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Rare Events and Phase Transition in Reaction–Diffusion Systems. Alex Kamenev,. in collaboration with. Vlad Elgart, Virginia Tech. PRE 70 , 041106 (2004); PRE 74 , 041101 (2006);. Ann Arbor, June, 2007. Binary annihilation. Lotka-Volterra model. Reaction–Diffusion Models. - PowerPoint PPT Presentation
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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