rare events and phase transition in reaction–diffusion systems

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