Boundary-induced nonequilibrium phase transitions into an absorbing - - PowerPoint PPT Presentation

Boundary-induced nonequilibrium phase transitions into an absorbing state

Andre Cardoso Barato ICTP

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References

Simple Absorbing-state transition

• O. Deloubrière and F

. van Wijland

• Phys. Rev. E 65, 046104 (2002).

Boundary-induced nonequilibrium phase transition into an absorbing state

• A. C. Barato and H. Hinrichsen
• Phys. Rev. Lett. 100, 165701 (2008)

Nonequilibrium phase transition in a spreading process on a timeline

• A. C. Barato and H. Hinrichsen
• J. Stat. Mech. (2009) P02020

Simplest nonequilibrium phase transition into an absorbing state

• A. C. Barato, C. E. Fiore, J. A. Bonachela, H. Hinrichsen and M. A. Muñoz
• Phys. Rev. E 79, 041130 (2009)

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What is this talk about?

Boundary induced phase transitions into an absorbing state and its mapping onto a 0−dimensional non-Markovian process.

position 2000

6

10 random walk in the bulk time CP−like dynamics at leftmost site

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The Contact Process in 1D

1 λ λ λ /2 /2

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The Contact Process

infection seed dominates spontaneous recovery dominates infectious spreading

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Phase Transition in the Contact Process

Transition from active into absorbing phase belongs to the universality class of Directed Percolation (DP).

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Universality and Critical Exponents

ρ ∼ |λ − λc|β ξ⊥ ∼ |λ − λc|−ν⊥ ξ ∼ |λ − λc|−ν exponent d = 1 d = 2 d = 3 d ≥ 4 β 0.276 0.58 0.81 1 ν⊥ 1.097 0.73 0.58 1/2 ν 1.734 1.29 1.11 1

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Boundary-induced phase transitions

α β 1 1 1

0.5 1 1.5

α

0.5 1 1.5

β J= 1/4 J= β(1−β) J= α(1−α)

low density phase high density phase maximal current phase

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Model Definition

• max. one particle per site

use random-sequential dynamics p, the probability of creating a new particle at site 1 if 0 is occupied is the control parameter.

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System exhibits a Phase Transition

ρ0 is the density of particles at the leftmost site Ps(t) is the survival probability, i.e., the probability of not entering the absorbing state at least until time t. N(t) is the average total number of particles in the system.

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Critical Point and Exponents

Critical point: pc = 0.74435(15). ρ0(t) ∼ t−α , α = 2β/ν = 0.500(5). The exponent α = 1/2 is an exact result of the field theory and is related to diffusion in the bulk. ρsat ∼ (p − pc)2β , β = 0.71(2). The −expansion, until first order gives: β = 5/8 = 0.625. Ps(t) ∼ t−δ , δ = 0.165(3). Unusual value, δ = β/ν = 1/4. Conjecture: δ = 1/6

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Effectively like a random walk

1 2 3 4 5 6

x/t

1/2

0,5 1

ρ(x,t) t

1/2

10 10

1

10

2

10

3

10

4

x

10

• 5

10

• 4

10

• 3

10

• 2

10

• 1

10

x/t

1/2 10

• 2

10

• 1

10

ρ(x,t) t

1/2

ρ(x,t0) ρ

pair(x,t0)

slope -1/2

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Bosonic versions

First bosonic version: all sites are allowed to have an infinity number of particles. ρ0(t) = 2 √ πt + 4∆ exp(4∆2t)erf(−2∆ √ t) β = 1, α = 1/2, δ = 1/4. Second bosonic version: just the site at the boundary and it’s neighbor have the fermionic constraint. Same critical behavior as the full model. The fermionic constraint is relevant only at the boundary.

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Mapping onto a zero-dimensional process

Bulk Particle

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Relation to a one site process

In 1D random-walkers return to the origin after finite time ∆t distributed as

P(∆t) ∼ (∆t)−3/2.

Toy model

• 1. Select the lowest t for which s(t) = 1.
• 2. With probability µ generate waiting time ∆t and set

s(t + ∆t) := 1.

• 3. Otherwise set s(t) := 0.

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Toy model

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Toy model

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Toy model

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Toy model

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Toy model

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Toy model

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Toy model

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Toy model

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Toy model

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Toy model

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Toy model

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Toy model

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Toy model

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Toy model

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Toy model

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Toy model

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Toy model

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Toy model

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Toy model

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Toy model

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Result

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Directed Levy flights

Algebraically distributed waiting times ∆t with probability distribution P(∆t) ∼ ∆t−1−κ are generated by an operator called fractional derivative ∂κ

t

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Fractional Derivative

Action in momentum space: ∂κ

t eiωt = (iω)κeiωt

Integral kernel representation: ∂κ

t ρ(t) =

1 N(κ) ∞ dt t−1−κ[ρ(t) − ρ(t − t)] , with N(κ) = −Γ(−κ). New propagator: G0(ω) = 1 (−iω)κ − a

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Langevin equation

∂tρ(x, t) = D∇2ρ(x, t) + aρ(x, t) − ρ(x, t)2 +

• ρ(x, t)ξ(t) .

∂tρ(x, t) = D∇2ρ(x, t)+

• aρ(x, t)−ρ(x, t)2+
• ρ(x, t)ξ(t)
• δd(x) .

∂κ

t ρ(t) = aρ(t) − ρ(t)2 +

• ρ(t)ξ(t) .

d = 2 − 2κ

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Field theory and simulations

0.2 0.4 0.6 0.8 1

κ

2 3 4

ν||

0.2 0.4 0.6 0.8 1

κ

0.1 0.2

δ

0.2 0.4 0.6 0.8 1

κ

0,5 1

β

δ(κ) = κ/2 δ(κ) = κ − κ2 1 + κ β/ν = 1 − κ 2 β = 1 − 9/4 = κ − 1/3

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Conclusion

• 1. We found a new universality class of phase transitions

into an absorbing state and it is the simplest one.

• 2. Some critical exponents are non-trivial and others are

related to a random walk.

• 3. Model can be related to non-Markovian single-site

process.

• 4. Nonequilibrium phase transitions are possible in zero

dimensions.

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References

Simple Absorbing-state transition

• O. Deloubrière and F

. van Wijland

• Phys. Rev. E 65, 046104 (2002).

Boundary-induced nonequilibrium phase transition into an absorbing state

• A. C. Barato and H. Hinrichsen
• Phys. Rev. Lett. 100, 165701 (2008)

Nonequilibrium phase transition in a spreading process on a timeline

• A. C. Barato and H. Hinrichsen
• J. Stat. Mech. (2009) P02020

Simplest nonequilibrium phase transition into an absorbing state

• A. C. Barato, C. E. Fiore, J. A. Bonachela, H. Hinrichsen and M. A. Muñoz
• Phys. Rev. E 79, 041130 (2009)

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