# 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

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/
 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
 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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