Random Diffusivity from Stochastic Equations:Two Models in Comparison for Brownian Yet

Non-Gaussian Diffusion.

Vittoria Sposini1,2 Aleksei V. Chechkin1,4 Gianni Pagnini2,3

Ralf Metzler1.

1Institute for Physics & Astronomy, University of Potsdam, 14476 Potsdam-Golm2Basque Center for Applied Mathematics, 48009 Bilbao, Spain

3Ikerbasque - Basque Foundation for Science4Akhiezer Institute for Theoretical Physics, 61108 Kharkov, Ukraine

Vittoria Sposini University of Potsdam

Outline

1 Introduction

2 A set of stochastic equations for random diffusivity

3 Generalised grey Brownian motion with random diffusivity

4 A diffusing diffusivities model

5 The two models in comparison

6 Conclusions

Vittoria Sposini University of Potsdam

Outline

1 Introduction

2 A set of stochastic equations for random diffusivity

3 Generalised grey Brownian motion with random diffusivity

4 A diffusing diffusivities model

5 The two models in comparison

6 Conclusions

Vittoria Sposini University of Potsdam

Outline

1 Introduction

2 A set of stochastic equations for random diffusivity

3 Generalised grey Brownian motion with random diffusivity

4 A diffusing diffusivities model

5 The two models in comparison

6 Conclusions

Vittoria Sposini University of Potsdam

Outline

1 Introduction

2 A set of stochastic equations for random diffusivity

3 Generalised grey Brownian motion with random diffusivity

4 A diffusing diffusivities model

5 The two models in comparison

6 Conclusions

Vittoria Sposini University of Potsdam

Outline

1 Introduction

2 A set of stochastic equations for random diffusivity

3 Generalised grey Brownian motion with random diffusivity

4 A diffusing diffusivities model

5 The two models in comparison

6 Conclusions

Vittoria Sposini University of Potsdam

Outline

1 Introduction

2 A set of stochastic equations for random diffusivity

3 Generalised grey Brownian motion with random diffusivity

4 A diffusing diffusivities model

5 The two models in comparison

6 Conclusions

Vittoria Sposini University of Potsdam

Introduction

Brownian Yet Non-GaussianDiffusion

1 linear trend of the MSD;

2 non Gaussian PDF.

Biological systems(Wang, B. et al. 2009, Proc.

Nat. Acad, Sci. U.S.A., 106);

Materials with glassydynamics(Chaudhuri, P. et al. 2007,

Phys. Rev. Lett., 99);

Ecological processes(Hapca, S. et al. 2009 , J. R.

Soc. Interface, 6).

Not well described by stan-dard models for anomalous diffusion(CTRWs, fBM). Other models havebeen proposed:

Correlated random walks(Codling, E. A. et al. 2008, J. R.

Soc. Interface, 5);

Superstatisical Brownianmotion(Beck, C. 2006, Prog. Theor.

Phys. Suppl, 162);

Diffusing Diffusivities(Chubynsky, M.V. & Slater, G.W.

2014, Phys. Rev. Lett., 113).

Vittoria Sposini University of Potsdam

Introduction

Brownian Yet Non-GaussianDiffusion

1 linear trend of the MSD;

2 non Gaussian PDF.

Biological systems(Wang, B. et al. 2009, Proc.

Nat. Acad, Sci. U.S.A., 106);

Materials with glassydynamics(Chaudhuri, P. et al. 2007,

Phys. Rev. Lett., 99);

Ecological processes(Hapca, S. et al. 2009 , J. R.

Soc. Interface, 6).

Not well described by stan-dard models for anomalous diffusion(CTRWs, fBM). Other models havebeen proposed:

Correlated random walks(Codling, E. A. et al. 2008, J. R.

Soc. Interface, 5);

Superstatisical Brownianmotion(Beck, C. 2006, Prog. Theor.

Phys. Suppl, 162);

Diffusing Diffusivities(Chubynsky, M.V. & Slater, G.W.

2014, Phys. Rev. Lett., 113).

Vittoria Sposini University of Potsdam

Introduction

Brownian Yet Non-GaussianDiffusion

1 linear trend of the MSD;

2 non Gaussian PDF.

Biological systems(Wang, B. et al. 2009, Proc.

Nat. Acad, Sci. U.S.A., 106);

Materials with glassydynamics(Chaudhuri, P. et al. 2007,

Phys. Rev. Lett., 99);

Ecological processes(Hapca, S. et al. 2009 , J. R.

Soc. Interface, 6).

Not well described by stan-dard models for anomalous diffusion(CTRWs, fBM). Other models havebeen proposed:

Correlated random walks(Codling, E. A. et al. 2008, J. R.

Soc. Interface, 5);

Superstatisical Brownianmotion(Beck, C. 2006, Prog. Theor.

Phys. Suppl, 162);

Diffusing Diffusivities(Chubynsky, M.V. & Slater, G.W.

2014, Phys. Rev. Lett., 113).

Vittoria Sposini University of Potsdam

Stochastic Equations for Random Diffusivity: Assumptions

{D(t) = y2(t)

dy = a(y)dt + σ dW (t).

a(y) can be defined through thedistribution of y(t) by means of :

a(y) =σ2

2 py (y)

dpy (y)

dy. (1)

Given the Y = g(X ), for sufficientlygood functions g(x), we have:

pY (y) = pX(g−1(y)

)∣∣∣ ddy

g−1(y)∣∣∣. (2)

An ad hoc set of stochastic equations for D(t) can be built

pD(D)Eq. (2)−→ py (y)

Eq. (1)−→ a(y).

Vittoria Sposini University of Potsdam

Stochastic Equations for Random Diffusivity: Assumptions{D(t) = y2(t)

dy = a(y)dt + σ dW (t).

a(y) can be defined through thedistribution of y(t) by means of :

a(y) =σ2

2 py (y)

dpy (y)

dy. (1)

Given the Y = g(X ), for sufficientlygood functions g(x), we have:

pY (y) = pX(g−1(y)

)∣∣∣ ddy

g−1(y)∣∣∣. (2)

An ad hoc set of stochastic equations for D(t) can be built

pD(D)Eq. (2)−→ py (y)

Eq. (1)−→ a(y).

Vittoria Sposini University of Potsdam

Stochastic Equations for Random Diffusivity: Assumptions{D(t) = y2(t)

dy = a(y)dt + σ dW (t).

a(y) can be defined through thedistribution of y(t) by means of :

a(y) =σ2

2 py (y)

dpy (y)

dy. (1)

Given the Y = g(X ), for sufficientlygood functions g(x), we have:

pY (y) = pX(g−1(y)

)∣∣∣ ddy

g−1(y)∣∣∣. (2)

An ad hoc set of stochastic equations for D(t) can be built

pD(D)Eq. (2)−→ py (y)

Eq. (1)−→ a(y).

Vittoria Sposini University of Potsdam

Stochastic Equations for Random Diffusivity: Assumptions{D(t) = y2(t)

dy = a(y)dt + σ dW (t).

a(y) can be defined through thedistribution of y(t) by means of :

a(y) =σ2

2 py (y)

dpy (y)

dy. (1)

Given the Y = g(X ), for sufficientlygood functions g(x), we have:

pY (y) = pX(g−1(y)

)∣∣∣ ddy

g−1(y)∣∣∣. (2)

An ad hoc set of stochastic equations for D(t) can be built

pD(D)Eq. (2)−→ py (y)

Eq. (1)−→ a(y).

Vittoria Sposini University of Potsdam

Stochastic Equations for Random Diffusivity: Assumptions{D(t) = y2(t)

dy = a(y)dt + σ dW (t).

a(y) can be defined through thedistribution of y(t) by means of :

a(y) =σ2

2 py (y)

dpy (y)

dy. (1)

Given the Y = g(X ), for sufficientlygood functions g(x), we have:

pY (y) = pX(g−1(y)

)∣∣∣ ddy

g−1(y)∣∣∣. (2)

An ad hoc set of stochastic equations for D(t) can be built

pD(D)Eq. (2)−→ py (y)

Eq. (1)−→ a(y).

Vittoria Sposini University of Potsdam

Stochastic Equations for Random Diffusivity: Assumptions{D(t) = y2(t)

dy = a(y)dt + σ dW (t).

a(y) can be defined through thedistribution of y(t) by means of :

a(y) =σ2

2 py (y)

dpy (y)

dy. (1)

Given the Y = g(X ), for sufficientlygood functions g(x), we have:

pY (y) = pX(g−1(y)

)∣∣∣ ddy

g−1(y)∣∣∣. (2)

An ad hoc set of stochastic equations for D(t) can be built

pD(D)Eq. (2)−→ py (y)

Eq. (1)−→ a(y).

Vittoria Sposini University of Potsdam

Stochastic Equations for Random Diffusivity: Definition

γgenν,η (D) =η

Dν? Γ(ν/η)

Dν−1e−(D/D∗)η , 〈Dn〉 = Dn?

Γ(ν+nη

)Γ(νη

) ,

where D? = y2? , ν and η are positive constants.

dy = σ2

2 y

[2ν − 1− 2 η

(yy?

)2η]dt + σ dW (t)

D(t) = y2(t),

For ν = 0.5 and η = 1 we obtain an Ornstein-Uhlenbeck (OU)process for the variable y(t).

Vittoria Sposini University of Potsdam

Stochastic Equations for Random Diffusivity: Definition

γgenν,η (D) =η

Dν? Γ(ν/η)

Dν−1e−(D/D∗)η , 〈Dn〉 = Dn?

Γ(ν+nη

)Γ(νη

) ,

where D? = y2? , ν and η are positive constants.

dy = σ2

2 y

[2ν − 1− 2 η

(yy?

)2η]dt + σ dW (t)

D(t) = y2(t),

For ν = 0.5 and η = 1 we obtain an Ornstein-Uhlenbeck (OU)process for the variable y(t).

Vittoria Sposini University of Potsdam

Stochastic Equations for Random Diffusivity: Definition

γgenν,η (D) =η

Dν? Γ(ν/η)

Dν−1e−(D/D∗)η , 〈Dn〉 = Dn?

Γ(ν+nη

)Γ(νη

) ,

where D? = y2? , ν and η are positive constants.

dy = σ2

2 y

[2ν − 1− 2 η

(yy?

)2η]dt + σ dW (t)

D(t) = y2(t),

For ν = 0.5 and η = 1 we obtain an Ornstein-Uhlenbeck (OU)process for the variable y(t).

Vittoria Sposini University of Potsdam

Stochastic Equations for Random Diffusivity: Definition

γgenν,η (D) =η

Dν? Γ(ν/η)

Dν−1e−(D/D∗)η , 〈Dn〉 = Dn?

Γ(ν+nη

)Γ(νη

) ,

where D? = y2? , ν and η are positive constants.

dy = σ2

2 y

[2ν − 1− 2 η

(yy?

)2η]dt + σ dW (t)

D(t) = y2(t),

For ν = 0.5 and η = 1 we obtain an Ornstein-Uhlenbeck (OU)process for the variable y(t).

Vittoria Sposini University of Potsdam

Stochastic Equations for Random Diffusivity: Definition

γgenν,η (D) =η

Dν? Γ(ν/η)

Dν−1e−(D/D∗)η , 〈Dn〉 = Dn?

Γ(ν+nη

)Γ(νη

) ,

where D? = y2? , ν and η are positive constants.

dy = σ2

2 y

[2ν − 1− 2 η

(yy?

)2η]dt + σ dW (t)

D(t) = y2(t),

For ν = 0.5 and η = 1 we obtain an Ornstein-Uhlenbeck (OU)process for the variable y(t).

Vittoria Sposini University of Potsdam

Stochastic Equations for Random Diffusivity: Numerics

3 2 1 0 1 2 30.0

0.1

0.2

0.3

0.4

0.5

0.6

f Y(Y

)

η= 1. 3, ν= 0. 5

0.0 0.5 1.0 1.5 2.0 2.5 3.00.0

0.5

1.0

1.5

2.0

2.5

3.0

f D(D

)

η= 1. 3, ν= 0. 5

4 3 2 1 0 1 2 3 40.0

0.1

0.2

0.3

0.4

0.5

0.6

f Y(Y

)

η= 1. 3, ν= 1. 5

0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.00.00.10.20.30.40.50.60.70.8

f D(D

)

η= 1. 3, ν= 1. 5

Numerical evaluation of the correlation time by means of:

computation of the integral: τcorr = 1ACF (0)

∫∞0

ACF (τ)dτ ;

exponential fit: ACF (τ) = ACF (0) e−τ/τcorr .

0 5 10 15 20

t

0.00

0.05

0.10

0.15

0.20

0.25

ACFD(t

)

η= 1. 3, ν= 0. 5

τINTcor = 0. 79

τFITcor = 0. 80

0 2 4 6 8 10

t

0.0

0.1

0.2

0.3

0.4

0.5

ACFD(t

)

η= 1. 3, ν= 1. 5

τINTcor = 0. 68

τFITcor = 0. 67

Vittoria Sposini University of Potsdam

Stochastic Equations for Random Diffusivity: Numerics

3 2 1 0 1 2 30.0

0.1

0.2

0.3

0.4

0.5

0.6f Y

(Y)

η= 1. 3, ν= 0. 5

0.0 0.5 1.0 1.5 2.0 2.5 3.00.0

0.5

1.0

1.5

2.0

2.5

3.0

f D(D

)

η= 1. 3, ν= 0. 5

4 3 2 1 0 1 2 3 40.0

0.1

0.2

0.3

0.4

0.5

0.6

f Y(Y

)

η= 1. 3, ν= 1. 5

0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.00.00.10.20.30.40.50.60.70.8

f D(D

)

η= 1. 3, ν= 1. 5

Numerical evaluation of the correlation time by means of:

computation of the integral: τcorr = 1ACF (0)

∫∞0

ACF (τ)dτ ;

exponential fit: ACF (τ) = ACF (0) e−τ/τcorr .

0 5 10 15 20

t

0.00

0.05

0.10

0.15

0.20

0.25

ACFD(t

)

η= 1. 3, ν= 0. 5

τINTcor = 0. 79

τFITcor = 0. 80

0 2 4 6 8 10

t

0.0

0.1

0.2

0.3

0.4

0.5

ACFD(t

)

η= 1. 3, ν= 1. 5

τINTcor = 0. 68

τFITcor = 0. 67

Vittoria Sposini University of Potsdam

Stochastic Equations for Random Diffusivity: Numerics

3 2 1 0 1 2 30.0

0.1

0.2

0.3

0.4

0.5

0.6f Y

(Y)

η= 1. 3, ν= 0. 5

0.0 0.5 1.0 1.5 2.0 2.5 3.00.0

0.5

1.0

1.5

2.0

2.5

3.0

f D(D

)

η= 1. 3, ν= 0. 5

4 3 2 1 0 1 2 3 40.0

0.1

0.2

0.3

0.4

0.5

0.6

f Y(Y

)

η= 1. 3, ν= 1. 5

0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.00.00.10.20.30.40.50.60.70.8

f D(D

)

η= 1. 3, ν= 1. 5

Numerical evaluation of the correlation time by means of:

computation of the integral: τcorr = 1ACF (0)

∫∞0

ACF (τ)dτ ;

exponential fit: ACF (τ) = ACF (0) e−τ/τcorr .

0 5 10 15 20

t

0.00

0.05

0.10

0.15

0.20

0.25

ACFD(t

)

η= 1. 3, ν= 0. 5

τINTcor = 0. 79

τFITcor = 0. 80

0 2 4 6 8 10

t

0.0

0.1

0.2

0.3

0.4

0.5

ACFD(t

)

η= 1. 3, ν= 1. 5

τINTcor = 0. 68

τFITcor = 0. 67

Vittoria Sposini University of Potsdam

Stochastic Equations for Random Diffusivity: Numerics

3 2 1 0 1 2 30.0

0.1

0.2

0.3

0.4

0.5

0.6f Y

(Y)

η= 1. 3, ν= 0. 5

0.0 0.5 1.0 1.5 2.0 2.5 3.00.0

0.5

1.0

1.5

2.0

2.5

3.0

f D(D

)

η= 1. 3, ν= 0. 5

4 3 2 1 0 1 2 3 40.0

0.1

0.2

0.3

0.4

0.5

0.6

f Y(Y

)

η= 1. 3, ν= 1. 5

0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.00.00.10.20.30.40.50.60.70.8

f D(D

)

η= 1. 3, ν= 1. 5

Numerical evaluation of the correlation time by means of:

computation of the integral: τcorr = 1ACF (0)

∫∞0

ACF (τ)dτ ;

exponential fit: ACF (τ) = ACF (0) e−τ/τcorr .

0 5 10 15 20

t

0.00

0.05

0.10

0.15

0.20

0.25

ACFD(t

)

η= 1. 3, ν= 0. 5

τINTcor = 0. 79

τFITcor = 0. 80

0 2 4 6 8 10

t

0.0

0.1

0.2

0.3

0.4

0.5ACFD(t

)η= 1. 3, ν= 1. 5

τINTcor = 0. 68

τFITcor = 0. 67

Vittoria Sposini University of Potsdam

Generalised Grey Brownian Motion: Definition

It is possible to define ggBM model through the stochasticrepresentation

XggBM =√

ΛXg ,

where Λ is an independent non-negative random variable and Xg isa Gaussian process.

The PDF of the stochastic variable XggBM can be evaluated bymeans of the integral

PggBM(x , t) =

∫ ∞0

pXg

( x

λ1/2

)pΛ(λ)

dλ

λ1/2

where pXg and pΛ are the distributions of Xg and Λ respectively.

[Mura, A. & Pagnini, G. 2008, J. Phys. A: Math. Theor., 41]

Vittoria Sposini University of Potsdam

Generalised Grey Brownian Motion: Definition

It is possible to define ggBM model through the stochasticrepresentation

XggBM =√

ΛXg ,

where Λ is an independent non-negative random variable and Xg isa Gaussian process.

The PDF of the stochastic variable XggBM can be evaluated bymeans of the integral

PggBM(x , t) =

∫ ∞0

pXg

( x

λ1/2

)pΛ(λ)

dλ

λ1/2

where pXg and pΛ are the distributions of Xg and Λ respectively.

[Mura, A. & Pagnini, G. 2008, J. Phys. A: Math. Theor., 41]

Vittoria Sposini University of Potsdam

Generalised Grey Brownian Motion: Definition

It is possible to define ggBM model through the stochasticrepresentation

XggBM =√

ΛXg ,

where Λ is an independent non-negative random variable and Xg isa Gaussian process.

The PDF of the stochastic variable XggBM can be evaluated bymeans of the integral

PggBM(x , t) =

∫ ∞0

pXg

( x

λ1/2

)pΛ(λ)

dλ

λ1/2

where pXg and pΛ are the distributions of Xg and Λ respectively.

[Mura, A. & Pagnini, G. 2008, J. Phys. A: Math. Theor., 41]

Vittoria Sposini University of Potsdam

Generalised Grey Brownian Motion & Random Diffusivity

Particles diffusing in Brownian fashion in a complex randommedium;

environment properties independent on the diffusing particlesand represented by a random diffusivity.

XggBM =√

2D(t)W (t), W (t) =∫ t

0 ξ(s)ds.

Assuming a generalised Gamma distribution for the randomdiffusivity

γgenν,η (D) = ηDν? Γ(ν/η)D

ν−1e−(D/D∗)η , 〈Dn〉 = Dn?

Γ(ν+nη

)Γ(νη

) ,where D? , ν and η are positive constants.

Vittoria Sposini University of Potsdam

Generalised Grey Brownian Motion & Random Diffusivity

Particles diffusing in Brownian fashion in a complex randommedium;

environment properties independent on the diffusing particlesand represented by a random diffusivity.

XggBM =√

2D(t)W (t), W (t) =∫ t

0 ξ(s)ds.

Assuming a generalised Gamma distribution for the randomdiffusivity

γgenν,η (D) = ηDν? Γ(ν/η)D

ν−1e−(D/D∗)η , 〈Dn〉 = Dn?

Γ(ν+nη

)Γ(νη

) ,where D? , ν and η are positive constants.

Vittoria Sposini University of Potsdam

Generalised Grey Brownian Motion & Random Diffusivity

Particles diffusing in Brownian fashion in a complex randommedium;

environment properties independent on the diffusing particlesand represented by a random diffusivity.

XggBM =√

2D(t)W (t), W (t) =∫ t

0 ξ(s)ds.

Assuming a generalised Gamma distribution for the randomdiffusivity

γgenν,η (D) = ηDν? Γ(ν/η)D

ν−1e−(D/D∗)η , 〈Dn〉 = Dn?

Γ(ν+nη

)Γ(νη

) ,where D? , ν and η are positive constants.

Vittoria Sposini University of Potsdam

Generalised Grey Brownian Motion & Random Diffusivity

Particles diffusing in Brownian fashion in a complex randommedium;

environment properties independent on the diffusing particlesand represented by a random diffusivity.

XggBM =√

2D(t)W (t), W (t) =∫ t

0 ξ(s)ds.

Assuming a generalised Gamma distribution for the randomdiffusivity

γgenν,η (D) = ηDν? Γ(ν/η)D

ν−1e−(D/D∗)η , 〈Dn〉 = Dn?

Γ(ν+nη

)Γ(νη

) ,where D? , ν and η are positive constants.

Vittoria Sposini University of Potsdam

Generalised Grey Brownian Motion & Random Diffusivity

Particles diffusing in Brownian fashion in a complex randommedium;

environment properties independent on the diffusing particlesand represented by a random diffusivity.

XggBM =√

2D(t)W (t), W (t) =∫ t

0 ξ(s)ds.

Assuming a generalised Gamma distribution for the randomdiffusivity

γgenν,η (D) = ηDν? Γ(ν/η)D

ν−1e−(D/D∗)η , 〈Dn〉 = Dn?

Γ(ν+nη

)Γ(νη

) ,where D? , ν and η are positive constants.

Vittoria Sposini University of Potsdam

Generalised Grey Brownian Motion & Random Diffusivity

fggBM(x , t) =

∫ ∞0

G

(x√D, t

)pD(D)

dD√D

=

∫ ∞0

γgenν,η (D)G(x , t|D)dD

' 1

Γ(ν/η)√

4πD?t

(x2

4D?t

) 2ν−η−12(η+1)

exp

[−η + 1

ηη

1η+1

(x2

4D?t

) ηη+1

].

6 4 2 0 2 4 6

x

10-3

10-2

10-1

100

101

f XggBM(x

)

η= 1. 3, ν= 0. 5

t= 0. 05

t= 0. 5

t= 1. 5

30 20 10 0 10 20 30

x

10-4

10-3

10-2

10-1

100

f XggBM(x

)

η= 1. 3, ν= 0. 5

t= 5

t= 25

t= 50

〈x2ggBM(t)〉 =

∫ +∞

−∞x2 fggBM(x , t)dx

= 2t

∫ ∞0

D γgenν,η (D) dD = 2 〈D〉stt.

0 10 20 30 40 50

t

0

5

10

15

20

25

⟨ X2 ggBM

⟩

η= 1. 3, ν= 0. 5

FIT∼ 0. 40 t

Vittoria Sposini University of Potsdam

Generalised Grey Brownian Motion & Random Diffusivity

fggBM(x , t) =

∫ ∞0

G

(x√D, t

)pD(D)

dD√D

=

∫ ∞0

γgenν,η (D)G(x , t|D)dD

' 1

Γ(ν/η)√

4πD?t

(x2

4D?t

) 2ν−η−12(η+1)

exp

[−η + 1

ηη

1η+1

(x2

4D?t

) ηη+1

].

6 4 2 0 2 4 6

x

10-3

10-2

10-1

100

101

f XggBM(x

)

η= 1. 3, ν= 0. 5

t= 0. 05

t= 0. 5

t= 1. 5

30 20 10 0 10 20 30

x

10-4

10-3

10-2

10-1

100

f XggBM(x

)

η= 1. 3, ν= 0. 5

t= 5

t= 25

t= 50

〈x2ggBM(t)〉 =

∫ +∞

−∞x2 fggBM(x , t)dx

= 2t

∫ ∞0

D γgenν,η (D) dD = 2 〈D〉stt.

0 10 20 30 40 50

t

0

5

10

15

20

25

⟨ X2 ggBM

⟩

η= 1. 3, ν= 0. 5

FIT∼ 0. 40 t

Vittoria Sposini University of Potsdam

Generalised Grey Brownian Motion & Random Diffusivity

fggBM(x , t) =

∫ ∞0

G

(x√D, t

)pD(D)

dD√D

=

∫ ∞0

γgenν,η (D)G(x , t|D)dD

' 1

Γ(ν/η)√

4πD?t

(x2

4D?t

) 2ν−η−12(η+1)

exp

[−η + 1

ηη

1η+1

(x2

4D?t

) ηη+1

].

6 4 2 0 2 4 6

x

10-3

10-2

10-1

100

101

f XggBM(x

)

η= 1. 3, ν= 0. 5

t= 0. 05

t= 0. 5

t= 1. 5

30 20 10 0 10 20 30

x

10-4

10-3

10-2

10-1

100

f XggBM(x

)

η= 1. 3, ν= 0. 5

t= 5

t= 25

t= 50

〈x2ggBM(t)〉 =

∫ +∞

−∞x2 fggBM(x , t)dx

= 2t

∫ ∞0

D γgenν,η (D) dD = 2 〈D〉stt.

0 10 20 30 40 50

t

0

5

10

15

20

25

⟨ X2 ggBM

⟩

η= 1. 3, ν= 0. 5

FIT∼ 0. 40 t

Vittoria Sposini University of Potsdam

Generalised Grey Brownian Motion & Random Diffusivity

fggBM(x , t) =

∫ ∞0

G

(x√D, t

)pD(D)

dD√D

=

∫ ∞0

γgenν,η (D)G(x , t|D)dD

' 1

Γ(ν/η)√

4πD?t

(x2

4D?t

) 2ν−η−12(η+1)

exp

[−η + 1

ηη

1η+1

(x2

4D?t

) ηη+1

].

6 4 2 0 2 4 6

x

10-3

10-2

10-1

100

101

f XggBM(x

)

η= 1. 3, ν= 0. 5

t= 0. 05

t= 0. 5

t= 1. 5

30 20 10 0 10 20 30

x

10-4

10-3

10-2

10-1

100

f XggBM(x

)

η= 1. 3, ν= 0. 5

t= 5

t= 25

t= 50

〈x2ggBM(t)〉 =

∫ +∞

−∞x2 fggBM(x , t)dx

= 2t

∫ ∞0

D γgenν,η (D) dD = 2 〈D〉stt.

0 10 20 30 40 50

t

0

5

10

15

20

25

⟨ X2 ggBM

⟩

η= 1. 3, ν= 0. 5

FIT∼ 0. 40 t

Vittoria Sposini University of Potsdam

Generalised Grey Brownian Motion & Random Diffusivity

fggBM(x , t) =

∫ ∞0

G

(x√D, t

)pD(D)

dD√D

=

∫ ∞0

γgenν,η (D)G(x , t|D)dD

' 1

Γ(ν/η)√

4πD?t

(x2

4D?t

) 2ν−η−12(η+1)

exp

[−η + 1

ηη

1η+1

(x2

4D?t

) ηη+1

].

6 4 2 0 2 4 6

x

10-3

10-2

10-1

100

101

f XggBM(x

)

η= 1. 3, ν= 0. 5

t= 0. 05

t= 0. 5

t= 1. 5

30 20 10 0 10 20 30

x

10-4

10-3

10-2

10-1

100

f XggBM(x

)

η= 1. 3, ν= 0. 5

t= 5

t= 25

t= 50

〈x2ggBM(t)〉 =

∫ +∞

−∞x2 fggBM(x , t)dx

= 2t

∫ ∞0

D γgenν,η (D) dD = 2 〈D〉stt.

0 10 20 30 40 50

t

0

5

10

15

20

25

⟨ X2 ggBM

⟩

η= 1. 3, ν= 0. 5

FIT∼ 0. 40 t

Vittoria Sposini University of Potsdam

Generalised Grey Brownian Motion & Random Diffusivity

fggBM(x , t) =

∫ ∞0

G

(x√D, t

)pD(D)

dD√D

=

∫ ∞0

γgenν,η (D)G(x , t|D)dD

' 1

Γ(ν/η)√

4πD?t

(x2

4D?t

) 2ν−η−12(η+1)

exp

[−η + 1

ηη

1η+1

(x2

4D?t

) ηη+1

].

6 4 2 0 2 4 6

x

10-3

10-2

10-1

100

101

f XggBM(x

)

η= 1. 3, ν= 0. 5

t= 0. 05

t= 0. 5

t= 1. 5

30 20 10 0 10 20 30

x

10-4

10-3

10-2

10-1

100

f XggBM(x

)

η= 1. 3, ν= 0. 5

t= 5

t= 25

t= 50

〈x2ggBM(t)〉 =

∫ +∞

−∞x2 fggBM(x , t)dx

= 2t

∫ ∞0

D γgenν,η (D) dD = 2 〈D〉stt.

0 10 20 30 40 50

t

0

5

10

15

20

25

⟨ X2 ggBM

⟩η= 1. 3, ν= 0. 5

FIT∼ 0. 40 t

Vittoria Sposini University of Potsdam

A minimal model for Diffusing Diffusivities

XDD(t) =

∫ t

0

√2D(s) ξ(s)ds,

D(t) = Y2(t),

Y(t) n-dimensional OU process.

t � τD

Brownian yet non-GaussianDiffusion.

t � τD

Gaussian Diffusion.

[Chechkin, A. V. et. al. 2017, Phys. Rev. X, 7]Vittoria Sposini University of Potsdam

A minimal model for Diffusing Diffusivities

XDD(t) =

∫ t

0

√2D(s) ξ(s)ds,

D(t) = Y2(t),

Y(t) n-dimensional OU process.

t � τD

Brownian yet non-GaussianDiffusion.

t � τD

Gaussian Diffusion.

[Chechkin, A. V. et. al. 2017, Phys. Rev. X, 7]Vittoria Sposini University of Potsdam

A minimal model for Diffusing Diffusivities

XDD(t) =

∫ t

0

√2D(s) ξ(s)ds,

D(t) = Y2(t),

Y(t) n-dimensional OU process.

t � τD

Brownian yet non-GaussianDiffusion.

t � τD

Gaussian Diffusion.

[Chechkin, A. V. et. al. 2017, Phys. Rev. X, 7]Vittoria Sposini University of Potsdam

A minimal model for Diffusing Diffusivities

XDD(t) =

∫ t

0

√2D(s) ξ(s)ds,

D(t) = Y2(t),

Y(t) n-dimensional OU process.

t � τD

Brownian yet non-GaussianDiffusion.

t � τD

Gaussian Diffusion.

[Chechkin, A. V. et. al. 2017, Phys. Rev. X, 7]Vittoria Sposini University of Potsdam

A generalise minimal model for Diffusing Diffusivities

XDD(t) =

∫ t

0

√2D(s) ξ(s)ds,

D(t) = y2(t),

dy =σ2

2 y

[2ν − 1− 2 η

(y

y?

)2η]dt + σ dW (t).

For t � τcorr

f STDD (x , t) '∫ +∞

0

pD(D)G(x , t|D)dD

6 4 2 0 2 4 6

x

10-3

10-2

10-1

100

101

f XDD(x

)

η= 1. 3, ν= 0. 5

t= 0. 05

t= 0. 5

t= 1. 5

For t � τcorr

f LTDD (x , t) ' 1√4π〈D〉stt

exp

(− x2

4〈D〉stt

)

20 10 0 10 20

x

10-4

10-3

10-2

10-1

100

f XDD(x

) η= 1. 3, ν= 0. 5

t= 5

t= 25

t= 50

Vittoria Sposini University of Potsdam

A generalise minimal model for Diffusing Diffusivities

XDD(t) =

∫ t

0

√2D(s) ξ(s)ds,

D(t) = y2(t),

dy =σ2

2 y

[2ν − 1− 2 η

(y

y?

)2η]dt + σ dW (t).

For t � τcorr

f STDD (x , t) '∫ +∞

0

pD(D)G(x , t|D)dD

6 4 2 0 2 4 6

x

10-3

10-2

10-1

100

101

f XDD(x

)

η= 1. 3, ν= 0. 5

t= 0. 05

t= 0. 5

t= 1. 5

For t � τcorr

f LTDD (x , t) ' 1√4π〈D〉stt

exp

(− x2

4〈D〉stt

)

20 10 0 10 20

x

10-4

10-3

10-2

10-1

100

f XDD(x

) η= 1. 3, ν= 0. 5

t= 5

t= 25

t= 50

Vittoria Sposini University of Potsdam

A generalise minimal model for Diffusing Diffusivities

XDD(t) =

∫ t

0

√2D(s) ξ(s)ds,

D(t) = y2(t),

dy =σ2

2 y

[2ν − 1− 2 η

(y

y?

)2η]dt + σ dW (t).

For t � τcorr

f STDD (x , t) '∫ +∞

0

pD(D)G(x , t|D)dD

6 4 2 0 2 4 6

x

10-3

10-2

10-1

100

101

f XDD(x

)

η= 1. 3, ν= 0. 5

t= 0. 05

t= 0. 5

t= 1. 5

For t � τcorr

f LTDD (x , t) ' 1√4π〈D〉stt

exp

(− x2

4〈D〉stt

)

20 10 0 10 20

x

10-4

10-3

10-2

10-1

100

f XDD(x

) η= 1. 3, ν= 0. 5

t= 5

t= 25

t= 50

Vittoria Sposini University of Potsdam

A generalise minimal model for Diffusing Diffusivities

XDD(t) =

∫ t

0

√2D(s) ξ(s)ds,

D(t) = y2(t),

dy =σ2

2 y

[2ν − 1− 2 η

(y

y?

)2η]dt + σ dW (t).

For t � τcorr

f STDD (x , t) '∫ +∞

0

pD(D)G(x , t|D)dD

6 4 2 0 2 4 6

x

10-3

10-2

10-1

100

101

f XDD(x

)

η= 1. 3, ν= 0. 5

t= 0. 05

t= 0. 5

t= 1. 5

For t � τcorr

f LTDD (x , t) ' 1√4π〈D〉stt

exp

(− x2

4〈D〉stt

)

20 10 0 10 20

x

10-4

10-3

10-2

10-1

100

f XDD(x

) η= 1. 3, ν= 0. 5

t= 5

t= 25

t= 50

Vittoria Sposini University of Potsdam

A generalise minimal model for Diffusing Diffusivities

XDD(t) =

∫ t

0

√2D(s) ξ(s)ds,

D(t) = y2(t),

dy =σ2

2 y

[2ν − 1− 2 η

(y

y?

)2η]dt + σ dW (t).

For t � τcorr

f STDD (x , t) '∫ +∞

0

pD(D)G(x , t|D)dD

6 4 2 0 2 4 6

x

10-3

10-2

10-1

100

101

f XDD(x

)

η= 1. 3, ν= 0. 5

t= 0. 05

t= 0. 5

t= 1. 5

For t � τcorr

f LTDD (x , t) ' 1√4π〈D〉stt

exp

(− x2

4〈D〉stt

)

20 10 0 10 20

x

10-4

10-3

10-2

10-1

100

f XDD(x

) η= 1. 3, ν= 0. 5

t= 5

t= 25

t= 50

Vittoria Sposini University of Potsdam

A generalise minimal model for Diffusing Diffusivities

For t � τcorr

〈x2DD(t)〉ST =

∫ +∞

−∞x2 f STDD (x , t)dx

= 2t

∫ ∞0

D γgenν,η (D)dD = 2 〈D〉st t.

For t � τcorr

〈x2DD(t)〉LT =

∫ +∞

−∞x2 f LTDD (x , t)dx

=

∫ +∞

−∞x2G (x , t|〈D〉st)dx = 2〈D〉st t.

0 10 20 30 40 50

t

0

5

10

15

20

25

⟨ X2 DD

⟩

η= 1. 3, ν= 0. 5

FIT∼ 0. 41 t

Vittoria Sposini University of Potsdam

A generalise minimal model for Diffusing Diffusivities

For t � τcorr

〈x2DD(t)〉ST =

∫ +∞

−∞x2 f STDD (x , t)dx

= 2t

∫ ∞0

D γgenν,η (D)dD = 2 〈D〉st t.

For t � τcorr

〈x2DD(t)〉LT =

∫ +∞

−∞x2 f LTDD (x , t)dx

=

∫ +∞

−∞x2G (x , t|〈D〉st)dx = 2〈D〉st t.

0 10 20 30 40 50

t

0

5

10

15

20

25

⟨ X2 DD

⟩

η= 1. 3, ν= 0. 5

FIT∼ 0. 41 t

Vittoria Sposini University of Potsdam

A generalise minimal model for Diffusing Diffusivities

For t � τcorr

〈x2DD(t)〉ST =

∫ +∞

−∞x2 f STDD (x , t)dx

= 2t

∫ ∞0

D γgenν,η (D)dD = 2 〈D〉st t.

For t � τcorr

〈x2DD(t)〉LT =

∫ +∞

−∞x2 f LTDD (x , t)dx

=

∫ +∞

−∞x2G (x , t|〈D〉st)dx = 2〈D〉st t.

0 10 20 30 40 50

t

0

5

10

15

20

25

⟨ X2 DD

⟩

η= 1. 3, ν= 0. 5

FIT∼ 0. 41 t

Vittoria Sposini University of Potsdam

A generalise minimal model for Diffusing Diffusivities

For t � τcorr

〈x2DD(t)〉ST =

∫ +∞

−∞x2 f STDD (x , t)dx

= 2t

∫ ∞0

D γgenν,η (D)dD = 2 〈D〉st t.

For t � τcorr

〈x2DD(t)〉LT =

∫ +∞

−∞x2 f LTDD (x , t)dx

=

∫ +∞

−∞x2G (x , t|〈D〉st)dx = 2〈D〉st t.

0 10 20 30 40 50

t

0

5

10

15

20

25

⟨ X2 DD

⟩

η= 1. 3, ν= 0. 5

FIT∼ 0. 41 t

Vittoria Sposini University of Potsdam

The Two Models in Comparison

Generalised grey Brownianmotion (ggBM)

XggBM =√

2D(t)W (t),

X t+1ggBM =

√2Dt+1 W t+1.

0 10 20 30 40 50

t

20

15

10

5

0

5

10

15

20

XggBM

(t)

η= 1. 3, ν= 0. 5

0.0 0.5 1.0

1

0

1

Diffusing Diffusivities(DD)

XDD(t) =

∫ t

0

√2D(s)ξ(s)ds,

X t+1DD = X t

DD +√

2Dt dW t .

0 10 20 30 40 50

t

20

15

10

5

0

5

10

15

XDD(t

)

η= 1. 3, ν= 0. 5

0.0 0.5 1.01

0

1

Vittoria Sposini University of Potsdam

The Two Models in Comparison

Generalised grey Brownianmotion (ggBM)

XggBM =√

2D(t)W (t),

X t+1ggBM =

√2Dt+1 W t+1.

0 10 20 30 40 50

t

20

15

10

5

0

5

10

15

20

XggBM

(t)

η= 1. 3, ν= 0. 5

0.0 0.5 1.0

1

0

1

Diffusing Diffusivities(DD)

XDD(t) =

∫ t

0

√2D(s)ξ(s)ds,

X t+1DD = X t

DD +√

2Dt dW t .

0 10 20 30 40 50

t

20

15

10

5

0

5

10

15

XDD(t

)

η= 1. 3, ν= 0. 5

0.0 0.5 1.01

0

1

Vittoria Sposini University of Potsdam

The Two Models in Comparison

Generalised grey Brownianmotion (ggBM)

XggBM =√

2D(t)W (t),

X t+1ggBM =

√2Dt+1 W t+1.

0 10 20 30 40 50

t

20

15

10

5

0

5

10

15

20

XggBM

(t)

η= 1. 3, ν= 0. 5

0.0 0.5 1.0

1

0

1

Diffusing Diffusivities(DD)

XDD(t) =

∫ t

0

√2D(s)ξ(s)ds,

X t+1DD = X t

DD +√

2Dt dW t .

0 10 20 30 40 50

t

20

15

10

5

0

5

10

15

XDD(t

)

η= 1. 3, ν= 0. 5

0.0 0.5 1.01

0

1

Vittoria Sposini University of Potsdam

The Two Models in Comparison

Generalised grey Brownianmotion (ggBM)

XggBM =√

2D(t)W (t),

X t+1ggBM =

√2Dt+1 W t+1.

0 10 20 30 40 50

t

20

15

10

5

0

5

10

15

20

XggBM

(t)

η= 1. 3, ν= 0. 5

0.0 0.5 1.0

1

0

1

Diffusing Diffusivities(DD)

XDD(t) =

∫ t

0

√2D(s)ξ(s)ds,

X t+1DD = X t

DD +√

2Dt dW t .

0 10 20 30 40 50

t

20

15

10

5

0

5

10

15

XDD(t

)

η= 1. 3, ν= 0. 5

0.0 0.5 1.01

0

1

Vittoria Sposini University of Potsdam

The Two Models in Comparison

Generalised grey Brownianmotion (ggBM)

XggBM =√

2D(t)W (t),

X t+1ggBM =

√2Dt+1 W t+1.

0 10 20 30 40 50

t

20

15

10

5

0

5

10

15

20

XggBM

(t)

η= 1. 3, ν= 0. 5

0.0 0.5 1.0

1

0

1

Diffusing Diffusivities(DD)

XDD(t) =

∫ t

0

√2D(s)ξ(s)ds,

X t+1DD = X t

DD +√

2Dt dW t .

0 10 20 30 40 50

t

20

15

10

5

0

5

10

15

XDD(t

)

η= 1. 3, ν= 0. 5

0.0 0.5 1.01

0

1

Vittoria Sposini University of Potsdam

The Two Models in Comparison

In common

non-Gaussian dynamics forshort times;

ubiquitous linear trend ofthe variance.

Not in common

DD approaches a standarddiffusion at long times;

ggBM does not displaychanges in its trend.

0 10 20 30 40 50

t

2

4

6

8

10

12

Kurtosis

η=1. 3, ν=0. 5

XDD XggBM

K = 3Γ(ν+2η

)Γ(νη

)Γ(ν+1η

)2

Vittoria Sposini University of Potsdam

The Two Models in Comparison

In common

non-Gaussian dynamics forshort times;

ubiquitous linear trend ofthe variance.

Not in common

DD approaches a standarddiffusion at long times;

ggBM does not displaychanges in its trend.

0 10 20 30 40 50

t

2

4

6

8

10

12

Kurtosis

η=1. 3, ν=0. 5

XDD XggBM

K = 3Γ(ν+2η

)Γ(νη

)Γ(ν+1η

)2

Vittoria Sposini University of Potsdam

The Two Models in Comparison

In common

non-Gaussian dynamics forshort times;

ubiquitous linear trend ofthe variance.

Not in common

DD approaches a standarddiffusion at long times;

ggBM does not displaychanges in its trend.

0 10 20 30 40 50

t

2

4

6

8

10

12

Kurtosis

η=1. 3, ν=0. 5

XDD XggBM

K = 3Γ(ν+2η

)Γ(νη

)Γ(ν+1η

)2

Vittoria Sposini University of Potsdam

The Two Models in Comparison

In common

non-Gaussian dynamics forshort times;

ubiquitous linear trend ofthe variance.

Not in common

DD approaches a standarddiffusion at long times;

ggBM does not displaychanges in its trend.

0 10 20 30 40 50

t

2

4

6

8

10

12

Kurtosis

η=1. 3, ν=0. 5

XDD XggBM

K = 3Γ(ν+2η

)Γ(νη

)Γ(ν+1η

)2

Vittoria Sposini University of Potsdam

Conclusions

Broad spectrum of distributions for the random diffusivity thatallows us to reproduce a broad spectrum of distributions forthe particles displacement also.

Proposal of two models suitable for different physical and/orbiological systems exhibiting Brownian yet non-gaussiandiffusion.

Development, generalisation and comparison of two modelsalready known in literature.

Introduction of non equilibrium initial condition for thediffusivity dynamics.

Possibility to generalise the study to fractional Gaussian noise,introducing thus an anomalous spreading.

Vittoria Sposini University of Potsdam

Conclusions

Broad spectrum of distributions for the random diffusivity thatallows us to reproduce a broad spectrum of distributions forthe particles displacement also.

Proposal of two models suitable for different physical and/orbiological systems exhibiting Brownian yet non-gaussiandiffusion.

Development, generalisation and comparison of two modelsalready known in literature.

Introduction of non equilibrium initial condition for thediffusivity dynamics.

Possibility to generalise the study to fractional Gaussian noise,introducing thus an anomalous spreading.

Vittoria Sposini University of Potsdam

Conclusions

Broad spectrum of distributions for the random diffusivity thatallows us to reproduce a broad spectrum of distributions forthe particles displacement also.

Proposal of two models suitable for different physical and/orbiological systems exhibiting Brownian yet non-gaussiandiffusion.

Development, generalisation and comparison of two modelsalready known in literature.

Introduction of non equilibrium initial condition for thediffusivity dynamics.

Possibility to generalise the study to fractional Gaussian noise,introducing thus an anomalous spreading.

Vittoria Sposini University of Potsdam

Conclusions

Broad spectrum of distributions for the random diffusivity thatallows us to reproduce a broad spectrum of distributions forthe particles displacement also.

Proposal of two models suitable for different physical and/orbiological systems exhibiting Brownian yet non-gaussiandiffusion.

Development, generalisation and comparison of two modelsalready known in literature.

Introduction of non equilibrium initial condition for thediffusivity dynamics.

Possibility to generalise the study to fractional Gaussian noise,introducing thus an anomalous spreading.

Vittoria Sposini University of Potsdam

Conclusions

Broad spectrum of distributions for the random diffusivity thatallows us to reproduce a broad spectrum of distributions forthe particles displacement also.

Proposal of two models suitable for different physical and/orbiological systems exhibiting Brownian yet non-gaussiandiffusion.

Development, generalisation and comparison of two modelsalready known in literature.

Introduction of non equilibrium initial condition for thediffusivity dynamics.

Possibility to generalise the study to fractional Gaussian noise,introducing thus an anomalous spreading.

Vittoria Sposini University of Potsdam

Conclusions

Broad spectrum of distributions for the random diffusivity thatallows us to reproduce a broad spectrum of distributions forthe particles displacement also.

Proposal of two models suitable for different physical and/orbiological systems exhibiting Brownian yet non-gaussiandiffusion.

Development, generalisation and comparison of two modelsalready known in literature.

Introduction of non equilibrium initial condition for thediffusivity dynamics.

Possibility to generalise the study to fractional Gaussian noise,introducing thus an anomalous spreading.

Vittoria Sposini University of Potsdam

Vittoria Sposini University of Potsdam