for large Δp,
namely, in Levy jump region. It is clear that the power law decay is much slower
than the exponential decay at a large argument; consequently, the contribution by
Levy jumps is modeled in the probability density.
It is useful to know the property of Eq. (9.3.16) for α ¼ 1. In the case of α ¼ 1,
(9.3.16) is the equation of convection to the positive direction in p > 0,and the
negative direction for p < 0. Some physical quantities are transported on the flow
with the velocity D, so (9.3.16) is also the equation to describe such convective
transport.
9.4 Time Evolution of Distribution and Fractal Index α
Let us derive the time evolution of the distribution function by external stochastic
force governed by the FFPE in (9.3.16) for a simple condition. In the case of α ¼ 2,
the solution of the diffusion equation is well-known to be given in the form for the
initial distribution of the delta function f(p,0) ¼ δ(p):
f p, t
ð Þ ¼
1
ffiffiffiffiffiffiffiffiffiffiffiffi
πD Ã t
p
exp À
p
2
D Ã t
,
D
Ã
¼ 4D
ð9:4:1Þ
For arbitral α, the solution is given as the inverse-Fourier integral form for FFPE
in the form with use of (9.3.15):
9.4 Time Evolution of Distribution and Fractal Index α
345