Rare Events and Phase Transition in Reaction–Diffusion systems by sdfwerte

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									    Rare Events and Phase Transition in
       Reaction–Diffusion Systems

        Alex Kamenev,

 in collaboration with

                       Vlad Elgart, Virginia Tech.

                       Baruch Meerson, Jerusalem

                       Michael Assaf,    Jerusalem


Cambridge, Dec. 2008
    Reaction–Diffusion Models

                        Examples:

                    Binary annihilation

                        A A 
                    SIR: susceptible-
                      infected-recovered

                       S
                       S 
                      
 Dynamical rules     
                       S  I  2I
 Discreteness         I  R; S ; 
                      
                Outline:


   Hamiltonian formulation


 Rare events calculus (   Freidlin-Wentzell (?))



 Phase transitions and their classification
          Example: Branching-Annihilation
                        Rate equation:
  Reaction     A 2A                               
                          n                 ns 
   rules:           
               2 A           n   n2          
                          t
      n
 n0                                PDF:
n(t )
 ns
                                  Extinction time
                          time
           t
                                            
           Master Equation    A 2 A ;   2 A




• Generating Function (GF):

• Normalization:                         extinction

                     n
• Multiply ME by p
  and sum over n:
                                                 
          Hamiltonian            A 2 A ;    2 A




• Imaginary time “Schrodinger” equation:




                        Hamiltonian is normally ordered,
                        but non-Hermitian
               Hamiltonian           nA mA


For arbitrary reaction:   nA mA




  Conservation of probability

  If no particles are created from the vacuum
   Semiclassical (WKB) treatment

               G( p, t )  exp S ( p, t )

• Assuming:         S ( p, t )  1    (rare events !)

 S        S 
     H R  p, 
           p 
                              Hamilton-Jacoby equation
 t           
• Hamilton equations:            • Boundary conditions:

   q   p H R ( p, q )
                                     q(0)  n0
                                      
   p   q H R ( p, q)
                                      p(t )  p
                                                           
 Branching-Annihilation A  2 A ; 2 A  


   (2 p  1)q  pq2
q                                      p 1 
p   ( p  p 2 )   (1  p 2 )q
                                    t q   q   q2
                                    • Rate equation !

                                                   
                                              ns 
                                                   
                                           Long times:
                                           zero energy
                                           trajectories !
                                            
Extinction time          A 2 A ;      2 A


        H R   (1  p)(1  p)q  ns p q
                                          
                           q ( 0)  n s 
                                          
                           p (t )  0
                            0  exp{ S0}
                                  t


                                 S0   qdp
                                 2(1  ln 2)ns
  Time Dependent Rates (e.g. a Catastrophe)
                               
  A 2A ;     2A                   A   B
• Temporary drop in the
  reproduction rate                         t

                 q

             A

         B

                     1
                     1       p
  Susceptible (S) – Infected (I) model
      N            I
   S
       
 S  
         
        
S  I  2 I
 I  
                                  N
                                         S
               pI                        I
                        pS




                                         S
                             Diffusion
                                                      
                                             p  p( x )
                                                    
                                            q  q ( x )
                                   “Quantum Mechanics” 
                                          “QFT “
                                           
                                   H   d x [H R ( p, q)  Dpq]
• Equations of Motion:                   • Rate Equation:
q  D 2 q   p H R ( p, q)
                                              p 1 


 p   D 2 p   q H R ( p, q)         q  D2 q   p H R ( q) p 1
                                         
                                                  
                  Refuge           A 2 A ;    2 A

                      q( boundary, t)  0;
                                    
                      q(x,0)  n 0 ( x );     Instanton
                                             solution
                      p ( x,  )  0
  R




  D/
   Lifetime:
 d  exp{ Sd }
                                            
 Phase Transitions         A 2 A ;    2 A


          Thermodynamic limit
          Extinction time vs. diffusion time




                                      Hinrichsen 2000

  c      c           c
         Critical exponents

                                                        
                                    ns  (    c )


     |   c |      |   c |   Hinrichsen 2000




                                       |    c |     


                                                     ||
                                     || |    c |


  c              c
       Critical Exponents (cont)

         d=1     d=2   d=3     d>4

        0.276   0.584 0.811   1

 ||     1.734   1.296 1.106   1     Hinrichsen 2000




 How to calculate critical exponents analytically?
 What other reactions belong to the same
       universality class?

 Are there other universality classes
       and how to classify them?
                Equilibrium Models
                                     
• Landau Free Energy: F [j (x)]   d x [V (j )  D(j )2 ]
    (Lagrangian field theory)
                V(j)
                                   Ising universality class:

                                   V (j )  m j  u j
                                                  2        4


                                      critical parameter
                          j

    Critical dimension d c  4
    Renormalization group,  (i.e.    4  d )-expansion
             Reaction-diffusion models
• Hamiltonian field theory:
                            
  S[ p( x, t), q( x, t)]   dx dt [ pq  H R ( p, q)  Dpq]
                                      
                   q
                                                          V(j)




                       1
                       1        p                                    j
                                              V (j )  mj 2  uj 4
  H R ( p, q)  pm  up  vq q
                                                critical parameter
                 Directed Percolation
               
 S[ p, q]   dx dt [ p( q  D2 q)  mpq  up 2 q  vpq2 ]
                         

• Reggeon field theory        Janssen 1981, Grassberger, Cardy 1982


   Critical dimension       dc  4
   Renormalization group,


    -expansion   1   / 6          cf.     0.81 in d=3
          What are other universality classes (if any)?
               k-particle processes

• `Triangular’ topology is stable!




Effective Hamiltonian: H [ p, q]  p(m  up  vq)qk

     All reactions start from at least k particles

• Example: k = 2 Pair Contact         2 A               4
                                                     dc 
                                      2 A  3 A
  Process with Diffusion (PCPD)                            k
Reactions with additional symmetries
 Parity conservation:
       2 A  0
       
        A  3A
      Cardy, Tauber, 1995
                                dc  2
 Reversibility:
       2 A  A
       
       A  2A
                                dc  2
First Order Transitions




                   A  
        • Example: 
                   2 A  3 A
         Wake up !


 Hamiltonian formulation and
    and its semiclassical limit.

 Rare events as trajectories
    in the phase space

 Classification of the phase
    transitions according to the
    phase space topology

								
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