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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
Cambridge, Dec. 2008
Michael Assaf, Jerusalem Michael Assaf, Jerusalem
Baruch Meerson, Jerusalem Baruch Meerson, Jerusalem
Reaction–Diffusion Models
;;
2
SRI
IIS
S
S
SIR: susceptible-infected-recovered
Examples:
AA
Binary annihilation
Dynamical rules
Discreteness
Outline:Outline:
Hamiltonian formulation
Rare events calculus (Freidlin-Wentzell (?))
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
npn
• Multiply ME by and sum over :
• Normalization: extinction
Hamiltonian Hamiltonian
AAA 2 ; 2
• Imaginary time “Schrodinger” equation:
Hamiltonian is normally ordered,but 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
Long times: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
Time Dependent Rates (e.g. a Catastrophe)Time Dependent Rates (e.g. a Catastrophe)
• Temporary drop in the
reproduction rate
p
q
11
AAA 2 ; 2
t
A
A
B
B
Susceptible (S) – Infected (I) modelSusceptible (S) – Infected (I) model
SpIp
SN
I
I
S
I
IIS
S
SN
2
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?
>
Hinrichsen 2000
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, Cardy 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
Cardy, Tauber, 1995
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
