Reaction/Aggregation Processes. Clément Sire

Transcription

1 Reaction/Aggregation Processes Clément Sire Laboratoire de Physique Théorique CNRS & Université Paul Sabatier Toulouse, France

2 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

3 But Also Two-dimensional decaying turbulence Epidemics spreading (S+H S+S, S H, or S Ø ) Formation of river and vascular networks

4 And Even Galaxy or bacteria clusters Poker tournaments (players aggregate, chips are conserved ) and the reverse and similar problem of fracture (A A+A)

5 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 z and is often scale invariant: P( m, t) t f ( m / t )

6 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 )

7 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)

8 Reaction-Diffusion Processes The reaction A A Ø Mean-field approach: d dt k, ( t) ~ 1 2kt 2kt (0) In fact, there is a strong depletion effect near surviving particles: d d/2 ( t) ~ L ( t) ~ t, d 2 ln t ( t) ~, d 2 t And the mean-field approximation is exact in d d c 2 The mean-field decay is the fastest and can be realized by stirring the solution (hence eliminating spatial correlations)

9 Reaction-Diffusion Processes The reaction A+B Mean-field approach: Ø da dt k 1 1 A ( t) ~, if A (0) B(0) 1 kt kt (0) A In fact, when A( t) B( t) there is again a very strong depletion effect giving rise to segregation of the particles A, ( t) (0)exp kt (0) (0), if (0) (0) A A B A A B B

10 Reaction-Diffusion Processes The reaction A+B Ø, for (0) (0) A B A physical argument: Initially, in a region of linear size L, (0) (0) ~ A B L d /2 The species in excess will dominate in this region, and L is ultimately the diffusion length, L(t)~t 1/2. Hence, d /4 ( t) ~ t, d 4 ln t ( t) ~, d 4 t And the mean-field approximation is exact in d d c 4

11 Reaction-Diffusion Processes The reaction A A A Mean-field approach: d k, ( t) ~ dt 1 kt kt (0) Same depletion effect as for A A Ø: d d/2 ( t) ~ L ( t) ~ t, d 2 ln t ( t) ~, d 2 t And the mean-field approximation is again exact in d d c 2

12 Reaction-Diffusion Processes The reaction A A A with conservation of mass Mean-field approach (Smoluchowsky s equation): d m ( m, t) k ( m s, t) ( s, t) ds 2 k( m, t) ( t), dt 0 with () t ( m, t) dm, and m m( m, t) dm 0 tot 0 Mean-field solution: ( mt, ) m exp mtotkt 2 m ( kt) tot Only correct in d d c 2

13 Reaction-Diffusion Processes The reaction A A A with conservation of charge and constant injection of particles (mass distribution Im ( )) Mean-field approach (Smoluchowsky s equation): d ( m, t) k ( m s, t) ( s, t) ds 2 k( m, t) ( t) Im ( ), dt with m tot m( m, t) dm It General scaling solution: z ( m, t) m f ( m / t ), with f (0) 1, and f ( x) exp( x) x0 3 1 I 0 : zmf 2, zd 1, and 2 2 z 3 2 I 0 : zmf 1, zd 1, and 3 4 z x Polydispersity ( dc 2)

14 Reaction-Diffusion Processes Application to the formation of a river network Fractal dimension: d f 4 3 The next river is ~ 3 x longer than the previous one V=R 2 V=pR 2 V=R 3 V=R 4/3

15 General Mean-Field Approach A general physical aggregation process is described by: Conservation law(s): m m m Nature of the collision/aggregation physical process: the merging Kernel, Km (, m ) 1 2 Ex: d-dimensional cross-section ¾~R d and m~r D Intrinsic growth of clusters Deposition: Fracture Kernel K( m ) m m m v( m, t) 1/ D 1/ D 1, m d I( m, t) m m m

16 General Mean-Field Approach A general physical aggregation process is described by: ( m, t) v( m, t) ( m, t) K( m s, s) ( m s, t) ( s, t) ds t m 2 ( m, t) K( m, s) ( s, t) ds I( m, t) Fracture... Can describe many physical situations and can lead to mono and polydispersity and even gelation

17 General Mean-Field Approach The general case of pure aggregation ( v( m, t) 0, I( m, t) 0) Assume that the collision Kernel satisfies K( m, m ) K( m, m ) K( m, m ) ~ m m, for m m Then (Van Dongen and Ernst), there is a precise z mono/polydispersity criterion: ( m, t) m f ( m / t ); f (0) ~ 1 0 : the mass distribution is monodisperse ( 0) 0 : the mass distribution is polydisperse with 1 0 : the mass distribution is polydisperse with 1 2 x f ( x) dx 0 Ex: Km ( ) can be determined b y pert urbative and non- perturbative methods 1/ D 1/ D 1, m2 m d 1 m2 d / D, 0

18 Scaling in Poker Tournaments Scaling distribution of fortunes (stacks): Scaling equation (integro-differential and non-linear; no parameter):

19 Internet & WPT Data ( $ buy-in)

20 Dynamical Phase Transitions Consider the reaction/breakdown process for diffusing particles: A+A Ø (rate k), A (n+1)a (rate p) Mean-field approach: d 2 2 k np, dt n (, ) ( p pc), with pc 0, and 1 2k ( t, p ) ~ t, with 1 c 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

21 Dynamical Phase Transitions Odd n: the directed percolation (DP) universality class Absorbing State Active State d 4; for d 1, (6)... and (6)... c Best theoretical estimate in d 1 : 11/ 5 / ! Field theoretical methods available for d close to d c 2 ( 1 / ; 4 d)

22 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 CO 2 ) Growing interfaces Flowing granular matter Porous media Turbulent liquid crystals Douady & Daer

23 Dynamical Phase Transitions DP universality class observed in d=2+1 (time) in turbulent liquid crystals (intermittent regimes between to dynamic scattering modes DSM)

24 Dynamical Phase Transitions DP universality class observed in d=2+1 (time) in turbulent liquid crystals (scaling function) 1/ ( t) ~ t f t ; p pc f (0) 1; f ( x) ~ x ; / ; f ( x) ~ exp( x ) x x

25 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

26 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 ) Thank you for your attention