fate of a particle in a (vanishing) sea of sub diffusive traps

Fate of a particle in a (vanishing) sea of SUBdiffusive

traps

S. B.Yuste & J.J. Ruiz-Lorenzo, UEx

K. Lindenberg, UCSDBad Honnef 2006

3. Static particle in a sea of vanishing subdiffusive traps

1. An example: biological system with particles of very different mobility

2. Trapping with mobile particle and mobile traps

First system

Second system

Subdiffusion limited reactions

“Visualization and Tracking of Single Protein Molecules in the Cell Nucleus”, Kues, Peters, and Kubitscheck, Biophysical Journal, 80 (2001) 2954

Very different mobilities!

“Visualization and Tracking of Single Protein Molecules in the Cell Nucleus”, Kues, Peters, and Kubitscheck, Biophysical Journal, 80 (2001) 2954

“Using the comparatively “inert” recombinant protein, P4K, we observed only diffusional motion. However, within this category a range of modes was detected. Thus one population of molecules appeared to be immobile ... while at least three different mobile fractions were detected. Two of the mobile fractions, fmob,1 and fmob, 2, were analyzed quantitatively in terms of time dependent diffusion coefficients.... However, the diffusion coefficient of fraction fmob,2 was also decreasing with time. The third mobile fraction, fmob,3, of ;5% moves very fast with jump...”

subdiffusion!

The general problem: What are the consequences of the subdiffusive nature of the reactants

on the reaction dynamics of (sub)diffusion limited reactions?

● A bumpy road: generalization of the usual reaction-diffusion equations

Example: A → B in one dimension (Sokolov, Schmidt, Sagués, PRE, 73 (2006))κ

Diffusive reactants Subdiffusive reactants

Generalization



● A bumpy road: generalization of the usual reaction-diffusion equations

diffusion limited reactions subdiffusion limited reactionst! t

h x2 i » t h x2 i » t

...but not always true:Long-time reaction rate in the subdiffusive trapping problem » 1/t

Long-time reaction rate in the diffusive trapping problem » exp(- t1/3)

? Donsker-Varadhan

● An easy (but qualitative) answer: Subordination :

What is relevant in reaction dynamics is h x2i so that...

Particle in a 1D sea of subdiffusive traps

1. Target problem: A(static)+T→T

Yuste&Lindenberg, PRE 72, 061103 (2005) ☞ 3. A+T→T

Yuste&Acedo Physica A 336, 334 (2004)2. Trapping problem: A+T(static) →T(static)

(A & T mobile)

☑

☑

Yuste&Acedo Physica A 336, 334 (2004)

☞ Objective: reaction dynamics ⇔ survival probability P(t)

Particle in a 1D sea of subdiffusive traps :

1D subdiffusion limited reactions where A

exp³¡ ¸t1=2

´Donsker-Varadhan Bramson-Lebowitz

Bray-Blythe

Particle in a sea of traps

1D Long-time reaction dynamics

Recapitulation

A mobileT static

exp³¡ ¸t1=3

´

???????

Diffusion

Subdiffusion

A mobileT mobile

Blumen-Klafter-Zumofen

1=t°0

☞ Our first problem

A rigorous approach ...to the problem of

SUBdiffusive particle in a sea of SUBdiffusive traps

To find lower and upper bounds for P(t):

PL(t) ≤ P(t) ≤ PU(t)

where, for t→∞,

PL(t) → P(t) ←PU(t)

A successful approach for normal diffusion

(Bray & Blythe, PRL 2001)

For a subdiffusive particle ( 0< γ’ < 1 ) and subdiffusive or diffusive traps ( 0< γ ≤ 1 )

0

2¡ (1+ °=2)

· ¡lnP (t)

p½2K t°

·2

¡ (1 + °=2)+

2lnhp

½2K 0t°0i+ 2 + ln[2¡ (1¡ °0)]

p½2K t°

+:: :

t → ∞

For a diffusive particle (γ’ = 1) and subdiffusive traps with 2/3 < γ ≤ 1

2¡ (1+ °=2)

· ¡lnP (t)

p½2K t°

·2

¡ (1+ °=2)+ 3

µ¼2½

¶2=3 D01=3

K 1=2 t1=3¡ °=2 + :: :

t → ∞

0

subordination ☑ P(t)» exp(-4 /1/2 D1/2 t1/2) t ! tγ

P (t) » exp

Ã

¡

p4½2K t°

¡ (1+ °=2)

!

; t ! 1Target

problem

P(t) !

Yuste&Lindenberg, PRE 72, 061103 (2005)

Normal diffusion

Diffusive particle (γ’=1) in a sea of strongly (γ<2/3) SUBdiffusive traps

1. Exact result for the green region

Survival probability of a (SUB)diffusive particle in a sea of (SUB)diffusive traps

P(t) » exp

Ã

¡

p4½2K t°

¡ (1+ °=2)

!

Donsker-Varadhan

P (t) » exp(¡ ¸t1=3)

P (t) » exp(¡ 4=¼1=2½D1=2t1=2)Bramson-Lebowitz, Bray-Blythe

Normal diffusion

1. Exact result for the green region


P(t) » exp

Ã

¡

p4½2K t°

¡ (1+ °=2)

!P (t) » exp(¡ ¸t1=3)

λ bounded but unknown

P (t) » exp(¡ 4=¼1=2½D1=2t1=2)Bramson-Lebowitz, Bray-Blythe


Donsker-Varadhan

P (t) » exp(¡ ¸t1=3)

For a diffusive particle (γ’ = 1) and subdiffusive traps with γ < 2/3

PL(t) ≤ P(t) ≤ PU(t)

2¡ (1+ °=2)

· ¡lnP (t)

p½2K t°

·2

¡ (1 + °=2)+ 3

µ¼2½

¶2=3 D01=3

K 1=2 t1=3¡ ° =2

∞

subordination ?

P(t) » ? ; t ! 1

t → ∞

P (t) = ?


Exact result for the green region

No exact prediction for the orange line


P(t) » exp

Ã

¡

p4½2K t°

¡ (1+ °=2)

!

Donsker-Varadhan

P (t) » exp(¡ ¸t1=3)

Normal diffusionP (t) » exp(¡ 4=¼1=2½D1=2t1=2)

Bramson-Lebowitz, Bray-Blythe

P (t) » exp(¡ ¸t1=3)

λ bounded but unknown

?orP (t) » exp(¡ ¸t1=3)

P (t) » exp(¡ ¸t° =2)

P (t) » exp(¡ ¸tµ)

γ

γ’0

1

10

□(0.4,0.4)

□(0.8,0.8)

□(0.5,1)

□(0.6,0.4)

□(0.7,0.4)

□(0.9,0.5)

□(1.0,0.4)

□(0.4,0.8)

Simulation results for the exponent θ :

□(1.0,0.8)

□(0.5,0.4)

θ=1/2Bramson-Lebowitz

θ = γ/22/3

θ=1/3

θ = 1/3 ?γ/2 (subord.) ?

θ=1/3Donsker-Varadhan


ln[¡ ln(P (t)]

lnt

ρ=0.01γ=0.4 (trap)

γ’=0.4 (part.)L=10^4

θ=0.2 (Simu.)

θ=γ/2 (Theo.)

Case γ’=0.4 , γ=0.4 : Exponent θ

☑ Exponent θ

Subdiffusive particle ( 0< γ’ < 1 ) and subdiffusive traps ( 0< γ <1 )

ln[¡ ln(P (t)]= ¸ + µlntP (t) » exp(¡ ¸tµ) ⇔

P(t)=1.4E-06

P(t)=0.034

P(t)=6.2E-04

OK!

ln[¡ ln(P (t)]

lnt

ρ=0.01γ=1 (trap)

γ’=0.5 (part.)L=10^4 θ=0.5 (Simu.)

θ=γ/2 (Theo.)

Case γ’=0.5 , γ=1 : Exponent θ

☑ Exponent θ

Subdiffusive particle ( 0< γ’ < 1 ) and diffusive traps ( γ = 1 )

ln[¡ ln(P (t)]= ¸ + µlntP (t) » exp(¡ ¸tµ) ⇔

P(t)=5.1E-06

P(t)=0.37

OK!

6 8 10 12 14 16 18 200

1

2

3

ln[¡ ln(P (t)]= ¸ + µlnt


Subdiffusive particle ( 0< γ’ < 1 ) and subdiffusive traps ( 0< γ ≤ 1 )

P (t) » exp(¡ ¸tµ) ⇔

θ_teo=γ/2=0.4

OK!

rho_0=0.01

ln[¡ ln(P (t)]

lnt

P(t)=5.1E-06

P(t)=0.37

θ_simu=0.39

rho_0=0.1

θ_simu=0.38

ln[¡ ln(P (t)]=¸ + µlnt

Case γ=0.8, γ’=0.8 : Exponent θ


P (t) » exp(¡ ¸tµ) ⇔

θ_simu=0.38

OK!

rho_0=0.01

rho_0=0.1

θ_simu=0.37

θ_teo=γ/2=0.4

lnt

ln[¡ ln(P (t)]

γ

γ’0

1

10

☺(0.4,0.4)

☺(0.8,0.8)

☺(0.5,1)

□(0.6,0.4)

□(0.8,0.4)

□(0.7,0.4)

□(0.9,0.5)

□(1.0,0.4)

☺(0.4,0.8)

□(1.0,0.8)

□(0.5,0.4)

θ = γ/2


θ = 1/3 ?γ/2 (subord.) ?


θ=1/32/3

P (t) » exp(¡ ¸tµ)Simulation results for the exponent θ :





P (t) » exp(¡ ¸tµ) ⇔

θ_simu=0.15

?

rho_0=0.01

rho_0=0.1

θ_simu=0.18

θ_teo=γ/2=0.2

lnt

ln[¡ ln(P (t)]


Case γ’=1, γ=0.8 : Exponent θ

P (t) » exp(¡ ¸tµ) ⇔?

θ_teo=γ/2=0.4

θ_simu=0.45rho_0=0.01

Diffusive particle (γ’= 1 ) and subdiffusive traps with 2/3 < γ ≤ 1

lnt

ln[¡ ln(P (t)]

P(t)=5.1E-06

P(t)=0.37

γ

γ’0

1

10

☺(0.4,0.4)

☺(0.8,0.8)

☺(0.5,1)

□(0.6,0.4)

□(0.7,0.4)

□(0.9,0.5)

□(1.0,0.4)

☺(0.4,0.8)

(1.0,0.8)

(0.5,0.4)

θ = γ/2


2/3


θ=1/3

θ = 1/3 ?γ/2 (subord.) ?




Case γ’=0.6 , γ=0.4, : Exponent θ


P (t) » exp(¡ ¸tµ) ⇔

θ_simu=0.13

?

rho_0=0.01

rho_0=0.1

θ_simu=0.18

θ_teo=γ/2=0.2

lnt

ln[¡ ln(P (t)]

6 8 10 12 14 16 18 20 22

1.6

1.8

2.0

2.2

2.4

2.6

2.8

3.0




P (t) » exp(¡ ¸tµ) ⇔

θ_simu=0.12

?

rho_0=0.1

rho_0=0.5

θ_simu=0.18

θ_teo=γ/2=0.2

lnt

ln[¡ ln(P (t)]

5 10 15 20 250.5

1.0

1.5

2.0

2.5

3.0




P (t) » exp(¡ ¸tµ) ⇔

θ_simu=0.12

?

rho_0=0.01

rho_0=0.5

θ_simu=0.27

θ_teo=γ/2=0.25

θ_simu=0.19

rho_0=0.1

lnt

ln[¡ ln(P (t)]

P (t) » exp(¡ ¸tµ)

γ

γ’0

1

10

☺(0.4,0.4)

☺(0.8,0.8)

☺(0.5,1)

☹(0.6,0.4)

☹(0.7,0.4)

☹(0.9,0.5)

□(1.0,0.4)

☺(0.4,0.8)

Exponent θ :

(1.0,0.8)

(0.5,0.4)

θ = γ/2


θ = 1/3 ?γ/2 (subord.) ?



2/3 θ=1/3

0 2 4 6 8 10-1

0

1

2

3


Case γ=0.4, γ’=1 : Exponent θ

Diffusive particle ( γ’ = 1 ) and subdiffusive traps ( γ <2/3 )

P (t) » exp(¡ ¸tµ) ⇔

θ_simu=0.43

?

rho_0=0.1

rho_0=0.5

θ_simu=0.47

θ_teo={1/3,γ/2=0.2}

ln[¡ ln(P (t)]

lnt

γ

γ’0

1

10

☺(0.4,0.4)

☺(0.8,0.8)

☺(0.5,1)

☹(0.6,0.4)

☹(0.7,0.4)

☹(0.9,0.5)

(1.0,0.4)

☺(0.4,0.8)

(1.0,0.8)

(0.5,0.4)

θ = γ/2


Simulation´s Bermuda triangleOkay, you win, I can't drive a ship...

θ=1/32/3


θ = γ/2 (subord.) ?1/3 ?



Making sense (?) of simulations results for

Simulations for ’ ≤ are conclusive: • exponent θ =γ/2 robust

P (t) » exp(¡ ¸tµ)

Simulations for ’ > are inconclusive:

• exponent θ not well-defined

Why ?Why ?

γ

γ’0

1

10

Theo. ?

?

☹

☺

Fast particle’ >

Slow traps

Static particle

Mobile traps

for t→∞

Hard for simulations!!

Target problem

☞ Static particle in a 1D sea of vanishing (sub)diffusive traps

t1 : (t1)

t2 : (t2)< (t1)

What is the survival probability P(t) of the particle?

+

T → ØA + T → Tstatic

An exact integral equation for P(t)=exp[-μ0(t)]Bray, Majumdar, Blythe , PRE 67, 060102R (2003)

G(x;t) =1

p4¼K ° t°

H 1011

"jxj

pK ° t°

¯¯¯¯¯

(1¡ °=2;°=2)

(0;1)

#

G(0; t) =t¡ °=2

p4K ° ¡

¡1¡ °

2

¢

Probability density to find a trap at x=0 at

time t

Probability that a trap has met the particle in the time interval (t',t'+dt') for the first time

the probability density for this particular trap to be again at x=0 at

time t.the probability of a trap surviving till time t, given that it survives till time t’

For subdiffusive traps:

x

t’ t

xt’ t

½(t) =Z t

0dt0½(t)

½(t0)_¹ 0(t

0) G(0;t ¡ t0)

t=t-t’ t! 0

_¹ 0 = _P =P

(t) =_¹ 0(t)½(t)

(t) =_¹ 0(t)½(t)

=

p4K °

¡ (°=2)t° =2¡ 1

The final integral equation

is the famous (generalized) Abel’s integral equation.

p4K ° =

1¡ (1¡ °=2)

Z t

0dt0 (t0)

(t¡ t0)°=2

The solution is:

¡ lnP (t) = ¹ 0(t) =

p4K °

¡ (°=2)

Z t

0d¿ ½(¿) ¿° =2¡ 1

½(t) =¡ (°=2)p

4K °t1¡ ° =2

_PP

Inverse problem

prescribed

Tautochrone

Exponentially vanishing sea of traps

½(t) = ½0 exp(¡ t=¿)

P (t) = exp½

¡ `½0

µ1¡

¡ (°=2; t=¿)¡ (°=2)

¶¾

` ´p

4K ° ¿°

= trap’s mean timelife

typical length explored by a trap before vanishing

P (t ! 1 ) = exp¡¡ `=½¡ 1

0¢

0-1 = typical distance between traps at t=0

! 1 Classic target problem

Yuste&Acedo Physica A

336, 334 (2004)

Normal diffusive traps, =1

P (t) = expn

¡ `½0 erf³ p

t=¿´o

P(t) =exp

Ã½0

p4K ° t°

¡ (1+ °=2)

!

☑ Subordination: t! t

… eternal life!

Survival probability for normal diffusive traps with0 =0.001, τ=105

- ln P(t)

time

Survival probability for subdiffusive traps withγ= ½, ρ0 =0.01, τ=108

time

- ln P(t)

Power-law vanishing sea of traps

½(t) =½0

(1+ t=¿)¯ » t¡ ¯ for t ! 1

2<

¡ lnP (t) = ¹ 0(t) =½0

p4K ° ¿°

¡ (°=2)Bt=(¿ +t) (°=2; ¯ ¡ °=2)

¹ 0(t) = ½0p

4K ° ¿°

µ¡ (¯ ¡ °=2)

¡ (¯)¡

(t=¿)°=2¡ ¯

(¯ ¡ °=2)¡ (°=2)+ :: :

¶

2=

2>

¹ 0(t) =½0

p4K ° ¿°

¡ (°=2)ln(t=¿) + : ::

¹ 0(t) =½0

p4K ° ¿°

(°=2¡ ¯)¡ (°=2)

µt¿

¶° =2¡ ¯

+ :: :

typical length explored by a trap » t/2 mean distance between traps »-1» t

For t! 1 :

P (t) = exp(¡ ¸t° =2¡ ¯ )

P (t) » 1=t¢¢¢

Survival probability for subdiffusive traps withγ= 0.75, =0.4, =106, ρ0 =0.01

- ln P(t)

time

γ/2<


- ln P(t)

time

γ/2=


- ln P(t)

time

γ/2>

Remarks and conclusions:1. Systems with reactive particles of very disparate diffusivities do exist and are

worth studying.2. First steps into the study of subdiffusion-limited reactions with reactive species

with different anomalous diffusion exponents.3. Long-time reaction dynamics regime for subdiffusive particles in a sea of

subdiffusive traps inaccessible by simulation methods when γ’>γ.4. “Long-time” reaction dynamics regime for subdiffusive particles in a sea of

subdiffusive traps reached very soon when γ’<γ.5. .Exact solution for a class of a subdiffusion-limited reactions: the target problem

with arbitrary time-dependent density of traps.

... and some open problems

• What is the survival probability of a diffusive particle in a sea of strongly (γ < 2/3 ) SUBdiffusive traps?

• Reaction dynamics for a subdiffusive particle in a sea of vanishing subdiffusive traps.

• To improve simulations (better methods?)

• To extend the results to higher dimensions (d=2, d=3).• Intermediate-time reaction dynamics (specially for γ’>γ) .



● A bumpy road: generalization of the reaction-diffusion equation

Example: A → B in one dimension (Sokolov, Schmidt, Sagués, PRE, 73 (2006))κ

@A@t

= K ¢ A ¡ · A@A@t

= K ®T̂t(1¡ ®;· )¢ A ¡ · A

Tt(1¡ ®;· )f =µ

ddt

+ ·¶ Z t

0

e¡ · (t¡ t0)

(t ¡ t0)1¡ ®f (t0)dt0

@B@t

= K ¢ B ¡ · B

0D1¡ ®t = T̂t(1¡ ®;0)

@B@t

=K ® 0D1¡ ®t ¢ B + · A

+ K ®[ 0D1¡ ®t ¡ T̂t(1¡ ®;· )]¢ A

Diffusive reactants Subdiffusive reactants

Generalization

● An easy (but qualitative) answer: Subordination :

What is relevant is h x2i so that...

diffusion limited reactions subdiffusion limited reactionst! t

h x2 i » t h x2 i » t

...but not always true:Long-time reaction rate in the subdiffusive trapping problem » 1/t

Long-time reaction rate in the diffusive trapping problem » exp(- t1/3)

?Donsker-Varadhan

An idea...1. We have seen that

P(t)» exp(-4 /1/2 D1/2 t1/2) , t ! 1for the full diffusive case γ’ =γ=1

2. Qualtitatively, in many cases, one can get the subdiffusive behavior

from the diffusive one by means of the change t ! tγ (“subordination”)

Hypothesis:

P(t)» exp (-const. K1/2 tγ/2) , t ! 1

So that forSUBdiffusive particle in a sea of SUBdiffusive traps …

Black line: erfc(t/2)Blue line: 1¡ ¡ (°=2;t)=¡ (°=2)

An open problem:Diffusive particle (γ’=1) in a sea of strongly (γ < 2/3) SUBdiffusive traps

θ

γ12/31/3

1/2

1/3µ= °=2 θ =1/3

Donsker-Varadhan exponent for the trapping

problem: diffusive particle in a sea of

static (γ→0) traps0

0

●?

¡ lnP (t) » tµ

Subordin

ation

Kinetics of Trapping Reactions with a Time Dependent Density of Traps

Alejandro D. Sánchez, Ernesto M. Nicola, and Horacio S. Wio

Phys. Rev. Lett. 78, 2244–2247 (1997)

Last minute message

A closely related problem studied in :

• Involve three normal diffusive species• Different approach• Approximate expressions• No subdiffusion

... but different:

γ=0.4 (trap)

γ’=0.4 (part.)L=10^4

Case γ’=0.4 , γ=0.4 : Prefactor λ

☑ Prefactor λ

¡ln(P (t)

Â

Â = ½µ

2K °

¡ (1 + °)t°

¶1=2

= ½hx2i1=2

Â

ρ=0.1 (red)

ρ=0.01 (green)

Subdiffusive particle ( 0< γ’ < 1 ) and subdiffusive traps ( 0< γ < 1 )

ρ=0.01 γ=1 (trap)

γ’=0.5 (part.)L=10^4

Case γ’=0.5 , γ=1 : Prefactor λ

☑ Prefactor λ

¡ln(P (t)

Â

Â = ½µ

2K °

¡ (1 + °)t°

¶1=2

= ½hx2i1=2

Â

Subdiffusive particle ( 0< γ’ < 1 ) and diffusive traps ( 0< γ ≤ 1 )

-6 -4 -2 0

0

1

2

3

ln[¡ ln(P (t)]

Case γ=0. 4, γ’=1 : Exponent θ

Diffusive particle ( γ’ = 1 ) and subdiffusive traps ( γ <2/3 )

rho_0=0.5

θ_simu=0.47

ln½2t2µ (µ= °=2)

rho_0=0.1

θ_simu=0.44

⇔ ln[¡ ln(P (t)]= ^̧+ ln½tµP (t) » exp(¡ ^̧½tµ)

-1 0 1 2 3 41.0

1.5

2.0

2.5

3.0

Case γ’=0.9 , γ=0. 5 : Exponent θ


θ_teo=γ/2=0.25

?

rho_0=0. 01 (red)

rho_0=0.5 (green)

θ_simu=0.27

θ_simu=0.12

rho_0=0. 1 (blue) θ_simu=0.19

ln[¡ ln(P (t)]

P(t)=5.1E-06

P(t)=0.066

ln½2t2µ (µ= °=2)

⇔ ln[¡ ln(P (t)]= ^̧+ ln½tµP (t) » exp(¡ ^̧½tµ)

particle in a sea of traps:

D’,K’

+

D, K

Diffusive

Subdiffusive

diffusive

subdiffusive

diffusion coefficient of the trapdiffusion coefficient of the particle

☞ First problem: Long time reaction dynamics of …

•Conjectured by Blythe&Bray, PRL 2002•Proved by Bray, Majumdar&Blythe, PRE 2003•Proved by Moreau, Oshanin, Bénichou&Coppey, PRE 2003Proved by S. B. Yuste, K. Lindenberg, PRE 72, 061103 (2005)

Upper bound PU(t)

The sole cause of man's unhappiness is that he does not know how to stay quietly in his room. Pensées (1670)

Pascal principlefor diffusion:

Pascal principlefor subdiffusion:

“Target problem”

PU(t) is a upper bound because the best a particle can do for surviving is to stay quietly in his place → Pascal principle

P(t) ≤ PU(t) Prob. of trapping of a static particle

in a sea of mobile trapsProb. of trapping of a diffusive particle in a sea of mobile traps ≤

Lower bound PL(t)P(t)≥ PL(t)=Probability that a particle remains inside a

box of size and that all the traps remain outside this box until time t

K’ , γ’ K , γ

time=0

time=t

time=t NO considered

PL(t)=Q1Q2Q3

• Q1=prob. a box of size contains no traps

• Q2=prob. no traps enter a box of size until

time t=PU(t) (target problem)

• Q3=prob. that the particle has no left the box

of size up to time t

is chosen so that PL(t) is maximized

fate of a particle in a (vanishing) sea of sub diffusive traps

Documents

d sea of subdiffusive

subdiffusive nature

mobile particle

sea of vanishing subdiffusive

sea of subdiffusive

mobile traps

ttarget problem pt

static particle