Clément Sire

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Reaction/Aggregation Processes

Clément Sire

Laboratoire de Physique Théorique

CNRS & Université Paul Sabatier Toulouse, France

www.lpt.ups-tlse.fr

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A Few Classical Examples

 Ubiquitous in physics and chemistry

 Atoms diffusing on a surface and Diffusion Limited Aggregation

 Atomic clusters

 Chemical reactions (A+B → C)

 Smoke in air/latex particles at the surface of water

 Breath figures

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But Also…

 Two-dimensional decaying turbulence

 Epidemics spreading (S+H → S+S, S → H, or S → Ø…)

 Formation of river and vascular networks

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And Even…

 Galaxy… or bacteria clusters

 Poker tournaments (players “aggregate”, chips are conserved…)

 … and the reverse and similar problem of fracture (A → A+A)

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Important Features

 Importance of conservation laws: mass, energy, momentum, or more exotic quantities (flow of a river, chips in a poker tournament…)

 Important quantities: distribution of cluster masses (or joint distribution of conserved quantities), number of

remaining clusters…

 This distribution of “mass” can be mono or polydisperse and is often scale invariant: P m t( , )  t^^ f m t( / ^z)

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Important Features

 Importance of the diffusive or ballistic motion of clusters

 Possibility of obtaining compact or fractal clusters

 The mean-field approach (neglecting spatial correlations) is often incorrect in low effective dimensions

(typically d· 2 or even d· 4 for diffusion processes)

 Out of equilibrium reaction/aggregation processes can lead to dynamical phase transitions (directed

percolation…)

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A Few Selected Topics

 Simple reaction-diffusion processes:

A+A → Ø; A+B → Ø; A+A → A

(application to a river network, in the latter case)

 Monodispersity (bell-shaped mass distribution)

& polydispersity (power-law mass distribution)

in the framework of Smoluchowsky’s approach

(application to breath figures and… poker tournaments)

 Dynamical phase transitions:

A+A → Ø, A → (n+1)A

(n odd: directed percolation; n even: parity conserving)

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Reaction-Diffusion Processes

 The reaction

Mean-field approach:

In fact, there is a strong depletion effect near surviving particles:

And the mean-field approximation is exact in

The mean-field decay is the fastest and can be realized by stirring the solution (hence eliminating spatial correlations)

A  A  Ø

2 1 1

( ) ~

1 2

(0) 2

2 , t

kt kt

d k

dt 



 



  

 

( ) ~ ( ) ~ /2, 2 ( ) ~ ln , 2

d d

t L t t d

t t d

t



  



c 2 d  d 

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Reaction-Diffusion Processes

In fact, when

there is again a very strong depletion effect giving rise to segregation of the particles

A+B  Ø

A ,

A B

d k

dt

     

1 1

( ) ~ , (0) (0)

1 (0)

A A B

A

if

t kt

kt

  



 



 

 

( ) (0) exp (0) (0) , (0) (0)

A t A kt B A if A B

         

( ) ( )

A t B t

  

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Reaction-Diffusion Processes

A physical argument:

Initially, in a region of linear size L,

The species in excess will dominate in this region, and L is ultimately the diffusion length, L(t)~t^1/2. Hence,

And the mean-field approximation is exact in , _A(0) _B(0) A+B  Ø for   

( ) ~

/4

, 4 ( ) ~ ln , 4

t t

d

t t d

t









c 4 d  d  (0) (0) | /2

| _A  _B ~ L^^d

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Reaction-Diffusion Processes

Same depletion effect as for :

And the mean-field approximation is again exact in A  A  A

2 1 1

( ) ~

1 (0) d ,

d k t

t

t k

kt



 



  



( ) ~ ( ) ~ /2, 2 ( ) ~ ln , 2

d d

t L t t d

t t d

t



  



c 2 d  d  A  A  Ø

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Reaction-Diffusion Processes

 The reaction with conservation of “mass”

Mean-field approach (Smoluchowsky’s equation):

Mean-field solution:

Only correct in

A  A  A

0

0 0

( , ) ( , ) ( , ) 2 ( , ) ( ),

( , ) , ( , )

with ( ) _tot

d m

m t k m s t s t ds k m t t dt

m t dm and m t dm

t m m

    

 ^  ^ 











 

c 2 d  d 

2

ex

( ) )

( p

,

^tot

tot

m m kt m t m

 kt

 

 

  



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Reaction-Diffusion Processes

 The reaction with conservation of “charge” and constant injection of particles (mass distribution )

Mean-field approach (Smoluchowsky’s equation):

General scaling solution:

Polydispersity

A  A  A

( , ) ( , ) ( , ) 2 ( , ) ( ) ),

( , )

( with _tot

d m t k m s t s t ds k m t

m m

dt t I m

I m t dm t

    











  



  







0

(0) 1, ( ) exp( )

( , ) ( / ),

with and

x x

z

f f x x

m t m ^ f m t



 





1

3 1

0 : 2, , 2

2

0 : 1, 3 , 3

4

2

MF

and and

d

I z z

z

I z z

z





      

      

( ) I m

(d_c  2)

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Reaction-Diffusion Processes

 Application to the formation of a river network

 Fractal dimension:

 The “next” river is

~ 3 x longer than the previous one

V=pR² V=R²

V=R³ V=R^4/3

4

f

3 d 

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General Mean-Field Approach

A general physical aggregation process is described by:

 Conservation law(s):

 Nature of the collision/aggregation physical process: the merging Kernel,

Ex: d-dimensional “cross-section” ¾~R^d and m~R^D

 Intrinsic growth of clusters

 Deposition:

 Fracture Kernel

 …

1 2

m  m  m

1 2) ( ,

K m m

( , ) I m t

 ( , ) m  v m t



^1/ ^1/



1, 2 1 2

( m ) ^D ^D ^d

K m  m  m

1 2

m  m  m

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General Mean-Field Approach

A general physical aggregation process is described by:

Can describe many physical situations and can lead to mono and polydispersity and even gelation…



^{( , )}



⁽ ^{, )}

( ( ,

, )

) ( , ) ( , ) ( , )

2 ( , ) ( , )

( , ) Fracture ...

m t v m t m t K m s s m s t s t ds

t m

m t s t ds

t

m m

s I

K

   

 









  

  

 



  



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General Mean-Field Approach

The general case of pure aggregation

 Assume that the collision Kernel satisfies

 Then (Van Dongen and Ernst), there is a precise mono/polydispersity criterion:

( ( , )v m t  0, ( , )I m t  0)

1 2 1 2

1 2 1 2 2 1

( , ) ( , )

( , ) ~ , for

m m m m

m

K K

m m m

K m m



  

  





0

0 : 0)

0 : 1

2 ( )

the mass distribution is

the mass distribution is with the mass distribution is with

can be determined b

(

y

x^{ } f x dx

 

  

 ^ ^

 

  

  

 





monodisperse polydisperse polydisperse

perturbative and non - perturbative methods

( , )m t m ^ f m t( / ^z); f (0) ~ 1

  ^



^1/ ^1/



1, 2 1 2

( )

/ , 0

D D d

m m m

d K m

 D 

 

 

Ex:

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Scaling distribution of fortunes (stacks):

Scaling equation (integro-differential and non-linear; no parameter):

Scaling in Poker Tournaments

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Internet & WPT Data

(20-10000$ buy-in)

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Dynamical Phase Transitions

Consider the reaction/breakdown process for diffusing particles: A+A → Ø (rate k), A → (n+1)A (rate p)

Mean-field completely fails in describing the fact that for d<d_c, one has p_c>0, and different universality classes depending on the parity of n

2

( , ) ) , 0,

2 ,

2 1

( , ) ~ , 1

( with and

with

c c

c

d k np

n p p

k t

t

p t

d

p ^





 











    

 









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Dynamical Phase Transitions

Odd n: the directed percolation (DP) universality class

4 ; for 1, 0.276486(6) ... and 0.159464(6 )...

d

c

 d     



^{1 1/ 5 / 2}



^0.276393..

1 . !

Best theoretical estimate in d  :    



Field theoretical methods available for close to d d_c



( 1 / 6 0.01128 ...;  2   4 d)

Absorbing State

Active State

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Dynamical Phase Transitions

Odd n: the directed percolation (DP) universality class

 DP universality class is ubiquitous in out of equilibrium physics and is thus very robust (adding the reaction A+A → A does not change anything…),… except in the presence of disorder

 In principle, many possible experimental realizations:

 Catalytic reactions on a surface (CO+O → CO₂)

 Growing interfaces

 Flowing granular matter

 Porous media

 Turbulent liquid crystals

 … Douady & Daer

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Dynamical Phase Transitions

DP universality class observed in d=2+1 (time) in turbulent liquid crystals

(intermittent regimes between to dynamic scattering modes – DSM)

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Dynamical Phase Transitions

DP universality class observed in d=2+1 (time) in turbulent liquid crystals (scaling function)

 

^1/

( ) ~t t ^ f t ^ ; p p_c

 ^    



^{(0) 1;} ^{( ) ~} ^; ^/ ^; ( ) ~ exp( | |)



x x

f f x x^    f x x

 

  

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Dynamical Phase Transitions

Even n: the parity conserving (PC) universality class (a “theorist’s curiosity”)

d_c>1, but d_c· 5/3<2 !

Different critical exponents from DP (in d=1, ¯¼ 0.4, ±¼ 0.2)

Strong numerical corrections to scaling (DP ???)

No analytical (even approximate) results available

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Conclusion

Reaction/aggregation processes:

Are ubiquitous in Nature

Appear at all spatial and temporal scales

Offer rich physical properties (dynamical phase diagram, fractals, dynamical scaling…)

Involve all the modern tools of theoretical physics

(field theory, renormalization group, perturbative and non perturbative methods, sophisticated numerical methods…)