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

Statistical Properties of Functionals of 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 Ref. Phys. Rev. E 73 , 051102 (2006). Alain Comtet 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 x ( τ ) = η ( τ ) , d τ where � η ( τ ) � = 0 and � η ( τ ) η ( τ ′ ) � = δ ( τ − τ ′ ) . The functional � t θ ( x ( τ )) θ ( x ( τ )) d τ T = 0 is known as Occupation time. T 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 x ( τ ) = η ( τ ) , d τ where � η ( τ ) � = 0 and � η ( τ ) η ( τ ′ ) � = δ ( τ − τ ′ ) . The functional � t θ ( x ( τ )) θ ( x ( τ )) d τ T = 0 is known as Occupation time. T For a fixed t , the value of T is random – depends on [ { x ( τ ) } , 0 ≤ τ ≤ t ] . for P ( T | t ) =? 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 3 � T �� � P ( T ′ | t ) dT ′ = 2 T t P(T|t) arcsin 2 π t 0 1 Probability density function 1 P ( T | t ) = , 0 < T < t . 0 � π T ( t − T ) 0 0.5 1 T/t The scaling form: � � P ( T | t ) = 1 1 T , where , 0 < x < 1 . f f ( x ) = � t t π x ( 1 − x ) 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 ) Intensity (a.u.) θ ∼ 1 / 2 0 100 200 300 400 500 600 Time (s) [ 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 ) Intensity (a.u.) θ ∼ 1 / 2 0 100 200 300 400 500 600 Time (s) [ 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 ) Intensity (a.u.) θ ∼ 1 / 2 0 100 200 300 400 500 600 Time (s) [ 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 ) Intensity (a.u.) θ ∼ 1 / 2 0 100 200 300 400 500 600 Time (s) [ 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 ) Intensity (a.u.) θ ∼ 1 / 2 0 100 200 300 400 500 600 Time (s) [ 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 0 θ ( h ( x , t ′ )) dt ′ . T ( x , t ) = [ 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 θ ( x ( τ )) d τ T = 0 where x ( τ ) is the position of a particle diffusing in an external random medium. U ( x ) ② x P ( T | t ) =? P ( T | t ) differs from sample to sample. 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. Results for occupation time and inverse occupation time. 3 Outline of the formalism. 4 Summary. 5 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 = − d U ( x ( τ )) + η ( τ ) , d τ dx where � η ( τ ) � = 0 and � η ( τ ) η ( τ ′ ) � = δ ( τ − τ ′ ) . U ( x ) = U d ( x ) + U r ( 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 = − d U ( x ( τ )) + η ( τ ) , d τ dx where � η ( τ ) � = 0 and � η ( τ ) η ( τ ′ ) � = δ ( τ − τ ′ ) . U ( x ) = U d ( x ) + U r ( x ) → → − − − − − deterministic random ( quenched ) U d ( 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 = − d U ( x ( τ )) + η ( τ ) , d τ dx where � η ( τ ) � = 0 and � η ( τ ) η ( τ ′ ) � = δ ( τ − τ ′ ) . U ( x ) = U d ( x ) + U r ( x ) → → − − − − − deterministic random ( quenched ) Sinai potential = √ σξ ( x ) − dU r ( x ) U d ( x ) = − µ | x | dx � 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 U ( x ) µ > 0 dx = − d x ② U ( x ( τ )) + η ( τ ) , d τ dx where � η ( τ ) � = 0 and � η ( τ ) η ( τ ′ ) � = δ ( τ − τ ′ ) . U ( x ) U ( x ) = U d ( x ) + U r ( x ) µ = 0 → → − − − − − ② deterministic random ( quenched ) x Sinai potential = √ σξ ( x ) − dU r ( x ) U d ( x ) = − µ | x | U ( x ) µ < 0 dx � ξ ( x ) ξ ( x ′ ) � = δ ( x − x ′ ) ② 0 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 U ( x ) µ > 0 dx = − d x ② U ( x ( τ )) + η ( τ ) , d τ dx where � η ( τ ) � = 0 and � η ( τ ) η ( τ ′ ) � = δ ( τ − τ ′ ) . U ( x ) U ( x ) = U d ( x ) + U r ( x ) µ = 0 → → − − − − − ② deterministic random ( quenched ) x Sinai potential = √ σξ ( x ) − dU r ( x ) U d ( x ) = − µ | x | U ( x ) µ < 0 dx � ξ ( x ) ξ ( x ′ ) � = δ ( x − x ′ ) � t ② P ( T | t ) =? θ ( x ( τ )) d τ T = 0 0 x S. Sabhapandit (Universit´ e Paris-Sud) Functionals of a particle in 1D random potential 7 / 24

Inverse Occupation Time � t T = θ ( x ( τ )) d τ x ( τ ) 0 For fixed T and given { x ( τ ) } , the time t is called inverse occupation time. I ( t | T ) =? θ ( x ( τ )) For Brownian motion √ T I ( t | T ) = π t √ I ( t | T ) t − T T 0 t T What is I ( t | T ) for particle in random potential? τ t 1 t 2 S. Sabhapandit (Universit´ e Paris-Sud) Functionals of a particle in 1D random potential 8 / 24