Nucleation time in coagulation-fragmentation models

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Nucleation time in coagulation-fragmentation models

Romain Yvinec, Julien Deschamps, Erwan Hingant

To cite this version:

Romain Yvinec, Julien Deschamps, Erwan Hingant. Nucleation time in coagulation-fragmentation

models. Journées Modélisation Aléatoire et Statistique de la SMAI (Société de Mathématiques Ap-

pliquées et Industrielles), Société de Mathématiques Appliquées et Industrielles (SMAI). FRA., Jun

2015, Les Karellis, France. �hal-02794818�

Nucleation Time in Coagulation-Fragmentation models

Romain Yvinec

, Julien Deschamps

and Erwan Hingant

1BIOS group INRA Tours, France.

2DIMA, Universita degli Studi di Genova, Italy.

3CI²MA, Universidad de Concepci´on, Chile

Becker-D¨ oring model for Nucleation

Reversible one-step agregation

C

` C

ÝÝá âÝÝ

bi`1

C

(1)

The nucleation time is given by the following First Passage Time, T

:“ inftt ě 0 : C

ptq “ 1 | C

pt “ 0q “ Mδ

u . (2) More generally, we can look at

T

:“ inftt ě 0 : C

ptq ě ρM

| C

pt “ 0q “ M δ

u . (3) for given positive constant ρ and 0 ď h ď 1.

Remark

C

ptq `

i

iC

ptq ” M .

“Large number limit” of the nucleation time

What are the dependencies of the nucleation time with respect to the model parameters ?

Here, we ask : what is the nucleation time for very large initial quantity M and nucleus size N ?

M,NÑ8

lim T

(4)

Premi` ere approche ”standard”

“petits M ,N ”

on peut d´ ecrire l’ensemble du syst` eme, ´ ecrire la matrice de passage entre les ´ etats et r´ esoudre l’´ equation lin´ eaire correspondante.

(8,0,0)

(6,1,0)

(4,2,0)

(2,3,0) (5,0,1)

(3,1,1)

(1,2,1) (2,0,2) (0,4,0)

(0,1,2) (7,0,0)

(5,1,0)

(3,2,0)

(1,3,0) (4,0,1)

(2,1,1)

(0,2,1) (1,0,2)

(a) (b)

Dimension du syst` eme

7tconfig. tC u,

N

ÿ

i“1

iC

“ M u «

M

N!

Simplified model 1) Constant Monomer formulation

The model with Constant C

is Linear, we have ’exact’ solution Proposition

E“

T

„

C

pa, Nq M

1

M

(5) where C

pa, nq is a constant that depends on api q, i ď N and N.

pC

q

follows a Poisson distribution of mean the solution of a linear ODE (constant monomer original Becker-D¨ oring formulation)

d c

dt “ Ac ` B, dc

dt “ c

c

,

(6)

for which we can show c

ptq «

´^N´1ź

k“1

apkq

¯

C

t

pN ´ 1q! ,

Simplified model 2) Single-cluster formulation

If we suppose that a single cluster can be present at a time, then the model is one-dimensional, ’exact’ solution and classical FPT theory gives (it’s a 1D discrete random walk)

Proposition

E“

T

“

N´1

ÿ

i“1 i

ÿ

j“1

b

b

. . . b

a

a

. . . a

1

pM ´ iq . . . pM ´ j qM

Remark

Similar results with “Pre-equilibrium” hypothesis when b Ñ 8.

General idea : rescaling strategy

Consider the deterministic irreversible aggregation model dc

dt “ a

c

pc

´ c

q, c

`

ÿ

i

ic

“ m,

t

“ inftt ě 0 : c

ptq “ ρm | c

p0q “ mu Then with c

“

, τ “ mt ,

dc

d τ “ a

c

pc

´ c

q, c

`

i

ic

“ 1,

τ

“ inf tt ě 0 : c

ptq “ ρ | c

p0q “ 1u This strategy shows that

t

“ C

m .

Asymptotic for finite N

The same results holds Proposition

E

“

T

„

C

pa, Nq M

1

M

(7) where C

pa, nq is a constant that depends on api q, i ď N and N.

Indeed,

is “close” to the solution of the SDE d c

“ Jpc

qcdt ` ?

εBpc

qcd w

(8)

where the variance is of order ε “

and the nucleation time can be comuted by

T

” inf tτ ě 0 : c

pτ q “ ρε

| c

pτ “ 0q “ 1δ

u . (9)

But ! There are complex behavior for ’intermediate’ M . Indeed T

may be

§

bimodal

§

nearly independent of M over several log.

log10(M)

1 3 5 7 9

log10(T)

-10 -8 -6 -4 -2 0 2 4

N=6

log10(M)

1 3 5 7 9

N=10

log10(M) 1 3 5 7 9 11

N=15

Asymptotic for large N

Wa want to prove behavior like T

?M,M

1,0

“ inf tt ě 0 : C

ptq “ 1 | C

pt “ 0q “ M δ

u

„

? M . (10)

We need a limit theorem when N, M Ñ 8...

Choose a fix state space

With ε “ 1 N “ 1

? M Ñ 0, we obtain a fix state space by looking at

µ

p. q “

iě2

ε

C

pε

tqδ

p. q P

pR

q (11)

The nucleation time is thus T

?M,M

1,0

“ inf tt ě 0 : µ

pt1uq “ ε

ą 0 | µ

“ 0u, (12)

The scaling exponents α, γ need to be adjusted in order to obtain

a non-trivial limit (and according to other scaling hypotheses on

a, b...).

Rescaled Equation (1)

ÿ

i

ϕpiεqC

ptq “

i

ϕpiεqC

p0q `

`

0

ϕp2εq

a

pC

psqq

´ b

C

psq

ds

`

0

ÿ

i

pϕpiε ` εq ´ ϕpiεqqa

C

ps qC

ps q ds

`

0

ÿ

i

pϕpiε ´ εq ´ ϕpiεqqb

C

psq .

xµ

, ϕy “ xµ

, ϕy `

0

ϕp2εq

a

pC

psqq

´ b

C

psq

ds

` ε

0

ż

∆

pϕqpa

pxqC

psq ´ b

pxqqµ

pdx q ds (13)

Rescaled Equation (2)

Hence, if the flux pa

pxqC

ps q ´ b

pxqq is of order ε

), we may expect everything to converge nicely, except the red term !

xµ

, ϕy “ xµ

, ϕy `

0

ϕp2εq

a

pC

psqq

´

ds

`

0

ż_`8

0

∆

pϕqpa

pxqC

psq ´ b

pxqqµ

pdx q ds (14)

We need to look at the term

!

Fast variable

The equations on C

involves C

, which involves C

and so on...and there are fast variables.

C

1

εa^εpεpi´1qqC₁^εC_i´1^ε

Ý ÝÝÝÝÝÝÝÝÝÝÝ á â ÝÝÝÝÝÝÝÝÝÝÝ Ý

1 εb^εpεiqC_i^ε

C

1

εa^εpεiqC₁^εC_i^ε

ÝÝÝÝÝÝÝÝÝÝá âÝÝÝÝÝÝÝÝÝÝ

1

εb^εpεpi`1qqC<sub>i`1</sub>^ε

C

,

We cannot hope a convergence in a standard function space.

We need a functional space that do not see the fast variations, such as

pR

, l

pR

qq, (with respect to the weak topology) for the occupation measure, defined by, for measurable sets U of l

,

Γ

´

r0, T s ˆ U

¯

:“

ż_t

0

1

i psqqiPUu

ds

Theorem

(Under some conditions...) pµ

, pC

qq converges in

pR

, p

p1 ` xqdxqq ˆ

pR

, l

pR

qq towards

xµ

, ϕy “ xµ

, ϕy `

0

ϕp0q“

a1pc1psqq²´b2c2psq‰

`

0

ż_`8

0

ϕ

pxqpapxqc

ps q ´ bpxqqµ

pdxq ds . xµ

,

y ` c

ptq “ m :“ xµ

,

y ` c

p0q

And pc

ptqq

is a stationary solution in l

of the following deterministic constant-monomer Becker-D¨ oring system ( for

’freezed’ c

“ c

ptq)

c

“ 0 “ ´

´

a

c

c

´ b

c

¯

, c

“ 0 “

´

a

c

c

´ b

c

¯

´

´

a

c

c

´ b

c

¯

.

where a

, b

depends on the behavior of a, b at 0

For apxq “ ax

` opxq, bpxq “ bx

` opxq, r ă 1, the above equation reduces to, for

,

xµ

, ϕy “ xµ

, ϕy `

0

ϕp0qa1pc1psqq²

`

0

ż_`8

0

ϕ

pxqpapxqc

ps q ´ bpxqqµ

pdxq ds . xµ

,

y ` c

ptq “ m :“ xµ

,

y ` c

p0q

Remark

The boundary condition is : flux at 0 = dimerization rate

Numerical illustration : SBD to LS

§

apxq ” 1, bpxq “ x,

§

Incoming charcateristics.

§

Video

0.0 0.5 1.0 1.5 2.0

Time 0.00.5

1.01.5 2.02.5 3.03.5 4.0

(Rescaled)Number Number of free particlec^ε₁(t)andc1(t)

0.0 0.5 1.0 1.5 2.0

Time 0

2 4 6 8 10 12

(Rescaled)Number Number of clustershµ^ε_t,1iandhµt,1i

ε= 10⁻² ε= 10⁻³ ε= 5·10⁻⁴ Deterministic

Numerical illustration : verifying the time-scale

10¹ 10² 10³ 10⁴ 10⁵ 10⁶ 10⁷ 10⁸ 10⁹ M

0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7

Time/√ M

First Passage TimeT_1,0^{√M ,M}

10⁴ 10⁵ 10⁶ 10⁷

M 01

23 45 67 8

Time/√ M

First Passage TimeT_ρ,h^{√M ,M},ρ= 0.01,h= 0.5

Numerical illustration and further work

§

apxq ” x, bpxq “ 1,

§

outgoing charcateristics.

§

Video : Metastability

0 50 100 150 200 250 300 350 400

Time 0.00.5

1.01.5 2.02.5 3.03.5

Number

Number of free particlec^ε₁(t),ε= 0.04

0 500 1000 1500 2000 2500 3000

Time 0.00.5

1.01.5 2.02.5 3.03.5

Number

Number of free particlec^ε₁(t),ε= 0.025

10² 10³ 10⁴ 10⁵

M= 3/ε² 10⁰

10¹ 10² 10³ 10⁴ 10⁵ 10⁶ 10⁷ 10⁸

Time

First Passage TimeT0

Merci !

§

First passage times in homogeneous nucleation and self-assembly,

R.Y., Maria D’Orsogna and Tom Chou

(Journal of Chemical Physics (2012) 137 :244107)

§

From a stochastic Becker-D¨ oring model to the

Lifschitz-Slyozov equation with boundary value,

Julien Deschamps, Erwan Hingant and R.Y.,

arXiv :1412.5025 (2014)

Consider a sequence ofp˜a_i^εq,p˜b_i^εq,pC˜_i^εp0qq,pM˜^εq. LetpC˜_i^εptqqbe the corresponding solution and define (supposea^ε₁,b₂^ε,a^ε,b^ε and

C₁^εp0q, µ^εp0,dxqconverges in an appropriate sense)

a^ε_i :“ ε^A˜a^ε_i , @iě2, b^ε_i :“ ε^Bb˜_i^ε, @iě3, a^ε₁ :“ ε^A1˜a^ε₁,

b^ε₂ :“ ε^B1b˜₂^ε.

χ^ε_i :“ 1_rpi´1{2qεβ,pi`1{2qεq, a^εpxq :“ ÿ

iě2

a^ε_iχ^ε_ipxq,

b^εpxq :“ ÿ

iě3

b^ε_iχ^ε_ipxq.

We then define the variables

C_i^ε “ ε^αC˜_i^ε,@i ě2, C₁^ε “ ε^θC˜₁^ε.

µ^εpt,dxq “ ÿ

iě2

C_i^εptqδ_iεβpdxq, M^ε “ ε^α`βM˜^ε.

The above results hold with the following choices

θ “ α`β , A “ ´α , B “ β ,

A₁ “ ´α´2β , B₁ “ 0,

a^ε_i „ aiε^raβ,@i ě2 b_i^ε „ biε^rbβ,@iě2,

0 ď minpra,rbq ă1.

Concrete Example

Consider a sequence of p˜ a

q, p b ˜

q, p C ˜

p0qq, p M ˜

q. Let p C ˜

ptqq be the corresponding solution and define (suppose a

, b

, a

, b

and C

p0q, µ

p0, dxq converges in an appropriate sense)

a

pxq :“ 1 ε

ÿ

iě2

˜

a

χ

pxq , b

pxq :“ ε

iě3

˜ b

χ

pxq .

a

:“ 1 ε

˜ a

, b

:“ b ˜

,

˜

a

„ a

ε

,

b ˜

„ b

ε

, minpr

, r

q ă 1 . We then define the variables

C

“ ε

C ˜

, µ

pt, dx q “ ε

iě2

C ˜

ptqδ

pdx q .