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reaction with mass-dependent diffusion rates

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					  Continuously varying dynamical exponents
in A+B→0 reactions with attractive interactions




               권성철
             윤수연, 김 엽
          복잡계 정보 물리학 연구실
              경희대학교
1. Review on diffusion-limited reactions A  B  0

2. The reaction with infinitely-ranged attractive
    interaction between A and B in one-dimension

3. The reaction with distance-dependent attractive
    interaction between A and B in one-dimension

4. Summary
    1. Review on diffusion-limited A  B  0 reactions
(0) Experimental phenomena

   Electron-hole recombination in semiconductors
   Soliton-antisoliton dynamics in quasi 1-d systems
   Evolution of competing species in biology
   Aerosol dynamics
   Star formation
   Polymerization
   Etc

   Extensive investigation in 1980~1990 after A. A. Ovchinnikov
    and Ya. B. Zeldovich, Chem.Phys. (1978)
(1) Model
             1 / 2  bA   1 / 2  bA     1 / 2  bB      1 / 2  bB




                    A B                       0
(2) Fluctuation-dominant kinetics
 Initial number of A and B in a volume V  Ld


       N A  N B   A (0) Ld   A Ld   for       A (0)   B (0)

   The fluctuation : N  N  
                        A    B                  A Ld
   Segregation of single-species domains
                            L


                                      t ~ Lz
                      LA            L AB


                                L
   Reactions occurs only at domain boundaries
   Slowing down of reactions
   Anomalous kinetic behavior which cannot be predicted by
    mean-field rate equations
   The kinetics depends on
       dimensions, differences of b A and bB ,
       hard-core or bosonic interaction
(3) Scaling of density and length in one dimension
 Isotropic diffusion : b  b  0
                         A   B


            A ~ 1 / L ~ t 1/ 4   L ~ t 1/ 2 ,       LAB ~ t 3 / 8 ,     LA ~ t 1 / 4
   Relative drift : bA  bB

           A ~ 1 / L ~ t 1/ 2    L ~ t1 ,        LAB ~ t 1/ 2 ,       LA ~ t 1 / 2
  Uniformly driven to one direction : bA  bB  0
- Without hard-core interaction : same as isotropic diffusion
- With hard-core interaction :
 S. A. Janowsky, PRE, 52, 2535 (’95) , Y. Shafrir and D. ben-
   Abraham, Phys. Lett. A(2001).

          A ~ 1/ LA ~ t 1/ 3 ,   L ~ t 7 /12 ,      LAB ~ t 3 / 8 ,      LA ~ t 1 / 3
S. Redner, PRE, 52, 2540 (’95)
           L ~ t 2/3                 A ~ 1 / L ~ t 1/ 3
(4) Universality classes in one dimension
                     without HC int.       with HC int.
    bA  0, bB  0      1/4                     1/4


      bA  bB  0      1/4                     1/3

       bA  bB         1/2                     1/2

 For uniform bias of bA  bB  0,
  the hard-core int. changes the universality class.
 Direct measurement of  is difficult due to slow approach to
  asymptotic scaling region
 S. A. Janowsky, Phys. Rev. E, 1995 ;

                   0.31(2) for L  4 10 6 , t  5 10 6
2. The reaction with infinitely-ranged attractive interaction between
   A and B in one-dimension

Configuration dependent attractive interaction

    Same neighbors : diffusive

Different neighbors : ballistic

    Bulk particles : isotropic diffusion
    Boundary particles : ballistic annihilations




    Pentagonal space-time trajectories
    Interplay of the isotropic diffusions and ballistic annihilations
                                            1 / 3
    lead to anomalous density decay of t
The critical behavior of infinitely-ranged attractive int. In one dimension


                L ~ t 2/3

                LAA ~ L ~ t 1/ 3 , LAB ~ L ~ t 2 / 3

                 A ~ 1 / L ~ t 1/ 3 ,
                 AA ~ 1 / LAA ~ t 1/ 3 ,  AB ~ 1 / L ~ t 2 / 3

  What is a more general case ?

                            Distance-dependent attractive int.
3. The reaction with distant-dependent attractive interaction
   between A and B in one-dimension
                     A                        B
                               rAB

            distance-dependent attractive interaction
                                                
               v A  vB  v(rAB ) ~ rAB              (  0)
                                         1
      Collision time :     t AB ~ rAB
 Physical system : oppositely charged particle systems.
 For conservative systems, the force is given as

      1 2              1 2
        mv0  V (r0 )  mv (r )  V (r )
      2                2                                F (r ) ~ r  ( 2 1)
                 dv d
            v(r )  V (r )   F (r )
                 dr dr
(1) Model
 Configuration dependent attractive interaction
 Hopping probabitity

                        P(r )  1  r  
                               1
      1  P( r )
                               2

                          r
 Bulk particles : isotropic diffusion         P( r )  1 / 2
 Boundary particles : Annihilations due to the Biased Diffusions



 Continuous varying exponents for -variation
 Interplay of the isotropic diffusions and ballistic annihilations
   lead to anomalous density decay
 0     0.1     1.25
    (2) Analytical approach
     Schematic trajectories of adjacent domains

                                                                L  domain size ,
               vA      vB       v A  v B  v ( r ) ~ r       N L  the number of particles
                                   v A  vB  v0
                                                                        in the domain
                                                                                   NL ~ L
     x              L AB L AA                                 1  LAB 1 ,
                               
t                                                             2   1  t2 with t2  ( LAA  LAB ) 1 ,
                                    1                2
                                                                               
                                              t2                                         n 1
                                                              n   n 1  tn with tn   (iLAA  LAB ) 1
                                                                                         i 1
                                                                               n
                                                              n ~ LAB1   ( LAB  yLAA ) 1 dy
                                                                      

    n   tn                                                                    1
   n  L  1
           AB
             1
            LAA
                       
                ( LAB  nLAA )  2  LAB2
                                         
                                                         for   0
                                                           n  n LAB  n(n  1) L AA / 2


      n  NL ~ L                 LAA ~ 1 /  (t ) ~ L

Assume     L  LAB ,


   L L  1
          AB
                1
                      ( LAB  L)  2  LAB2
                                            
                                                             L  N L LAB  N L L AA
                                                                                  2

                  L
     ~ LAB1  L 3 / 2 ~ L 3 / 2
          



                        L ~ L         3 / 2                       L ~ L3 / 2
                                        1
              (t ) ~ 1        , L~t        (  3 / 2 )   z   3
                                                                           2
                           L

                               1 /( 2  3)                      1
                 (t ) ~ t                                
                                                                2  3


                 LAA      ~ L~ 1 ~ t 1 ( 2  3 )
                                              

 lAB ?         d          1 1    1 1
                     k AB ~        1
                dt      t AB L t AB L l AB

                     (  1)2 z                                 1
           z AB               z ,                                z AB
                       2z 1                      LAB ~ t                  ~L
   Scaling of lengths

                                  L ~ t 1/ z

                  LAA ~ L ~ t1/ z AA , LAB ~ L ~ t1/ z AB
                                                                (  1)2 z
                 z   3       , z AA    3       , z AB 
                            2                    2                2z 1
   Scaling of densities :


                 ~ 1 / L ~ t  ,
                 AA ~ 1 / LAA ~ t  ,  AB ~ 1 / L ~ t 
                                         AA                              AB




                           1               1               2
                              ,  AA         ,  AB 
                         2  3          2  3          2  3
   Variations of exponents
              0  infite range attractive interaction
                isotropic diffusion

For   1 , z  zdiffusion  2.
          2                                     For 0    1         
                                                                  2
The interaction is irrelevant
                                                exponents of  , LAA , L
The diffusion motions dominate
                                                 continuous varying
the kinetics of the reaction

                     For LAB , z AB  z AB of diffusive  8 / 3
                     the diffusive motion is dominant.
                     Hence, for   7 / 6
                      continuous varying z AB
(3) Monte Carlo Simulations
  Initial density  A (0)   B (0)  0.1 with periodic boundary condition
   0
             * density                              * length




   prediction       0.3392(2)              prediction         1 / z  0.6831(5)
     1/ 3                                  1/ z  2 / 3
                   AA  0.3341(5)                              1 / z AB  0.662(2)
    AA  1 / 3                              1 / z AB  2 / 3
    AB  2 / 3    AB  0.662(2)            1 / z AA  1 / 3   1 / z AA  0.3397(5)
* density
                   1               1               2
                      ,  AA         ,  AB 
                 2  3          2  3          2  3
          0.1                                    1.25




prediction       0.316                   prediction       0.247
  0.313                                    0.25
                AA  0.312                                AA  0.238
 AA  0.313                                AA  0.25
 AB  0.626    AB  0.627                 AB  0.5      AB  0.483
* length
                                                               (  1)2 z
               z   3        , z AA    3       , z AB 
                           2                    2                2z 1

          0.1                                                  1.25




 prediction         1 / z  0.626                    prediction         1 / z  0.5
 1 / z  0.626                                       1 / z  0 .5
                    1 / z AB  0.623                 1 / z AB  0.375
                                                                        1 / z AB  0.375
 1 / z AB  0.625
 1 / z AA  0.313   1 / z AA  0.315                 1 / z AA  0.25    1 / z AA  0.262
            0.8                               
                                              AA

            0.7                               AB
    2/3                                       1/z
                                              1/zAA
            0.6
                                              1/zAB
exponents




            0.5                                             1/2

            0.4
                                                            3/8
   1/3
            0.3
                                                            1/4
            0.2


            0.1
                  0.0   0.5   1.0       1.5           2.0
                                    
4. Summary
 We investigate A  B  0 reaction with attractive interaction
    between opposite species.
 We analytically calculate dynamical exponents of densities and
   various lengths as
                       1               1               2
                          ,  AA         ,  AB 
                     2  3          2  3          2  3

                                                                (  1)2 z
                 z   3       , z AA    3       , z AB 
                            2                    2                2z 1
   There are two crossovers at  C  1 / 2 and 7/6
   For    C , diffusive motions of bulk particles dominate the
    kinetics of the reaction.

				
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posted:1/28/2013
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