Sinai diffusion (Random random walk)

Sinai diffusion and some of its descendants

descendants

G L E B O S H A N I N

É È

P H Y S I Q U E T H É O R I Q U E D E L A M A T I È R E C O N D E N S É E ( U M R C N R S 7 6 0 0 )

U N I V E R S I T É P I E R R E E T M A R I E C U R I E ( P A R I S 6 ) P A R I S , F R A N C E

P A R I S , F R A N C E

MECO36, LVIV, UKRAINE April 5 – 7, 2011

Outlook

y Sinai Diffusion <MSD> vs Probability currents

y Sinai Diffusion. <MSD> vs Probability currents.

y Generalized Sinai Diffusion. Brownian motion on Fractional Brownian Motion.

y Melting of a Heteropolymer

y Two Stock Options at the Races

Fi P f l f i d d B i i

y First Passages for a couple of independent Brownian motions

Sinai diffusion (Random random walk)

y Consider a 1D, infinite in both directions chain of integers and associate with each site « x » a random variable p_x (transition probability from x to x+1) with

di ib i i d ½ d b d d f 0 d 1

a distribution symmetric around ½ and bounded away from 0 and 1.

p_x and p_x’are uncorrelated

Si i Diff i R d W lk i R d E i t

Sinai Diffusion: Random Walk in a Random Environment P_n+1(x) = p_x-1 P_n(x - 1) + (1 - p_x+1) P_n(x+1)

y Continuous-space & time counterpart:

dx_t/dt = F(x_t) + ξ_t x_t– particle trajectory

F(x) – random, position-dependent, delta-correlated in space force with zero mean

mean

ξ_t – white noise in time

Random walk in a random potential which itself is a trajectory of a random walk Random walk in a random potential which itself is a trajectory of a random walk

Sinai diffusion (Random random walk)

) ( ln

⁴

2

t

x > ≈

< x

_t

> ≈ ln ( t )

(Sinai 1982)

<

(Sinai, 1982)

A simple argument: On a scale x a typical barrier due to a random potential is of order of √x. An (Arrhenius) time t needed to overcome such a barries

solely due to thermal activation is of order solely due to thermal activation is of order

t ≈ exp(√x)

Trajectories are strongly (logarithmically) confined

This logarithmic behavior is supported by typical realizations of disorder This logarithmic behavior is supported by typical realizations of disorder

Probability distribution (Kesten, 1986)y ( , 9 )

Sinai diffusion. Probability currents in finite samples

y Consider a finite Sinai chain with N+1 sites. Set the occupation of x=0 site equal to 1 and put a sink at site x=N.

y A probability current J_N for a given realization of disorder is J_N =1/τ_N pN

p p

p p

p p

1 p^{1 1 2 1 2 3 1}

y τ_N is (the so-called Kesten variable) the resistance offered by a chain to the

N N

N p

p p

p p

p p

p p p p

p p p p

p

⋅⋅ −

− ⋅ +

− +

− + −

− + −

+ −

= ... 1 1

1 1

1 1

1 1 1

1 1 3

3 2

2 1

1 2

2 1

1 1

τ 1

passage of a particle through the chainN

y J_N = hitting (splitting) probability = Probability that for a given

environment {p } a particle starting at x=1 will first hit x=N without having environment {p_x} a particle starting at x=1 will first hit x=N without having ever visited x=0.

y Continuous-space result, J_L = 1/τ_L

⎟⎟⎠

⎞

⎜⎜⎝

⎛ ⋅

⋅

=

∫

^L

∫

^x

L dx dx F x

0 0

) ' ( '

τ

exp

Sinai diffusion. Probability currents in finite samples

y Typical behavior exp ' ( ) exp( ), exp( )

0 0

L J

L x

F dx

dx _L

x L

L ⎟⎟ ∝ ∝ −

⎠

⎜⎜ ⎞

⎝

⎛ ⋅

⋅

=

∫ ∫

τ

y Typical current is very small, much smaller than the Fickian current in a homogeneous system, J_fick= 1/L (all p_x =1/2)

⎠

⎝

y Disorder-averaged current for a discrete Sinai chain J B

A ≤< >≤ _{(B l} k O h i & M 1992)

y Continuous-space & time : J N N

N ≤< ^N >≤

) (

ln^{2 (}Burlatsky, Oshanin & Moreau, 1992)

2 2 / T

σ

α

∝

) / 2

( 2 )

cosh(

4 ) 2 exp(

) exp(

) (

0

2

π α

α

τ π

Lx x K p

dx p

p =< − _L >= ⋅ − ⋅ ⋅ _ix

Φ ^∞

∫

0

⎟⎟⎠

⎜⎜ ⎞

⎝

⎛ + ⋅ ⋅

= ⋅ Φ

⋅

>=

<

∫ ∑

=

∞

1

2 2

0 !

) /

1 ( )

(

n

n n

L n

B L p L

dp

J

π α

π

α

(Oshanin & Moreau, 1993)

⎠

⎝ ⁼¹

0 n

Sinai diffusion. Probability currents in finite samples.

y Probability distribution

y Probability distribution

⎟⎞

⎜⎛

⎟⎞

⎜⎛ ln²( ) 1 ) 1

( J J

J

P ⁽Oshanin, Burlatsky & Moreau, 1993;

⎟⎟⎠

⎜⎜ ⎞

⎝

⎛−

⋅

⎟⎟⋅

⎠

⎜⎜ ⎞

⎝

⎛

− ⋅

⋅

∝

0 2

/

3 exp

1 4

) ( exp ln

) 1

( J

J J

L J J L

P α

( , y , ;

Monthus & Comtet, 1994)

y J₀ is independent of L

y Typical behavior stems from the log normal part disorder average out of the

y Typical behavior stems from the log-normal part, disorder-average – out of the tails of the distribution

y Moments of arbitrary order C

L J_Lⁿ >∝ Cⁿ

<

Sinai diffusion with global bias

y Random Force

<F(x)> = F₀, <F(x) F(x’)> = F₀²+ 2 D_w δ(x – x’)

y Key parameter – a Peclet-type number:y p yp μ = D₀ F₀/D_w

y D particle diffusion coefficient in absence of disorder

y D₀ – particle diffusion coefficient in absence of disorder

y μ < 1. Creep phase, <x(t)> ≈ t^μ

y 1 < μ < 2. Ballistic phase with anomalous dispersion, <x(t)> ≈ F₀ t + t^1/μ

y μ ≥ 2. Ballistic phase.

(de Gennes, 1974; Kesten, Kozlov & Spitzer, 1975; Derrida & Pomeau, 1983)

Generalized Sinai diffusion

y Sinai diffusion is a random walk (Brownian motion) in a random potential W(x) which itself is a trajectory of a random walk (Brownian motion)

W(x) which itself is a trajectory of a random walk (Brownian motion)

y Consider Brownian motion in a random quenched potential W_H(x) where W_H(x) is a fractional Brownian motion with the Hurst index H:

W_H(x) is a stationary Gaussian process such that

<W_H(x)> = 0, < W_H(x) W_H (x’)> =1/2 ( |x|^2H + |x’|^2H - |x – x’|^2H), 0 < H < 1

y if H = 1/2 then the process is in fact a Brownian motion

y if H > 1/2 then the increments of the process are positively correlated

if H 1/2 h h i f h i l l d

y if H < 1/2 then the increments of the process are negatively correlated.

With Alberto Rosso & Gregory Schehr (2011)

Generalized Sinai diffusion

y Disorder-averaged mean-square displacement

< x

_t²

>≈ ln

^2/H

( t )

y Typical probability current through a finite sample

1

( )

^H

L

L

J ≈ exp −

y Disorder – averaged probability current _θ

J

_L

L 1

>≈

<

y θ = 1 – H, like the persistence exponent for the fBm. Is it?

y Fix t and L and vary H from 1 to 0 MSD grows while the current becomes

y Fix t and L and vary H from 1 to 0. MSD grows while the current becomes smaller!

y Representative realizations of disorder (representative trajectories W_H(x))

With Alberto Rosso & Gregory Schehr (2011)

Generalized Sinai diffusion. Representative disorder

U b d L W [W (1) W (2) W (3) W (4) W (N)]

y Upper bound: Let

⇒

≥ +

+ +

+ +

=

^{W W W W N W}

N

1 e

^H(1)

e

^{H (2)}

e

^H(3)

e

^{H ( )}

e τ

)]

( ),...,

4 ( ),

3 ( ),

2 ( ),

1 (

max[W W W W W N

W = _{H H H H H}

W

N

e

J ≤

⁻

L b d

⇒

≥ +

+ +

+

N

1 + e e e ... e e

τ J

N

≤ e

dW e

W P

J

_N

>≤ ⋅

^W

⋅

< ∫ ^{( )}

^{− P(W) ≈ const}_L^{1−H +o⎜}_⎝^⎛ _L^{11−H ⎟}_⎠^⎞ (Molchan, 1991) y Lower bound:

Ω

⎭ ∈

⎬⎫

⎩⎨

≥ ⎧

⎭⎬

⎫

⎩⎨

>= ⎧

< _Ω ω

τ

τ ^{ω ,}

1 1

N N

N E E

J

y ω comprises such W_H(x) which obey W_H(x) ≤ f(x) =A – B ln(x)

const e

e e

e e

e

^{W W W N f f f N}

N

H H

H

+ + + ≤ + + + + →

+

= 1

^{(1) (2)}

...

^{( )}

1

^{(1) (2)}

...

^{( )}

τ

R t ti di d h t j t i W ( ) hi h b d d f b

const e

e e

e e

N

1 + e + + ... + ≤ 1 + + + ... + →

τ

{}

^H

E L const

J_N >≥ ⋅ ≈ = −

< 1 , 1

1 _θ θ

ω

y Representative disorder = such trajectories W_H(x) which are bounded from above by a slowly decaying function (constant current & sufficiently high measure)

With Alberto Rosso & Gregory Schehr (2011)

Helix or Coil? Fate of a melting heteropolymer

A heteropolymer of length L with a binary

alphabet (chemical groups A and B with melting temperatures T_A and T_B).

Initially some part Y of the heteropolymer is denaturated (coil state) and the Initially some part Y of the heteropolymer is denaturated (coil state) and the

remaining part is in a helix state

Dynamics of the boundary (de Gennes, 1974): c/h boundary diffuses in presence of a random, position-dependent force F(x):

•

ξ(t) white noise

) ( )

( x t

F

x

^•_t

= + ξ < F ( x ) >= F

₀

, < F ( x ) F ( x ' ) > − F

₀²

= σ

²

δ ( x − x ' )

ξ(t) – white noise.

For T=(T_A+T_B)/2, F₀ =0 – Sinai diffusion With Sid Redner (2009)

Helix or Coil? Fate of a melting heteropolymer

y E realization dependent hitting probability probability that the c/h boundary

y E₊ - realization-dependent hitting probability – probability that the c/h boundary starting at x = Y hits x=L first without ever having hit x=0:

−

+ =

= J τ

E

y Js are the currents from the starting point to the respective boundary, τ₊ and τ_-

+

−

− +

+ = +

= +

τ τ

J E J

are familiar objects – uncorrelated random variables:

∫ ∫

∫ ∫

⎟⎟⎞

⎜⎜

⎛− ⋅

⋅

⎟⎟ =

⎞

⎜⎜

⎛− ⋅

⋅

= ^Y dx ^x dx F x ^Ldx 1 ^x dx' F(x')

exp )

' ( 1 '

exp τ

τ

y In absence of disorder (Brownian motion):

∫ ∫

∫ ∫

⎟⎟

⎜⎜ ⎠

⎟⎟ ⎝

⎜⎜ ⎠

⎝ ⁺

−

Y Y

x F T dx

dx x

F T dx

dx exp ( ) , exp ( )

0 0

τ τ

E₊ = Y/L

y In presence of disorder, E₊ is a random variable. Probability distribution (over quenched disorder) P(E₊)?

With Sid Redner (2009)

Helix or Coil? Fate of a melting heteropolymer

y Exact result:

y Exact result:

⎟⎟

⎠

⎞

⎜⎜

⎝

⎛

= −

⎟⎟⎠

⎜⎜ ⎞

⎝

⎛ − −

⎟⎟⎠

⎜⎜ ⎞

⎝

⎛

= −

+

− +

− +

+ +

∞

∞

+^{) 1} _{(1 )}

∫

_cosh(^{cosh( )}₎^{cos exp} ₄ ^{, sinh} ₁ ^{cosh( )}

( ¹

2 2

2

3 u

E E u

u du u

E E

E

P η

α η α

α π α

π α η

α π

/ ( h /h b d l i h iddl )

⎠

⎠ ⎝

⎝

⎠

⎝ ^{+ + + +}

∞ + − +

−

+α E (1 E ) (η)

α π

T²Y

2

2

α₋ = σ ( )

2 ²

2

Y T L−

+ = α σ

y For Y=L/2 (the c/h boundary starts exactly in the middle),

y Is the mean value meaningful?

y There is a critical length L at which the distribution changes the shape 2 /

>≡1

< E₊

y There is a critical length L_c at which the distribution changes the shape

Lcrit

L<

Lcrit

L=

y Mean is the least probable for sufficiently long chains. Mean does not mean much Each realization has its unique fate

Lcrit

L>

much… Each realization has its unique fate.

With Sid Redner (2009)

Two (Asian) stock options at the races: Black-Scholes forecasts

I l i l Bl k S h l d l i S f t k b

y In classical Black-Scholes model, a price S_t of a stock obeys :

i d d h f h l ili f h

t t

t

t

z S dt S dW

dS = ⋅ + σ ⋅ ⋅

y z is a parameter dependent on the nature of the asset, σ – volatility of the market, dW_t are the increments of a Brownian trajectory

y Stratonovich Ito solution: ^{S = S ⋅exp}_⎜⎜^{⎛−σ μ ⋅t +σ ⋅W} _⎟⎟^⎞

2

y Stratonovich-Ito solution:

y μ = const, which has exactly the same meaning as the Peclet-type number in

th ti bi d Si i diff i A l f th k t +

⎟⎟⎠

⎜⎜⎝ ⋅ + ⋅

⋅

= _{t= t}

t S t W

S μ σ

exp 2

0

the section on biased Sinai diffusion. A slope of the market. + or -.

y An Asian (path-dependent) option is a time-average

⎟⎞

⎜⎛

∫

T

S σ ²

y Suppose one buys two very similar stocks (Freddie Mac & Fannie Mae) and is

⎟⎟⎠

⎜⎜ ⎞

⎝

⎛− ⋅ + ⋅

⋅

⋅

= ^t=

∫

^t

T dt t W

T

S σ μ σ

τ exp 2

0 0

pp y y ( )

curious how much, at time T, one of them will contribute to his wealth

) 2 ( )

1 (

) 1 (

E T

τ τ

τ

= +

With Gregory Schehr (2011)

T

T τ

τ +

Two (Asian) stock options at the races: Black-Scholes forecasts

∞

y Distribution P(E):

y Distribution of τ:

∫

^{⋅ ⋅Ψ ⋅ ⋅Ψ − ⋅}

=

0

) ) 1 ((

) (

)

(E dτ τ E τ E τ

P

− + + Γ

−

∝ −

Ψ ⁻

≤

≤

−

−

−

∑

^{( 1}₍₁^{) ( 2}₎^{) (1/ )}

)

( ²

2 / 0

) ( /

1 2

/ ατ

μ μ

τ τ_μ ^{ατ αn n μ τ n} L^μ_n ⁿ n

e n A e

L k b h ll b h i i i l l l d f l

+

≤ Γ

≤ _/2 (1 )

0 μ

τ _{n μ} n

) / 1 ( )

sinh(

2)

( 2 _{( 1)/2, /2}

2

0 2 / ) 1

( μ π ατ

τ ^{μ u Wμ iu}

i u du

B u

+

∞

+ ⋅ ⋅ Γ − + ⋅

+

∫

⁽Monthus & Comtet, 1994)

y Looks scary but the overall behavior is simple: log-normal decay for large τ, exponential decay when τ → 0 and a fat power-law tail inbetween.

y Rising market: ≤ 0 P(E) is a delta f nction at T 0 As T gro s P(E) broadens

y Rising market: μ ≤ 0. P(E) is a delta-function at T=0. As T grows, P(E) broadens, and undergoes at a certain T=T_c a continuous transition to an M-shaped form with a minimum at E=1/2 and maxima close to 0 and 1. It means that one of the options

b l t i hil th d i t h Th t i b k

becomes a complete winner while the second is trash. The symmetry is broken.

y Decreasing market. 0 < μ ≤ 1. A transition to a U-shaped form. The symmetry is broken.

y Decreasing market. μ > 1. P(E) stays a bell-shaped function with a maximum at E

= ½. Both stocks contribute proportionally.

With Gregory Schehr (2011)

Two (Asian) stock options at the races: Black-Scholes forecasts

y Correlated τ variables. We connect the increments by a spring of rigidity χ

) 1 ( )

1 ( )

1 ( )

1 (

t t

t

t z S dt S dW

dS = ⋅ +σ ⋅ ⋅

) 2 ( )

2 ( )

2 ( )

2 (

t t

t

t z S dt S dW

dS = ⋅ +σ ⋅ ⋅

No transition

Transition from a unimodal

to a bimodal M shaped distribution to a bimodal M-shaped distribution

Transition to a bimodal U-shaped distribution

With Gregory Schehr (2011)

U shaped distribution

First passages for a couple of independent Brownian motions

y Consider a semi-infinite, one-dimensional system with two BMs starting simul- taneously at x₀. BMs experience a constant, site-independent bias F towards the origin so that their drift velocities are = v < 0. Diffusion coefficients of both = D.g

y Let t₁ and t₂ be the time moments of their first passages to the origin. The distribution P(ω = t₁/(t₁+t₂))?

) 1

(

) ) 1

( /

) (

( ¹

ω ω

ω ω

ω μ

−

∝ K − P

y μ = x₀ |v|/2 D – the Peclet number

Th i hi h d k il ( f i bj i b bili h )

y The event in which two drunken sailors (a favorite object in probability theory), appearing simultaneously at the same position on a semi-infinite street with a bar at one end (which they « smell »), will enter the bar simultaneously is

- the most probable if μ > μ_c ≈ 1/3 - the least probable if μ < μ_c ≈ 1/3

With Gregory Schehr and Carlos Mejia-Monasterio (2011)

First passages for a couple of independent Brownian motions

y Consider a one-dimensional inteval [0,1] with an adsorbing boundary at x=0 and a reflecting boundary at x = 1. Let two BMs start simultaneously at xg y y ₀₀. Diffusion coefficients of both = D. No bias.

y Let t₁ and t₂ be the time moments of their first passages to the origin. The di t ib ti P( t /(t +t ))?

distribution P(ω = t₁/(t₁+t₂))?

y The event, in which two drunken sailors, appearing simultaneously at the same, , pp g y position on a bounded street with a bar at one end and a reflecting boundary at the other end, will enter the bar simultaneously is

- the most probable if x₀ > 2/π - the least probable if x₀ < 2/π

With Gregory Schehr and Carlos Mejia-Monasterio (2011)

Conclusion

y The symmetry between two i i d random variables whose distribution is truncated fromThe symmetry between two i.i.d random variables, whose distribution is truncated from both sides (and thus possesses all moments), is broken if the distribution has an

intermediate sufficiently « fat » tail and the characteristic cut-off is less than some critical value. One has to be cautious and not confuse typical, average, most probable.yp , g , p

y Such distributions are ubiquituous: (in addition to mentioned examples), interspike intervals for Integrate-and-Fire model of neuron dynamics, biased motion of solids on solid surfaces, SOC, first passages in finite systems or with an inward bias, chemo- and solid surfaces, SOC, first passages in finite systems or with an inward bias, chemo and infotaxis, and etc.

y Critical cut-off may be time, size, position, voltage, smell and etc.

y An agenda: Correlated variables (partially mentioned in the talk) e g Wigner proper

y An agenda: Correlated variables (partially mentioned in the talk), e.g. Wigner proper delay times in N-channel systems (with Gregory Schehr, 2011)

2 /

>≡1

<ω > 1/2 <ω >≡1/3

<ω <ω > 1/3