From Becker-Doring to Lifshitz-Slyozov: deriving the boundary condition

HAL Id: hal-02792548

https://hal.inrae.fr/hal-02792548

Submitted on 5 Jun 2020

HAL is a multi-disciplinary open access archive for the deposit and dissemination of sci- entific research documents, whether they are pub- lished or not. The documents may come from teaching and research institutions in France or abroad, or from public or private research centers.

L’archive ouverte pluridisciplinaire HAL, est destinée au dépôt et à la diffusion de documents scientifiques de niveau recherche, publiés ou non, émanant des établissements d’enseignement et de recherche français ou étrangers, des laboratoires publics ou privés.

From Becker-Doring to Lifshitz-Slyozov: deriving the boundary condition

Romain Yvinec, Julien Deschamps, Erwan Hingant

To cite this version:

Romain Yvinec, Julien Deschamps, Erwan Hingant. From Becker-Doring to Lifshitz-Slyozov: deriving the boundary condition. BIOMAT 2015, Instituto Nacional de Matemática Pura e Aplicada. BRA., Mar 2015, Cabo Frio, Brazil. �hal-02792548�

From Becker-D¨ oring to Lifshitz-Slyozov: deriving the boundary condition

Romain Yvinec¹,Julien Deschamps² andErwan Hingant³

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_i `C₁ ÝÝáâÝÝ^ai

bi`1

C<sub>i`1</sub> (1) o`uC_i “ 7tagregates of size iu.

Thenucleation timeis given by the following waiting time, t^˚ “infttě0 :C_Nptq “1| pC_ip0qq_iě1u, (2) Typical initial conditionC_ipt“0q “Mδ_i“1.

N is a size of the nucleus : it’s a parameter of the model (with M,ai,bi).

Remark

C₁ptq `ÿ

i

iC_iptq ”M.

This model is used to understand spontaneous protein

poylmerization experiments

“Large number limit” of the nucleation time

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

Analytical approximation and numerical simulations showed (R.Y., Maria R. D’Orsogna, Tom Chou, J. Chem. Phys. 2012) :

§ non-monotone behavior with respect to the detachment rate b

§ complex depencies with respect to the total mass M (logt^˚ is not proportional to logM)

§ discrete size effect due to specific configuration that are

“traps”.

Here, we ask : what is the nucleation time for very large nucleus

NÑ8lim t^˚ (3)

Choose a fix state space

Withε“ 1

N Ñ0, we obtain a fix state space by studying (for appropriately rescaledC_i^ε)

µ^ε_tp.q “ ÿ

iě2

C_i^εptqδ_iεp.q PM_bpR^`q (4)

The nucleation time is thus

t_ε^˚ “inftt ě0 :µ^ε_tpt1uq ą0|µ^ε₀u, (5)

Based on the flux at sizei, forC_i^ε, C_i´1^{ε J}

ε i´1

ÝÝáâÝÝC_i^{ε J}

iε

ÝáâÝC_i^ε_`1, (6) if

J_i^ε“ 1 ε

´

a^ε_iC₁^εC_i^ε´b_{i`1</sub><sup>ε</sup> C<sub>i`1}^ε

¯

, (7)

We will get for the limitµ“ lim

εÑ0

ÿ

iě2

C_i^εptqδiε

aLifhsitz-Slyozov equation(in weak form) Bµ_t

Bt ` B

´

Jpx,C₁qµ_t

¯

Bx “0ô dxµ_t, ϕy

dt “

ż₈

0

ϕ¹pxqJpx,C₁qµ_tpdxq. (8) withϕPC_cpR^`_˚qand

Jpx,C₁q “apxqC₁´bpxq andC₁ptq ` ż₈

0

xµ_tpdxq “m.

References :Lauren¸cot and Mischler 2002 J. Stat. Phys., Collet et al 2002 SIAM J. Appl. Math.

The limit is deterministic, finite or infinite

The nucleation time, for this limiting model Bµt

Bt ` B

´

papxqC1´bpxqqµt

¯

Bx “0,

is

t^˚“inftt ě0 :µtpt1uq ą0|µ0u, (9) which can be finite or infinite (but is “easier” to compute, at least numerically).

Howeverfor typical initial condition µ₀ “0, we need ap0qC1´bp0q ą0 and then aboundary condition!

Strategy of proof

To prove a convergence result, the strategy is the following

§ Write down equation on µ^ε

§ Prove compactness of ppµ^ε_tqtďTqεą0 (“choose” the right scaling for that)

§ Take a convergence subsequence ofµ^ε, and prove that all terms in the equation on µ^ε do converge (and find their limit)

§ Prove a uniqueness result fot the limit equation.

First problem : Topology !

Compacts ofM_bpRq are not so easy to handle in strong topology, so we use a weak-˚ topology.

Test functions :ϕPC_bpR^{`</sup>q to capture the boundary (and NOT ϕPC<sub>c</sub>pR<sup>`}_˚q).

Themass balancepropertyC₁ptq `ÿ

i

C_iptq ”M will translate into

C₁^εptq ` ż₈

0

xµ^ε_tpdxq ”M^ε. (10) so we actually need to take test functions of the form

p1`xqϕ withϕPC<sub>b</sub>pR<sup>`</sup>q.

(Long calculations...)

With the appropriate scaling, and within the weak topology on pM_b,p1`xqdxq,

Proposition

ppµ^ε_tqtďTqεą0 is compact in DpR<sup>`</sup>,pM<sub>b</sub>,p1`xqdxqq.

GREAT BUT IT’S NOT FINISHED !

Rescaled Equation (1)

So Let’s look at the equation onµ^ε, in a weak form, xµ^ε_t, ϕy “ xµ^ε_in, ϕy `O_t^ε,ϕ

ż_t

0

ϕp2εq“

a^ε₁pC₁^εpsqq²´b^ε₂xµ^ε_s,1_2εy‰ ds

` ż_t

0

ż

∆_εpϕqa^εpxqC₁^εpsqµ^ε_spdxqds

´ ż_t

0

ż

∆_εpϕqpxqb^εpxqµ^ε_spdxqds. (11)

Remark

Large monomer number / Slow-down agregation rates / speed up depolymerization rate /specific scaling for boundary termsa^ε₁,b₂^ε.

Rescaled Equation (2)

By compactness, everything converge nicely except the red term ! xµ^ε_t, ϕy “ xµ^ε_in, ϕy `O_t^ε,ϕ

ż_t

0

ϕp2εq“

a^ε₁pC₁^εpsqq²´b^ε₂xµ^ε_s,1_2εy‰ ds

` ż_t

0

ż_`8

0

∆_εpϕqa^εpxqC₁^εpsqµ^ε_spdxqds

´ ż_t

0

ż_`8

0

∆_εpϕqpxqb^εpxqµ^ε_spdxqds. (12)

We need to look at the termxµ^ε_s,1_2εy “. . .“C₂^ε!

Fast variable

The equations onC₂^ε involves C₃^ε, which involvesC₄^ε and so on...and there arefast variables.

C_i^ε_´1

1

εa^εpεpi´1qqC₁^εC_i´1^ε

ÝÝÝÝÝÝÝÝÝÝÝÝá âÝÝÝÝÝÝÝÝÝÝÝÝ

1 εb^εpεiqC_i^ε

C_i^ε

1

εa^εpεiqC₁^εC_i^ε

ÝÝÝÝÝÝÝÝÝÝá âÝÝÝÝÝÝÝÝÝÝ

1

εb^εpεpi`1qqC<sub>i`1</sub>^ε

C_i`1^ε ,

We cannot hope a convergence in a standard function space.

We need a functional space that do not see the fast variations, such asMpR^{`</sup>,l<sub>1</sub>pR<sup>`}qq, (with respect to the weak topology) for the occupation measure, defined by, for measurable setsU of l¹,

Γ^ε

´

r0,Ts ˆU

¯ :“

ż_t

0

1_tpC^ε

i psqqiPUuds

Parenthesis (Kurtz’ averaging theorem (1992))

LettpXn,Ynqua family of stochastic process, withsuitable

compactness conditions, such that for suitable test functionsf,g,

fpXnptqq “ żt

0

AfpXnpsq,Ynpsqqds`ε<sup>1,f</sup><sub>n</sub> ptq `M_t^1,f,n, gpYnptqq “

żt

0

αnBgpXnpsq,Ynpsqqds`ε<sup>2,g</sup><sub>n</sub> ptq `M_t^2,g,n,

whereM_t^1,f,n,M_t^2,g,n are martingales,αnÑ 8and

nÑ8lim E“ sup

tďT

|ε^1,f_n ptq |‰

“ 0,

nÑ8lim E“ sup

tďT

β^´1_n |ε^2,g_n ptq |‰

“ 0.

Assume that the operatorBx :DpBq ÑCbpE2q,Bxgpyq “Bgpx,yq, has a unique stationary distributionπx. Then any limiting pointX ofXnis solution of the martingale problem for

Cfpxq “ ż

Afpx,yqπxpdyq

Theorem

(Under some conditions...)pµ^ε,pC_i^εqqconverges in DpR<sup>`</sup>,pM,p1`xqdxqq ˆMpR^{`</sup>,l<sub>1</sub>pR<sup>`}qqtowards

xµt, ϕy “ xµin, ϕy ` ż_t

0

ϕp0q“

a1pC1psqq²´b2C2psq‰

` ż_t

0

ż_`8

0

ϕ¹pxqpapxqC₁psq ´bpxqqµ_spdxqds. xµ_t,idy `C<sub>1</sub>ptq “ m:“ xµ<sub>0</sub>,idy `C₁p0q

AndpCiptqqiě2 is a stationary solution in l1 of the following deterministic Becker-D¨oring system (for constant C1“C1ptq)

C92“0 “ ´

´

a2C1C2´b3C3

¯ , C9_i “0 “

´

a_i´1C₁C_i´1´b_iC_i

¯

´

´

a_iC₁C_i ´b_{i`1</sub>C<sub>i`1}

¯ . where a_i, b_i depends on the behavior of a,b at 0

Forapxq “ax^r `opxq,bpxq “bx<sup>r</sup> `opxq,r ă1, the above equation reduces to, forC₁ptq ď ^b_a,

xµt, ϕy “ xµ_in, ϕy

` ż_t

0

ż_`8

0

ϕ¹pxqpapxqC1psq ´bpxqqµspdxqds. xµt,idy `C1ptq “ m:“ xµ0,idy `C1p0q

and to, forC₁ptq ą^b_a,

xµt, ϕy “ xµ_in, ϕy ` żt

0

ϕp0qa1pC1psqq²

` ż_t

0

ż_`8

0

ϕ¹pxqpapxqC₁psq ´bpxqqµ_spdxqds. xµt,idy `C1ptq “ m:“ xµ0,idy `C1p0q

Remark

The boundary condition is : flux at 0 = dimerization rate

Numerical illustration

§ apxq ”1, bpxq “x,

§ Incoming charcateristics.

§ Video

0 2 4 6 8 10

Time 0

1 2 3 4 5 6

(Rescaled)Number

Polymerhµ^εt,1i

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

0 2 4 6 8 10

Time 0.0

0.5 1.0 1.5 2.0 2.5 3.0

(Rescaled)Number

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

10⁻⁴ 10⁻³ 10⁻² 10⁻¹ 10⁰

ε 10⁻⁴

10⁻³ 10⁻² 10⁻¹ 10⁰

(Rescaled)Number

Dimerp^ε_0,tvs.ε

t= 1 t= 5 t= 9 Slope1

Numerical illustration and further work

§ apxq ”x, bpxq “1,

§ outgoing charcateristics.

§ Video : Metastability

0 5 10 15 20

Time 0.0

0.5 1.0 1.5 2.0 2.5 3.0 3.5

(Rescaled)Number

MonomerCt^ε

0 5 10 15 20

Time 0

2 4 6 8 10

(Rescaled)Number

Polymerhµ^εt,1i

Obrigado !

§ 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 ofp˜a^ε_iq,pb˜^ε_iq,pC˜_i^εp0qq,pM˜^εq. LetpC˜_i^εptqqbe the corresponding solution and define (supposea^ε₁,b^ε₂,a^ε,b^ε and C₁^εp0q, µ^εp0,dxq converges in an appropriate sense)

a^εpxq :“ 1 ε

ÿ

iě2

˜

a^ε_iχ^ε_ipxq,

b^εpxq :“ εÿ

iě3

˜b_i^εχ^ε_ipxq.

a^ε₁ :“ 1 ε³˜a^ε₁, b^ε₂ :“ b˜^ε₂,

˜

a^ε_i „ a_iε^1`ra,

b˜^ε_i „ b_iε^rb´1, minpr_a,r_bq ă1. We then define the variables

C₁^ε “ε²C˜₁^ε, µ^εpt,dxq “εÿ

iě2

C˜_i^εptqδ_iεβpdxq.