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

Boundary-induced nonequilibrium phase transitions into an absorbing state Andre Cardoso Barato ICTP – p. 1/42

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) – p. 2/42

What is this talk about? Boundary induced phase transitions into an absorbing state and its mapping onto a 0 − dimensional non-Markovian process. 0 2000 position 0 random walk CP−like dynamics at leftmost site in the bulk 6 10 time – p. 3/42

The Contact Process in 1D /2 λ /2 λ λ 1 – p. 4/42

The Contact Process infection seed infectious spreading spontaneous recovery dominates dominates – p. 5/42

Phase Transition in the Contact Process Transition from active into absorbing phase belongs to the universality class of Directed Percolation (DP). – p. 6/42

Universality and Critical Exponents ρ ∼ | λ − λ c | β ξ ⊥ ∼ | λ − λ c | − ν ⊥ ξ � ∼ | λ − λ c | − 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 – p. 7/42

Boundary-induced phase transitions α β 1 1 1 1.5 J= 1/4 J = α (1 −α ) 1 maximal current phase β low density phase 0.5 J = β (1 −β ) high density phase 0 0 0.5 1 1.5 α – p. 8/42

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. – p. 9/42

System exhibits a Phase Transition ρ 0 is the density of particles at the leftmost site P s ( 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. – p. 10/42

Critical Point and Exponents Critical point: p c = 0 . 74435(15) . ρ 0 ( t ) ∼ t − α , α = 2 β/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. ρ sat ∼ ( p − p c ) 2 β , β = 0 . 71(2) . The � − expansion, until first order gives: β = 5 / 8 = 0 . 625 . P s ( t ) ∼ t − δ , δ = 0 . 165(3) . Unusual value, δ � = β/ν � = 1 / 4 . Conjecture: δ = 1 / 6 – p. 11/42

Effectively like a random walk -3 10 0 1 10 ρ (x,t 0 ) 1/2 1/2 ρ (x,t) t -1 10 slope -1/2 ρ (x,t) t -4 10 -2 10 0,5 pair (x,t 0 ) ρ -2 -1 0 10 10 10 -5 10 1/2 x/t 0 0 1 2 3 4 5 6 0 1 2 3 4 10 10 10 10 10 1/2 x x/t – p. 12/42

Bosonic versions First bosonic version: all sites are allowed to have an infinity number of particles. √ 2 πt + 4∆ exp(4∆ 2 t ) erf ( − 2∆ √ ρ 0 ( t ) = 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. – p. 13/42

Mapping onto a zero-dimensional process Particle Bulk – p. 14/42

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 . – p. 15/42

Toy model – p. 16/42

Toy model – p. 17/42

Toy model – p. 18/42

Toy model – p. 19/42

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Toy model – p. 22/42

Toy model – p. 23/42

Toy model – p. 24/42

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Toy model – p. 27/42

Toy model – p. 28/42

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Toy model – p. 31/42

Toy model – p. 32/42

Toy model – p. 33/42

Toy model – p. 34/42

Toy model – p. 35/42

Result – p. 36/42

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 – p. 37/42

Fractional Derivative Action in momentum space: t e iωt = ( iω ) κ e iωt ∂ κ Integral kernel representation: � ∞ 1 d t � t �− 1 − κ [ ρ ( t ) − ρ ( t − t � )] , ∂ κ t ρ ( t ) = N � ( κ ) 0 with N � ( κ ) = − Γ( − κ ) . New propagator: 1 G 0 ( ω ) = ( − i ω ) κ − a – p. 38/42

Langevin equation ∂ t ρ ( x, t ) = D ∇ 2 ρ ( x, t ) + aρ ( x, t ) − ρ ( x, t ) 2 + � ρ ( x, t ) ξ ( t ) . � � δ d ( x ) . � ∂ t ρ ( x, t ) = D ∇ 2 ρ ( x, t )+ aρ ( x, t ) − ρ ( x, t ) 2 + ρ ( x, t ) ξ ( t ) t ρ ( t ) = aρ ( t ) − ρ ( t ) 2 + ∂ κ � ρ ( t ) ξ ( t ) . d = 2 − 2 κ – p. 39/42

Field theory and simulations 0.2 4 1 ν || β δ 0.1 3 0,5 2 0 0 0 0.2 0.4 0.6 0.8 1 0 0.2 0.4 0.6 0.8 1 0 0.2 0.4 0.6 0.8 1 κ κ κ δ ( κ ) = κ − κ 2 δ ( κ ) = κ / 2 1 + κ β / ν � = 1 − κ 2 β = 1 − 9 / 4 � � = κ − 1 / 3 – p. 40/42

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. – p. 41/42

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) – p. 42/42