Rare Events and Phase Transition in Reaction–Diffusion Systems

Vlad Elgart, Virginia Tech. Vlad Elgart, Virginia Tech.

Alex Kamenev,Alex Kamenev,

in collaboration with

PRE 70, 041106 (2004); PRE 74, 041101 (2006);

Ann Arbor, June, 2007

Reaction–Diffusion Models

FFR

F

RR

2

2

Lotka-Volterra model

Examples:

AA

Binary annihilation

Dynamical rules

Discreteness

Outline:Outline:

Hamiltonian formulation

Rare events calculus

Phase transitions and their classification

Example: Branching-Annihilation

A

AA

2

2 Rate equation:

2 nnt

n

sn

n

time

)(tn

t

sn

0n

Reaction rules:

PDF:

Extinction time

Master Equation Master Equation

• Generating Function (GF):

AAA 2 ; 2

• GF properties:

npn

• Multiply ME by and sum over :

extinction probability

Hamiltonian Hamiltonian

AAA 2 ; 2

• Imaginary time “Schrodinger” equation:

Hamiltonian is non-Hermitian

Hamiltonian Hamiltonian

mAnAFor arbitrary reaction:

mAnA

Conservation of probability

If no particles are created from the vacuum

Semiclassical (WKB) treatment Semiclassical (WKB) treatment

),(exp ),( tpStpG

• Assuming: 1),( tpS

p

SpH

t

SR , Hamilton-Jacoby equation

(rare events !)

),(

),(

qpHp

qpHq

Rq

Rp

ptp

nq

)(

)0( 0

• Boundary conditions:• Hamilton equations:

Branching-Annihilation

AAA 2 ; 2

qpppp

pqqpq

)1()(

)12(22

2

2

1

qqq

p

t

• Rate equation !

sn

Zero energytrajectories !

Extinction timeExtinction time

}exp{ 00 S

qpnqppH sR )1()1(

AAA 2 ; 2

snq )0(

0)( tp

sn

qdpS

)2ln1(2

t

0

DiffusionDiffusion

)(

)(

xqq

xpp

“Quantum Mechanics”

“QFT “

][ ),( x qpDqpHdH R

),(

),(2

2

qpHpDp

qpHqDq

Rq

Rp

• Equations of Motion:

1

2 )(

1

pRp qHqDq

p

• Rate Equation:

Refuge

AAA 2 ; 2

R0),x(

);x(n,0)xq(

0;t)boundary,(

0

p

q

}exp{ dd S

/D

Lifetime:

Instantonsolution

Phase TransitionsPhase Transitions

AAA 2 ; 2

Thermodynamic limit

Extinction time vs. diffusion time

Hinrichsen 2000

c c c

Critical exponents Critical exponents

)( csn

Hinrichsen 2000|| c || c

||||

||

||

c

c

c c

Critical Exponents (cont)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||

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 Equilibrium Models

• Landau Free Energy:

V

][ 2)()( x )]x([ DVdF

42 )( umV

Ising universality class:

critical parameter

(Lagrangian field theory)

4cd Critical dimension

Renormalization group, -expansion)4 i.e.( d

Reaction-diffusion modelsReaction-diffusion models

• Hamiltonian field theory:

][ ),(dt xdt)],x(qt),,x([ qpDqpHqppS R

p

q

111

V

qqupmpqpHR v ),(

42)( umV

critical parameter

Directed PercolationDirected Percolation

][ 222 v)(dt xdq],[ pqqupmpqqDqppS

• Reggeon field theory Janssen 1981, Grassberger 1982

4cd Critical dimension

Renormalization group,

-expansion cf. in d=3 6/1 81.0

What are other universality classes (if any)?

k-particle processes k-particle processes

• `Triangular’ topology is stable!

Effective Hamiltonian: qqupmpqpH )v( ],[ k

All reactions start from at least k particles

• Example: k = 2 Pair Contact Process with Diffusion (PCPD)

AA

A

32

2

kdc

4

Reactions with additional symmetriesReactions with additional symmetries

Parity conservation:

AA

A

3

02

Reversibility:

AA

AA

2

2

2cd

2cd

First Order Transitions First Order Transitions

• Example:

AA

A

32

Wake up !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