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PHYSICAL CHARACTERISATION OF THE
GENERALISED GRAY BROWNIAN MOTION
Pham Minh Tuan
Belgorod National Research University, Russia
in collaboration with
Daniel Molina-Garcı́a and Gianni Pagnini
Ikebasque and Basque Center for Applied Mathematics
Bilbao 2015
Pham Minh Tuan Belgorod National Research University, Russia inPHYSICAL
collaboration
CHARACTERISATION
with Daniel Molina-Garcı́a
OF THE
and GENERALISED
Gianni Pagnini Ikeb
GR
Outline
1
Motivation
2
Definition of the ggBm with stationary increments
3
Definition of the ggBm with nonstationary increments
4
Physical characteristics of ggBm
a. Ensemble average mean square displacement
b. Ergodicity breaking parameter
c. Time average mean square displacement
5
Conclusion
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Motivation
Anomalous diffusion is observed in various physical systems like:
photonics, plasma physics, biological systems, etc.
1
Normal diffusion:
⟨X 2 (t)⟩ = Ct
2
Anomalous diffusion:
⟨X 2 (t)⟩ = Ct 𝛾
𝛾 ̸= 1
In order to provide a stochastic process that well describes an experiment
exhibiting anomalous diffusion, here we present an in-depth study of the
so-called ”generalised grey Brownian process with nonstationary
increments”.
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Definition of ggBm with stationary increments
Mura A. Pagnini G. 2008 Jour. of Phys. A: Math. & Theor.
√︀
X𝛽,H (t) = Λ𝛽 XH (t)
where Λ𝛽 is a positive random variable distributed according to the
one-side M-Wright/Mainardi function with 𝛽 ∈ (0, 1]
M𝛽 (x) =
∞
∑︁
(−1)k
k=0
k!
xk
,
Γ(1 − 𝛽(k + 1))
x ≥0
and XH (t) is the fBm with Hurst index H ∈ (0, 1) and
2 (t)⟩ = 2D t 2H
⟨X𝛽,H
H
[︂∫︁ t
1
)︀
XH (t) := (︀
(t − 𝜏 )H−1/2 dB(𝜏 )+
Γ H + 12
0
∫︁ 0 {︁
}︁
+
(t − 𝜏 )H−1/2 − (−𝜏 )H−1/2 dB(𝜏 )
−∞
⇒ the process has stationary increments
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PDF evolution equation
Pagnini G. 2012 Fractional Calculus and Applied Analysis
The one-point one-time PDF of the X𝛽,H (t) is
(︂ )︂
|x|
1
.
P(x, t) = H M𝛽/2
2t
tH
This PDF obeys the following evolution equation
2
𝜕P
2
𝛽−1,1−𝛽 𝜕 P
= t 2H−1 D2H/𝛽
,
𝜕t
𝛽
𝜕x 2
where D𝜂𝜎,𝜇 is the Erdélyi-Kober fractional derivative defined as
)︂
n (︂
∏︁
)︀
1 d (︀ 𝜎+𝜇,𝜂−𝜇
𝜎,𝜇
D𝜂 𝜙(t) =
𝜎+𝜇+ t
I𝜂
𝜙(t)
𝜂 dt
j=1
where
I𝜂𝜎,𝜇
is the Erdélyi-Kober fractional integral operator
∫︁
t −𝜂(𝜎+𝜇) t 𝜎𝜂 𝜂
𝜎,𝜇
𝜏 (t − 𝜏 𝜂 )𝜇−1 𝜙(𝜏 )d(𝜏 𝜂 )
I𝜂 𝜙(t) =
Γ(𝜇)
0
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Definition of ggBm with nonstationary increments
Molina, Pham, Pagnini, 2015 Proceedings of the NOLASC 15
X𝛼,𝛽,H (t) =
√︀
t 𝛼 Λ𝛽 XH (t) ,
𝛼 ∈ [−1, ∞)
where Λ𝛽 is a positive random variable distributed according to the
one-side M-Wright/Mainardi function with 𝛽 ∈ (0, 1]
M𝛽 (x) =
∞
∑︁
(−1)k
k=0
k!
xk
,
Γ(1 − 𝛽(k + 1))
x ≥0
and XH (t) is the fBm with Hurst index H ∈ (0, 1) and
2 (t)⟩ = 2D t 2H .
⟨X𝛽,H
H
⇒ this process has nonstationary increments
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Trajectories and PDF of ggBm with nonstationary
increments H = 0.3 , 𝛽 = 0.3, 𝛼 = 0.3
150
histogram (N=10000)
100
x(t)
50
0
-50
-100
-150
0
100 200 300 400 500 600 700 800 900 1000 0
t
0.005
0.01
0.015
0.02
0.025
P(x, t=1000)
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increments H = 0.3 , 𝛽 = 0.3, 𝛼 = −0.3
20
histogram (N=10000)
15
10
x(t)
5
0
-5
-10
-15
-20
0
100 200 300 400 500 600 700 800 900 1000 0
t
0.02 0.04 0.06 0.08 0.1 0.12 0.14 0.16
P(x, t=1000)
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increments H = 0.3 , 𝛽 = 0.3, 𝛼 = 0
histogram (N=10000)
40
f2H,β(x,t=1000)
x(t)
20
0
-20
-40
0
100 200 300 400 500 600 700 800 900 1000 0
t
0.01
0.02
0.03
0.04
0.05
0.06
P(x, t=1000)
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Physical characteristics
From physical point of view, the second moment of the process X𝛼,𝛽,H (t)
is one of the most crucial observables
EAMSD (Ensemble Average Mean Square Displacement)
⟨︀ 2
⟩︀
X𝛼,𝛽,H (t) =
2DH
t 2H+𝛼 ,
Γ(1 + 𝛽)
𝛼 ∈ [−1; ∞)
Remark:
In comparison with the well-known fBm, the power law of time is
perturbed by 𝛼.
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Ergodicity Breaking parameter
A parameter characterising the ergodicity of the studied process is the EB
(ergodicity breaking parameter).
Definition
)︁2 ⟩
𝛿X2 (ta , T , Δ)
⟨
⟩2 − 1
𝛿X2 (ta , T , Δ)
⟨(︁
EB = lim
T →∞
where T , Δ, ta stand for measurement time, lag time and aging time,
respectively. Remark:
Since this limit does not depend on the value of ta , so to simplify
notations, we consider 𝛿X2 (T , Δ) ≡ 𝛿X2 (ta , T , Δ).
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Temporal-Average MSD
The denominator of EB
TAMSD
∫︁
𝛿X2 (ta , T , Δ) =
⟨
⟩
𝛿X2 (ta , T , Δ)
≃
ta +T −Δ [︁
]︁2
X𝛼,𝛽,H (𝜏 + Δ) − X𝛼,𝛽,H (𝜏 ) d𝜏
ta
Δ≪ta ≪T
T −Δ
⎧
T𝛼
⎪
⎪
K
⎪
⎨ 1 + 𝛼 , −1 < 𝛼 < ∞
⎪
⎪
⎪
⎩ K lnT
T
,K =
2DH Δ2H
Γ(1 + 𝛽)
, 𝛼 = −1
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Comparison with numerical data H = 0.3, 𝛽 = 0.3,
𝛼 = −0.3
〈δ2 (T,∆) 〉
10
1
∆=1
∆=10
0.1 0
10
∆=20
∆=30
∆=40
∆=50
1
10
10
2
Tα
3
10
4
10
T
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Calculation for the numerator of the EB
From the definition of the TAMSD, we have
∫︁
⟨(︁
)︁2 ⟩
=
𝛿X2 (T , Δ)
T −Δ
∫︁
d𝜏1
0
T −Δ
d𝜏2 G (𝜏1 , 𝜏2 )
0
(T − Δ)2
where
⟨[︁
]︁2 [︁
]︁2 ⟩
G (𝜏1 , 𝜏2 ) = X (𝜏1 + Δ) − X (𝜏1 ) X (𝜏2 + Δ) − X (𝜏2 )
= G0 + G1 + G2 + G3
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Expressions for G0 and G1
⟨︀ ⟩︀
G0 = 4DH2 Δ4H Λ2𝛽 [𝜏1 𝜏2 (𝜏1 + Δ)(𝜏2 + Δ)]𝛼/2
⟨︀ ⟩︀ {︁
𝛼/2
+ 2DH2 Λ2𝛽 |𝜏1 + Δ − 𝜏2 |2H 𝜏2 (𝜏1 + Δ)𝛼/2
𝛼/2
+ |𝜏2 + Δ − 𝜏1 |2H 𝜏1 (𝜏2 + Δ)𝛼/2 − |𝜏1 − 𝜏2 |2H
[︁
]︁ }︁2
× (𝜏1 𝜏2 )𝛼/2 + (𝜏1 + Δ)𝛼/2 (𝜏2 + Δ)𝛼/2
𝛼/2
G1 = Δ2H 𝜏2
𝛼/2
+Δ2H 𝜏1 (𝜏1
[︁
]︁[︁
]︁
𝛼/2
2H+𝛼/2
−(𝜏1 +Δ)2H+𝛼/2 𝜏1 −(𝜏1 +Δ)𝛼/2 +
(𝜏2 +Δ)𝛼/2 𝜏1
+ Δ)
𝛼/2
[︁
]︁[︁
]︁
2H+𝛼/2
𝛼/2
2H+𝛼/2
𝛼/2
− (𝜏2 + Δ)
𝜏2
𝜏2 − (𝜏2 + Δ)
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Expressions for G2 and G3
{︁[︁ 2H+ 𝛼
]︁[︁ 𝛼
]︁
𝛼
𝛼
2
G2 = 𝜏1
− (𝜏1 + Δ)2H+ 2 𝜏22 − (𝜏2 + Δ) 2 +
[︁ 2H+ 𝛼
]︁[︁ 𝛼
]︁
𝛼
𝛼
2
+ 𝜏2
− (𝜏2 + Δ)2H+ 2 𝜏12 − (𝜏1 + Δ) 2 ×
{︁
(︁
)︁ 𝛼
(︁
)︁ 𝛼
2
2
2H
2H
× |𝜏1 + Δ − 𝜏2 |
+ |𝜏2 + Δ − 𝜏1 |
𝜏2 (𝜏1 + Δ)
𝜏1 (𝜏2 + Δ)
]︁}︁
[︁
𝛼
𝛼
𝛼
−|𝜏1 − 𝜏2 |2H (𝜏1 𝜏2 ) 2 + (𝜏1 + Δ) 2 (𝜏2 + Δ) 2
[︁
]︁ [︁
]︁
2H+𝛼/2 2
𝛼/2 2
G3 = (𝜏1 + Δ)2H+𝛼/2 − 𝜏1
(𝜏2 + Δ)𝛼/2 − 𝜏2
[︁
]︁ [︁
]︁
2H+𝛼/2 2
𝛼/2 2
+ (𝜏2 + Δ)2H+𝛼/2 − 𝜏2
(𝜏1 + Δ)𝛼/2 − 𝜏1
[︁
]︁[︁
]︁
2H+𝛼/2
𝛼/2
+ 4 𝜏1
− (𝜏1 + Δ)2H+𝛼/2 𝜏1 − (𝜏1 + Δ)𝛼/2
[︁
]︁[︁
]︁
2H+𝛼/2
𝛼/2
× 𝜏2
− (𝜏2 + Δ)2H+𝛼/2 𝜏2 − (𝜏2 + Δ)𝛼/2
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Approximation of G (𝜏1 , 𝜏2 )
It can be shown that
G0 ∼ T 4H+2𝛼 (Δ/𝜏 )4H
G1 , G2 ∼ T 4H+2𝛼 (Δ/𝜏 )2H+2
and
G3 ∼ T 4H+2𝛼 (Δ/𝜏 )4 .
This means that, due to 4H < 2 + 2H < 4, the following inequalities are
met
G0 ≫ G1 , G2 ≫ G3 .
Using the approximation Δ ≪ 𝜏1 , 𝜏2 , we obtain
[︁
⟨︀ 2 ⟩︀ 𝛼 𝛼 {︁ 2 4H
G (𝜏1 , 𝜏2 ) ≈
Λ𝛽 𝜏1 𝜏2 4DH Δ + 2DH2 |𝜏1 + Δ − 𝜏2 |2H +
Δ≪𝜏1 ,𝜏2
+|𝜏2 + Δ − 𝜏1 |2H − 2|𝜏1 − 𝜏2 |2H
]︁2 }︁
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Final result
Now, EB is given by
EB = lim
T →∞
⟨Λ2𝛽 ⟩
⟨Λ𝛽 ⟩2
[︂
]︂
(1 + 𝛼)2 Ψ
−1
1+
2T 2𝛼+2 Δ4H
Here, the integral Ψ is defined as
∫︁ T −Δ ∫︁ T −Δ
{︁
Ψ=
d𝜏1 d𝜏2 𝜏1𝛼 𝜏2𝛼 |𝜏1 − 𝜏2 + Δ|2H +
0
0
+ |𝜏2 − 𝜏1 + Δ|2H − 2|𝜏1 − 𝜏2 |2H
}︁2
Depending on the value of H,
⎧ 4H+1 2𝛼+1
T
, 0 < H < 43 ,
⎨ Δ
Ψ∼
⎩ 4 4H+2𝛼−2 3
Δ T
, 4 < H < 1.
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Final result
EB depends solely on 𝛽
EB =
⟨Λ2𝛽 ⟩
⟨Λ𝛽 ⟩2
−1=
𝛽Γ2 (𝛽)
−1
Γ(2𝛽)
We have presented here the analysis of the Ergodicity breaking parameter
of the defined ”nonstationary” ggBm. Our finding is that this parameter
depends only on 𝛽 in the same way like the standard ggBm.
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Comparison with numerical data
10
EB (T,∆)
Vn(2)(t)
2
n=9
n=8
n=7
n=6
5·10
2
4·10
3·102
2·102
1·102
0
0
2·103
4·103
t
6·103
1
∆=1
∆=10
101
∆=20
∆=30
102
∆=40
∆=50
103
104
T
[H = 0.3, 𝛽 = 0.3, 𝛼 = −0.3]
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Conclusion
1
The variance of the process X𝛼,𝛽,H (t) is mainly governed by H and
affected by 𝛼
2
The process is non-ergodic and 𝛽 is the only parameter that controls
the ergodic/non-ergodic transition. This mean that in this aspect,
there is no difference from ggBm with stationary and nonstationary
increments.
3
𝛼 is the only parameter that controls aging, driving its slope
according to its sign.
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Thank you for your attention!
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Talk Pham - BCAM - Basque Center for Applied

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