Statistical Properties of Functionals of the Paths of a Particle - - PowerPoint PPT Presentation

Statistical Properties of Functionals

• f the Paths of a Particle

Diffusing in a One-Dimensional Random Potential

(Occupation Time & Inverse Occupation Time) Sanjib Sabhapandit

Laboratoire de Physique Th´ eorique et Mod` eles Statistiques, Universit´ e Paris-Sud, France

Collaborators Satya N. Majumdar Alain Comtet

• Ref. Phys. Rev. E 73, 051102 (2006).
• S. Sabhapandit (Universit´

e Paris-Sud) Functionals of a particle in 1D random potential 1 / 24

Occupation Time of a one-dimensional Brownian motion

The position x(τ) evolves from x(0) = 0, via dx dτ = η(τ), where η(τ) = 0 and η(τ)η(τ′) = δ(τ − τ′). The functional T =

t θ (x(τ)) dτ

is known as Occupation time.

τ

T

θ (x(τ))

x(τ)

• S. Sabhapandit (Universit´

e Paris-Sud) Functionals of a particle in 1D random potential 2 / 24

Occupation Time of a one-dimensional Brownian motion

The position x(τ) evolves from x(0) = 0, via dx dτ = η(τ), where η(τ) = 0 and η(τ)η(τ′) = δ(τ − τ′). The functional T =

t θ (x(τ)) dτ

is known as Occupation time. For a fixed t, the value of T is random – depends on [{x(τ)}, for 0 ≤ τ ≤ t]. P(T|t) =?

τ

T

θ (x(τ))

x(τ)

• S. Sabhapandit (Universit´

e Paris-Sud) Functionals of a particle in 1D random potential 2 / 24

L´ evy’s arcsine law

[L´

evy, 1939]

For the occupation time of a one-dimensional Brownian motion: Integrated distribution

T

P(T ′|t) dT ′ = 2

π

arcsin

• T

t

• Probability density function

P(T|t) = 1

π

• T(t − T)

,

0 < T < t.

1 0.5 3 2 1

t P(T|t) T/t

The scaling form: P(T|t) = 1 t f

• T

t

• ,

where f (x) = 1

π

• x(1 − x)

,

0 < x < 1. Brownian particle “tends” to stay on one side of the origin.

• S. Sabhapandit (Universit´

e Paris-Sud) Functionals of a particle in 1D random potential 3 / 24

Why do physicist care about Occupation time?

Some applications of occupation time: Fluorescence intermittency in single cadmium selenide (CdSe)

• nanocrystals. (Blinking quantum dots).

ψ(τ) ∝ τ−(1+θ) θ ∼ 1/2

100 200 300 400 500 600 Time (s) Intensity (a.u.)

[Nirmal et al., Nature 383, 802 (1996)] [Brokmann et al., PRL 90, 120601 (2003)]

• S. Sabhapandit (Universit´

e Paris-Sud) Functionals of a particle in 1D random potential 4 / 24

Why do physicist care about Occupation time?

Some applications of occupation time: Fluorescence intermittency in single cadmium selenide (CdSe)

• nanocrystals. (Blinking quantum dots).

ψ(τ) ∝ τ−(1+θ) θ ∼ 1/2

100 200 300 400 500 600 Time (s) Intensity (a.u.)

[Nirmal et al., Nature 383, 802 (1996)] [Brokmann et al., PRL 90, 120601 (2003)] Persistence ∝ P(T|t), in the limit T → 0 or T → t.

• S. Sabhapandit (Universit´

e Paris-Sud) Functionals of a particle in 1D random potential 4 / 24

Why do physicist care about Occupation time?

Some applications of occupation time: Fluorescence intermittency in single cadmium selenide (CdSe)

• nanocrystals. (Blinking quantum dots).

ψ(τ) ∝ τ−(1+θ) θ ∼ 1/2

100 200 300 400 500 600 Time (s) Intensity (a.u.)

[Nirmal et al., Nature 383, 802 (1996)] [Brokmann et al., PRL 90, 120601 (2003)] Persistence ∝ P(T|t), in the limit T → 0 or T → t. Diffusion controlled reactions activated by immobile catalytic sites. A + B + C → B + C [B´

enichou et al., JPA 38, 7205 (2005)]

• S. Sabhapandit (Universit´

e Paris-Sud) Functionals of a particle in 1D random potential 4 / 24

Why do physicist care about Occupation time?

Some applications of occupation time: Fluorescence intermittency in single cadmium selenide (CdSe)

• nanocrystals. (Blinking quantum dots).

ψ(τ) ∝ τ−(1+θ) θ ∼ 1/2

100 200 300 400 500 600 Time (s) Intensity (a.u.)

[Nirmal et al., Nature 383, 802 (1996)] [Brokmann et al., PRL 90, 120601 (2003)] Persistence ∝ P(T|t), in the limit T → 0 or T → t. Diffusion controlled reactions activated by immobile catalytic sites. A + B + C → B + C [B´

enichou et al., JPA 38, 7205 (2005)]

Temperature fluctuations in weather records. [Majumdar & Bray, PRE 65, 051112 (2002)]

• S. Sabhapandit (Universit´

e Paris-Sud) Functionals of a particle in 1D random potential 4 / 24

Why do physicist care about Occupation time?

Some applications of occupation time: Fluorescence intermittency in single cadmium selenide (CdSe)

• nanocrystals. (Blinking quantum dots).

ψ(τ) ∝ τ−(1+θ) θ ∼ 1/2

100 200 300 400 500 600 Time (s) Intensity (a.u.)

[Nirmal et al., Nature 383, 802 (1996)] [Brokmann et al., PRL 90, 120601 (2003)] Persistence ∝ P(T|t), in the limit T → 0 or T → t. Diffusion controlled reactions activated by immobile catalytic sites. A + B + C → B + C [B´

enichou et al., JPA 38, 7205 (2005)]

Temperature fluctuations in weather records. [Majumdar & Bray, PRE 65, 051112 (2002)] Morphology of growing surfaces. T(x, t) =

t

0 θ(h(x, t′)) dt′.

[Toroczkai et al., PRE 60, R1115 (1999)]

• S. Sabhapandit (Universit´

e Paris-Sud) Functionals of a particle in 1D random potential 4 / 24

Subject of the talk

Consider the occupation time T =

t θ (x(τ)) dτ

where x(τ) is the position of a particle diffusing in an external random medium.

U(x) x

②

P(T|t) differs from sample to sample.

P(T|t) =?

• S. Sabhapandit (Universit´

e Paris-Sud) Functionals of a particle in 1D random potential 5 / 24

Outline

1

Definition of the model.

2

Inverse occupation time.

3

Results for occupation time and inverse occupation time.

4

Outline of the formalism.

5

Summary.

6

Open directions for future research.

• S. Sabhapandit (Universit´

e Paris-Sud) Functionals of a particle in 1D random potential 6 / 24

The model

The position x(τ) evolves from x(0) = 0, via dx dτ = − d dx U (x(τ)) + η(τ), where η(τ) = 0 and η(τ)η(τ′) = δ(τ − τ′). U(x) = Ud(x) + Ur (x) deterministic

− − →

random (quenched)

− − − →

• S. Sabhapandit (Universit´

e Paris-Sud) Functionals of a particle in 1D random potential 7 / 24

The model

The position x(τ) evolves from x(0) = 0, via dx dτ = − d dx U (x(τ)) + η(τ), where η(τ) = 0 and η(τ)η(τ′) = δ(τ − τ′). U(x) = Ud(x) + Ur (x) deterministic

− − →

random (quenched)

− − − →

Ud(x) = −µ|x|

• S. Sabhapandit (Universit´

e Paris-Sud) Functionals of a particle in 1D random potential 7 / 24

The model

The position x(τ) evolves from x(0) = 0, via dx dτ = − d dx U (x(τ)) + η(τ), where η(τ) = 0 and η(τ)η(τ′) = δ(τ − τ′). U(x) = Ud(x) + Ur (x) deterministic

− − →

random (quenched)

− − − →

Ud(x) = −µ|x| Sinai potential

− dUr (x)

dx = √σξ(x)

ξ(x)ξ(x′) = δ(x − x′)

• S. Sabhapandit (Universit´

e Paris-Sud) Functionals of a particle in 1D random potential 7 / 24

The model

The position x(τ) evolves from x(0) = 0, via dx dτ = − d dx U (x(τ)) + η(τ), where η(τ) = 0 and η(τ)η(τ′) = δ(τ − τ′). U(x) = Ud(x) + Ur (x) deterministic

− − →

random (quenched)

− − − →

Ud(x) = −µ|x| Sinai potential

− dUr (x)

dx = √σξ(x)

ξ(x)ξ(x′) = δ(x − x′)

µ > 0 U(x) x

②

µ = 0 U(x) x

②

µ < 0 U(x) x

②

• S. Sabhapandit (Universit´

e Paris-Sud) Functionals of a particle in 1D random potential 7 / 24

The model

The position x(τ) evolves from x(0) = 0, via dx dτ = − d dx U (x(τ)) + η(τ), where η(τ) = 0 and η(τ)η(τ′) = δ(τ − τ′). U(x) = Ud(x) + Ur (x) deterministic

− − →

random (quenched)

− − − →

Ud(x) = −µ|x| Sinai potential

− dUr (x)

dx = √σξ(x)

ξ(x)ξ(x′) = δ(x − x′)

T =

t θ (x(τ)) dτ

P(T|t) =?

µ > 0 U(x) x

②

µ = 0 U(x) x

②

µ < 0 U(x) x

②

• S. Sabhapandit (Universit´

e Paris-Sud) Functionals of a particle in 1D random potential 7 / 24

Inverse Occupation Time

T =

t θ (x(τ)) dτ

For fixed T and given {x(τ)}, the time t is called inverse occupation time. I(t|T) =?

T t I(t|T)

For Brownian motion I(t|T) =

√

T

πt√

t − T What is I(t|T) for particle in random potential?

τ

T

θ (x(τ))

x(τ) t1 t2

• S. Sabhapandit (Universit´

e Paris-Sud) Functionals of a particle in 1D random potential 8 / 24

Sinai potential

µ = 0 U(x) x

②

P(T|t) = RL(T|t) + RL(t − T|t) RL(T|t)

t→∞

− − − →

1 log t R(T)

t

lim t → ∞

log t P(T|t) T

t/2

• ←

− − − σ√

2

√ πT

1 2T

← − − − −

Inverse occupation time I(t|T)

T→∞

− − − − →

t>T

1 log T I3(t − T) T t log T I(t|T)

← − − − σ√

2

• π(t − T)

1 2(t − T)

← − − −

• S. Sabhapandit (Universit´

e Paris-Sud) Functionals of a particle in 1D random potential 9 / 24

Occupation time for unstable potential

σ = 0

P(T|t) = R(T|t) + R(t − T|t) R(T|t)

t→∞

− − − → R(T)

µ > 0 U(x) x

②

t

lim t → ∞

P(T|t) T

t/2

• ←

− − − µ√

2

√ πT √

2 9µ√π e−µ2T/2 T 3/2

← − − −

• S. Sabhapandit (Universit´

e Paris-Sud) Functionals of a particle in 1D random potential 10 / 24

Occupation time for unstable potential

σ = 0

P(T|t) = R(T|t) + R(t − T|t) R(T|t)

t→∞

− − − → R(T)

µ > 0 U(x) x

②

t

lim t → ∞

P(T|t) T

t/2

• ←

− − − µ√

2

√ πT √

2 9µ√π e−µ2T/2 T 3/2

← − − − σ > 0

P(T|t) = R(T|t) + R(t − T|t) R(T|t)

t→∞

− − − → R(T)   

R(T) ≈ µ√ 2

√ πT ,

for small T R(T) ∼ e−bT, for large T

• S. Sabhapandit (Universit´

e Paris-Sud) Functionals of a particle in 1D random potential 10 / 24

Inverse occupation time for unstable potential

σ = 0

I(t|T)

T→∞

− − − − →

t>T

I1(t − T) + 1 2δ(t − ∞)

µ > 0 U(x) x

②

I(t|T)

LARGE

t T ← − − − µ√

2

• π(t − T)

√

2 9µ√π e−µ2(t−T)/2 (t − T)3/2

← − − − σ > 0

I(t|T)

T→∞

− − − − →

t>T

I4(t − T) + 1 2δ(t − ∞)

    

I4(τ) ≈ µ√ 2

√πτ ,

for small τ. I4(τ) ∼ e−bτ, for large τ.

• S. Sabhapandit (Universit´

e Paris-Sud) Functionals of a particle in 1D random potential 11 / 24

Stable potential, without disorder (σ = 0)

System becomes ergodic at large t. p(x) = e−2U(x) Z

,

where Z =

∞

−∞

e−2U(x) dx.

• S. Sabhapandit (Universit´

e Paris-Sud) Functionals of a particle in 1D random potential 12 / 24

Stable potential, without disorder (σ = 0)

System becomes ergodic at large t. p(x) = e−2U(x) Z

,

where Z =

∞

−∞

e−2U(x) dx.

θ[x(t)] → Z+

Z where Z+ =

∞

e−2U(x) dx.

• S. Sabhapandit (Universit´

e Paris-Sud) Functionals of a particle in 1D random potential 12 / 24

Stable potential, without disorder (σ = 0)

System becomes ergodic at large t. p(x) = e−2U(x) Z

,

where Z =

∞

−∞

e−2U(x) dx.

θ[x(t)] → Z+

Z where Z+ =

∞

e−2U(x) dx.

T = t θ[x(t′)] dt′ →

• Z+

Z

• t.

[if U(x) is symmetric, T = t/2]

• S. Sabhapandit (Universit´

e Paris-Sud) Functionals of a particle in 1D random potential 12 / 24

Stable potential, without disorder (σ = 0)

System becomes ergodic at large t. p(x) = e−2U(x) Z

,

where Z =

∞

−∞

e−2U(x) dx.

θ[x(t)] → Z+

Z where Z+ =

∞

e−2U(x) dx.

T = t θ[x(t′)] dt′ →

• Z+

Z

• t.

[if U(x) is symmetric, T = t/2] For T near T, the random variables θ[x(t′)] − θ[x(t′)] at different times, become only weakly correlated. Therefore, when t ≫ correlation time P(T|t) ∼ exp

• − (T − T)2

2∆2

• ,

where ∆2 = T 2 − T2. For U(x) = |µ| |x|, ∆2 = t/4µ2.

• S. Sabhapandit (Universit´

e Paris-Sud) Functionals of a particle in 1D random potential 12 / 24

Stable potential, without disorder (σ = 0)

System becomes ergodic at large t. p(x) = e−2U(x) Z

,

where Z =

∞

−∞

e−2U(x) dx.

θ[x(t)] → Z+

Z where Z+ =

∞

e−2U(x) dx.

T = t θ[x(t′)] dt′ →

• Z+

Z

• t.

[if U(x) is symmetric, T = t/2] For T near T, the random variables θ[x(t′)] − θ[x(t′)] at different times, become only weakly correlated. Therefore, when t ≫ correlation time P(T|t) ∼ exp

• − (T − T)2

2∆2

• ,

where ∆2 = T 2 − T2. For U(x) = |µ| |x|, ∆2 = t/4µ2. Inverse occupation time I(t|T) = I2(t − T, T) θ(t − T) I2(τ, T)

large T

− − − − → |µ|T √

2πτ 3 exp

• −µ2(τ − T)2

2τ

• S. Sabhapandit (Universit´

e Paris-Sud) Functionals of a particle in 1D random potential 12 / 24

Stable potential with disorder (σ > 0,

ν = |µ|/σ)

µ < 0 U(x) x

Occupation time P(T|t)

t→∞,T→∞

− − − − − − − →

T/t fixed

1 t fo(T/t) Inverse occupation time I(t|T)

t→∞,T→∞

− − − − − − − →

t/T fixed

1 T go(t/T)

• S. Sabhapandit (Universit´

e Paris-Sud) Functionals of a particle in 1D random potential 13 / 24

Stable potential with disorder (σ > 0,

ν = |µ|/σ)

µ < 0 U(x) x

Occupation time P(T|t)

t→∞,T→∞

− − − − − − − →

T/t fixed

1 t fo(T/t) fo(y) = 1 B(ν, ν) [y(1 − y)]ν−1 (0 ≤ y ≤ 1)

0.5 1 1 2

y fo(y)

← − ν > 1

↑

ν < 1

Inverse occupation time I(t|T)

t→∞,T→∞

− − − − − − − →

t/T fixed

1 T go(t/T) go(x) = 1 B(ν, ν) (x − 1)ν−1 x2ν (x ≥ 1) 1 2 3 4 5 0.5 1 x go(x)

← − ν > 1 ← − ν < 1

(1) Phase transition at ν = 1. (2) ν = 1/2 (µ = 0, σ = 0)!

• S. Sabhapandit (Universit´

e Paris-Sud) Functionals of a particle in 1D random potential 13 / 24

General formalism

Langevin equation: dx dτ = F (x(τ)) + η(τ),

η(τ) = 0, η(τ)η(τ′) = δ(τ − τ′).

Starting position x(0) = y. Consider T =

t

V (x(τ)) dτ. [Occupation time: V (x) = θ(x)] How does one compute P(T|t, y)? [P(T|t, 0) ≡ P(T|t)]

• S. Sabhapandit (Universit´

e Paris-Sud) Functionals of a particle in 1D random potential 14 / 24

General formalism

Langevin equation: dx dτ = F (x(τ)) + η(τ),

η(τ) = 0, η(τ)η(τ′) = δ(τ − τ′).

Starting position x(0) = y. Consider T =

t

V (x(τ)) dτ. [Occupation time: V (x) = θ(x)] How does one compute P(T|t, y)? [P(T|t, 0) ≡ P(T|t)]

1

Laplace Transform: Qp(y, t) =

∞

e−pT P(T|t, y) dT =

• e−pT

x(0)=y

satisfies the Backward Fokker-Planck equation

∂Qp ∂t

= 1 2

∂2Qp ∂y 2

+ F(y)∂Qp

∂y − p V (y)Qp,

with Qp(y, 0) = 1.

• S. Sabhapandit (Universit´

e Paris-Sud) Functionals of a particle in 1D random potential 14 / 24

General formalism

Langevin equation: dx dτ = F (x(τ)) + η(τ),

η(τ) = 0, η(τ)η(τ′) = δ(τ − τ′).

Starting position x(0) = y. Consider T =

t

V (x(τ)) dτ. [Occupation time: V (x) = θ(x)] How does one compute P(T|t, y)? [P(T|t, 0) ≡ P(T|t)]

1

Laplace Transform: Qp(y, t) =

∞

e−pT P(T|t, y) dT =

• e−pT

x(0)=y

satisfies the Backward Fokker-Planck equation

∂Qp ∂t

= 1 2

∂2Qp ∂y 2

+ F(y)∂Qp

∂y − p V (y)Qp,

with Qp(y, 0) = 1.

2

Laplace transform: u(y) =

∞

e−αt Qp(y, t) dt 1 2 u′′(y) + F(y)u′(y) − [α + pV (y)] u(y) = −1,

• S. Sabhapandit (Universit´

e Paris-Sud) Functionals of a particle in 1D random potential 14 / 24

General formalism

Langevin equation: dx dτ = F (x(τ)) + η(τ),

η(τ) = 0, η(τ)η(τ′) = δ(τ − τ′).

Starting position x(0) = y. Consider T =

t

V (x(τ)) dτ. [Occupation time: V (x) = θ(x)] How does one compute P(T|t, y)? [P(T|t, 0) ≡ P(T|t)]

1

Laplace Transform: Qp(y, t) =

∞

e−pT P(T|t, y) dT =

• e−pT

x(0)=y

satisfies the Backward Fokker-Planck equation

∂Qp ∂t

= 1 2

∂2Qp ∂y 2

+ F(y)∂Qp

∂y − p V (y)Qp,

with Qp(y, 0) = 1.

2

Laplace transform: u(y) =

∞

e−αt Qp(y, t) dt 1 2 u′′(y) + F(y)u′(y) − [α + pV (y)] u(y) = −1,

3

Inverse Laplace transform: L−1

α,p{u(0)} = P(T|t)

• S. Sabhapandit (Universit´

e Paris-Sud) Functionals of a particle in 1D random potential 14 / 24

.....in the context of occupation time V (x) = θ(x)

1

    

y > 0 : 1 2 u′′

+(y) + F(y)u′ +(y) − (α + p) u+(y) = −1,

u+(∞) = 1

α + p ,

y < 0 : 1 2 u′′

−(y) + F(y)u′ −(y) − α u−(y) = −1,

u−(−∞) = 1

α,

Matching conditions at y = 0: (a) u+(0) = u−(0) = u(0), (b) u′

+(0) = u′

−(0).

• S. Sabhapandit (Universit´

e Paris-Sud) Functionals of a particle in 1D random potential 15 / 24

.....in the context of occupation time V (x) = θ(x)

1

    

y > 0 : 1 2 u′′

+(y) + F(y)u′ +(y) − (α + p) u+(y) = −1,

u+(∞) = 1

α + p ,

y < 0 : 1 2 u′′

−(y) + F(y)u′ −(y) − α u−(y) = −1,

u−(−∞) = 1

α,

Matching conditions at y = 0: (a) u+(0) = u−(0) = u(0), (b) u′

+(0) = u′

−(0).

Homogeneous equation

       

• u+(y) =

1

α + p

+ A v+(y) u−(y) = 1

α

+ B v−(y)

2

    

y > 0 : 1 2 v ′′

+(y) + F(y)v ′ +(y) − (α + p) v+(y) = 0,

v+(∞) = 0, y < 0 : 1 2 v ′′

−(y) + F(y)v ′ −(y) − α v−(y) = 0,

v−(−∞) = 0.

• S. Sabhapandit (Universit´

e Paris-Sud) Functionals of a particle in 1D random potential 15 / 24

.....in the context of occupation time V (x) = θ(x)

1

    

y > 0 : 1 2 u′′

+(y) + F(y)u′ +(y) − (α + p) u+(y) = −1,

u+(∞) = 1

α + p ,

y < 0 : 1 2 u′′

−(y) + F(y)u′ −(y) − α u−(y) = −1,

u−(−∞) = 1

α,

Matching conditions at y = 0: (a) u+(0) = u−(0) = u(0), (b) u′

+(0) = u′

−(0).

Homogeneous equation

       

• u+(y) =

1

α + p

+ A v+(y) u−(y) = 1

α

+ B v−(y)

2

    

y > 0 : 1 2 v ′′

+(y) + F(y)v ′ +(y) − (α + p) v+(y) = 0,

v+(∞) = 0, y < 0 : 1 2 v ′′

−(y) + F(y)v ′ −(y) − α v−(y) = 0,

v−(−∞) = 0. Matching conditions u(0) = 1

α

• z−(0)

z−(0) − z+(0)

• +

1

α + p

• −z+(0)

z−(0) − z+(0)

• ,

z±(y) = v ′

±(y)

v±(y).

• S. Sabhapandit (Universit´

e Paris-Sud) Functionals of a particle in 1D random potential 15 / 24

Example of Brownian motion

    

y > 0 : 1 2 v ′′

+(y) − (α + p) v+(y) = 0,

v+(∞) = 0, y < 0 : 1 2 v ′′

−(y) + −α v−(y) = 0,

v−(−∞) = 0. Solutions:

  

v+(y) = v+(0) exp

• −y
• 2(α + p)
• v−(y) = v−(0) exp
• y

√

2α

• z+(0) =

v ′

+(0)

v+(0) = −

• 2(α + p),

z−(0) = v ′

−(0)

v−(0) =

√

2α

∞

dt e−αt

t

dT e−pT P(T|t) = u(0) = 1

• α(α + p)

.

• L−1

s

• 1

√

s + b

• (τ) =

1

√πτ

e−bτ

• P(T|t) =

1

π

• T(t − T)

,

0 < T < t.

• S. Sabhapandit (Universit´

e Paris-Sud) Functionals of a particle in 1D random potential 16 / 24

Computing the disorder average

u(0) = 1

α

• z−(0)

z−(0) − z+(0)

• +

1

α + p

• −z+(0)

z−(0) − z+(0)

• ,

z±(y) = v ′

±(y)

v±(y). Riccati equations

• y > 0 :

z′

+(y) = −z2 +(y) − 2F(y)z+(y) + 2(α + p),

y < 0 : z′

−(y) = −z2 −(y) − 2F(y)z−(y) + 2α.

Disorder average

∞

dt e−αt

∞

dT e−pT P(T|t) = u(0) = 1

α

• z−

z− − z+

• +

1

α + p

• −z+

z− − z+

• P(z+) P(z−) dz+ dz−,

where z± ≡ z±(0).

• S. Sabhapandit (Universit´

e Paris-Sud) Functionals of a particle in 1D random potential 17 / 24

Disorder average for our random potential

F(x) = µ sign(x) + √σξ(x),

ξ(x) = 0

and ξ(x)ξ(x′) = δ(x − x′) Consider −∞ < y < 0: z′

−(y) = −z2 −(y) − 2[−µ + √σξ(y)]z−(y) + 2α.

• S. Sabhapandit (Universit´

e Paris-Sud) Functionals of a particle in 1D random potential 18 / 24

Disorder average for our random potential

F(x) = µ sign(x) + √σξ(x),

ξ(x) = 0

and ξ(x)ξ(x′) = δ(x − x′) Consider −∞ < y < 0: z′

−(y) = −z2 −(y) − 2[−µ + √σξ(y)]z−(y) + 2α.

Transformation: z− = eφ, and y = τ (time: −∞ → 0) dφ dτ = − dUcl dφ + ˜

ξ(τ)

˜

ξ(τ)˜ ξ(τ′) = 2(2σ)δ(τ−τ′)

[−∞ < τ < 0]. Ucl(φ) = eφ + 2αe−φ − 2µφ − (2α + 1).

• S. Sabhapandit (Universit´

e Paris-Sud) Functionals of a particle in 1D random potential 18 / 24

Disorder average for our random potential

F(x) = µ sign(x) + √σξ(x),

ξ(x) = 0

and ξ(x)ξ(x′) = δ(x − x′) Consider −∞ < y < 0: z′

−(y) = −z2 −(y) − 2[−µ + √σξ(y)]z−(y) + 2α.

Transformation: z− = eφ, and y = τ (time: −∞ → 0) dφ dτ = − dUcl dφ + ˜

ξ(τ)

˜

ξ(τ)˜ ξ(τ′) = 2(2σ)δ(τ−τ′)

[−∞ < τ < 0]. Ucl(φ) = eφ + 2αe−φ − 2µφ − (2α + 1). Equilibrium when τ → 0. Pst(φ) = 1 Z exp

• − 1

2σ Ucl(φ)

• .

Back transformation: φ = log z− and τ = y P (z−) = 1 Ω zµ/σ−1

−

exp

• − 1

2σ

• z− + 2α

z−

• ,

Ω = 2(2α)µ/2σKµ/σ

√

2α

σ

• .
• S. Sabhapandit (Universit´

e Paris-Sud) Functionals of a particle in 1D random potential 18 / 24

Summary

If the external potential is not stable, disorder makes the motion more stiff. For stable potential,

◮ phase transitions in the ergodicity at |µ|/σ = 1. ◮ |µ|/σ = 1/2

Brownian motion results for occupation time.

• S. Sabhapandit (Universit´

e Paris-Sud) Functionals of a particle in 1D random potential 19 / 24

Open directions

Distribution of P(T|t) with respect to disorder. O(1) random potential. Non-Markovian stochastic process in random media. Higher dimensions.

• S. Sabhapandit (Universit´

e Paris-Sud) Functionals of a particle in 1D random potential 20 / 24

END

• S. Sabhapandit (Universit´

e Paris-Sud) Functionals of a particle in 1D random potential 21 / 24

1

TABLE I: Flat potential. — Disorder averaged pdf’s of the local, inverse local, occupation and inverse occupation times of a particle starting at the origin, diffusing in the Sinai potential U(x) = √σB(x), where B(x) represents the trajectory of a Brownian motion in space with the initial condition B(0) = 0. PURE CASE (σ = 0) DISORDERED CASE (σ > 0) Ploc(T|t) = √ 2 √ πt exp » −T 2 2t – Ploc(T|t)

t→∞,T →∞

− − − − − − − − →

T/t fixed

1 t log2 tf1(T/t) f1(y) = 2 y e−y/σK0 (y/σ) Iloc(t|T) = T √ 2πt3 exp » −T 2 2t – Iloc(t|T)

t→∞,T →∞

− − − − − − − − →

t/T fixed

1 T log2 T g1(t/T) g1(x) = 2 x e−1/σxK0 (1/σx) Pocc(T|t) = 1 π p T(t − T) , 0 < T < t Pocc(T|t) = RL(T|t) + RL(t − T|t) RL(T|t)

t→∞

− − − → 1 log tR(T) R(T) ≈ √ 2σ √ πT , for small T R(T) ∼ 1 2T , for large T Iocc(t|T) = √ T πt √ t − T θ(t − T) Iocc(t|T)

T →∞

− − − − →

t>T

1 log T I3(t − T) I3(τ) ≈ √ 2σ √πτ , for small τ I3(τ) ∼ 1 2τ , for large τ

• S. Sabhapandit (Universit´

e Paris-Sud) Functionals of a particle in 1D random potential 22 / 24

2 TABLE II: Unstable potential. — Disorder averaged pdf’s of the local, inverse local, occupation and inverse occupation times

• f a particle starting at the origin, diffusing in the unstable random potential U(x) = −µ|x| + √σB(x), where µ > 0 and B(x)

represents the trajectory of a Brownian motion in space with the initial condition B(0) = 0. We denote ν = µ/σ. PURE CASE (σ = 0) DISORDERED CASE (σ > 0) Ploc(T|t)

t→∞

− − − → Ploc(T) , Ploc(T) = 2µe−2µT Ploc(T|t)

t→∞

− − − → Ploc(T), Ploc(T) = 2µ(1 + σT)−(2ν+1) Iloc(t|T) = T √ 2πt3 exp » −(T + µt)2 2t – + (1 − exp[−2µT]) δ(t − ∞) Iloc(t|T)

t→∞,T →∞

− − − − − − − − →

t/T fixed

1 T 2ν+1 g2(t/T) + ` 1 − [1 + σT]−2ν´ δ(t − ∞) g2(x) = » 2√π σ2ν−1Γ2(ν) – e−2/σx (σx)2ν+1 U(1/2, 1 + ν, 2/σx) Pocc(T|t) = RL(T|t) + RL(t − T|t) RL(T|t)

t→∞

− − − → RL(T) RL(T) =µ √ 2 exp » −µ2T 2 – × » 1 √ πT − 3µ √ 2 exp „9µ2 2 T « erfc „ 3µ √ 2 √ T «– RL(T) ≈ µ √ 2 √ πT , for small T RL(T) ≈ √ 2 9µ√π e−µ2T/2 T 3/2 , for large T Pocc(T|t) = RL(T|t) + RL(t − T|t) RL(T|t)

t→∞

− − − → RL(T) RL(T) ≈ µ √ 2 √ πT , for small T RL(T) ∼ e−bT , for large T b is given by the zero of Kν(√2p/σ) closest to origin in the left part of the complex–p plane. Iocc(t|T)

T →∞

− − − − →

t>T

I1(t − T) + 1 2δ(t − ∞) I1(τ) =µ √ 2 exp » −µ2τ 2 – × » 1 √πτ − 3µ √ 2 exp „9µ2 2 τ « erfc „ 3µ √ 2 √τ «– I1(τ) ≈ µ √ 2 √πτ , for small τ I1(τ) ≈ √ 2 9µ√π e−µ2τ/2 τ 3/2 , for large τ Iocc(t|T)

T →∞

− − − − →

t>T

I4(t − T) + 1 2δ(t − ∞) I4(τ) ≈ µ √ 2 √πτ , for small τ I4(τ) ∼ e−bτ, for large τ b is the same constant as above.

• S. Sabhapandit (Universit´

e Paris-Sud) Functionals of a particle in 1D random potential 23 / 24

3

TABLE III: Stable potential — Disorder averaged pdf’s of the local, inverse local, occupation and inverse occupation times

• f a particle starting at the origin, diffusing in the stable random potential U(x) = −µ|x| + √σB(x), where µ < 0 and B(x)

represents the trajectory of a Brownian motion in space with the initial condition B(0) = 0. We denote ν = |µ|/σ. PURE CASE (σ = 0) DISORDERED CASE (σ > 0) Ploc(T|t) ∼ exp » −tΦ „T t «– , ( ( ( t → ∞, T → ∞ T/t fixed Φ(r) = 1 2 (r − |µ|)2 , near r = |µ| Ploc(T|t)

t→∞,T →∞

− − − − − − − − →

T/t fixed

1 t f2(T/t) f2(y) = » 2√π σΓ2(ν) – “ y σ ”2ν−1 e−2y/σU(1/2, 1 + ν, 2y/σ) Iloc(t|T) = T √ 2πt3 exp » −(T − |µ|t)2 2t – Iloc(t|T)

t→∞,T →∞

− − − − − − − − →

t/T fixed

1 T g3(t/T) g3(x) = »2σ√π Γ2(ν) – e−2/σx (σx)2ν+1 U(1/2, 1 + ν, 2/σx) Pocc(T|t) ∼ exp » −tΦ „T t «– , ( ( ( t → ∞, T → ∞ T/t fixed Φ(r) = 2µ2 „ r − 1 2 «2 , near r = 1 2 Pocc(T|t)

t→∞,T →∞

− − − − − − − − →

T/t fixed

1 t fo(T/t) fo(y) = 1 B(ν, ν) [y(1 − y)]ν−1 , 0 ≤ y ≤ 1 Iocc(t|T) = I2(t − T, T) θ(t − T) I2(τ, T)

large T

− − − − → |µ|T √ 2πτ 3 exp » −µ2(τ − T)2 2τ – Iocc(t|T)

t→∞,T →∞

− − − − − − − − →

t/T fixed

1 T go(t/T) go(x) = 1 B(ν, ν) (x − 1)ν−1 x2ν , x > 1

• S. Sabhapandit (Universit´

e Paris-Sud) Functionals of a particle in 1D random potential 24 / 24